C15 09/22/2010 Page 568

Chapter 15

Adsorption, Ion Exchange, Chromatography,

and Electrophoresis

§15.0 INSTRUCTIONAL OBJECTIVES

After completing this chapter, you should be able to:

Explain why a few grams of porous adsorbent can have an adsorption area as large as a football field.

Differentiate between chemisorption and physical adsorption.

Explain how ion-exchange resins work.

Compare three major expressions (so-called isotherms) used for correlating adsorption-equilibria data.

List steps involved in adsorption of a solute, and which steps may control the rate of adsorption.

Describe major modes for contacting the adsorbent with a fluid containing solute(s) to be adsorbed.

Describe major methods for regenerating adsorbent.

Calculate vessel size or residence time for any of the major modes of slurry adsorption.

List and explain assumptions for ideal fixed-bed adsorption and explain the concept of width of mass-transfer zone.
Explain the concept of breakthrough in fixed-bed adsorption.

Calculate bed height, bed diameter, and cycle time for fixed-bed adsorption.

Compute separations for a simulated-moving-bed operation.

Calculate rectangular and Gaussian-distribution pulses in chromatography.

Describe electrophoresis of biomolecules, including factors that affect mobility as well as effects of electro-
osmosis and convective Joule heating

Distinguish different electrophoretic modes (native gel electrophoresis, SDS-PAGE, isoelectric focusing, isotacho-
phoresis, 2-D gel electrophoresis, and pulsed field gel electrophoresis) in terms of denaturants used, pH and elec-
trolyte content, and application of electric-field gradients.

Adsorption, ion exchange, and chromatography are sorp- pores are used, with adsorption occurring on the surface of
tion operations in which components of a fluid phase (sol- the pores.
utes) are selectively transferred to insoluble, rigid particles In an ion-exchange process, as in Figure 15.1b, ions of pos-
suspended in a vessel or packed in a column. Sorption, a gen- itive charge (cations) or negative charge (anions) in a liquid
eral term introduced by J.W. McBain [Phil. Mag., 18, 916– solution, usually aqueous, replace dissimilar and displaceable
935 (1909)], includes selective transfer to the surface and/or ions, called counterions, of the same charge contained in a
into the bulk of a solid or liquid. Thus, absorption of gas spe- solid ion exchanger, which also contains immobile, insoluble,
cies into a liquid and penetration of fluid species into a non- and permanently bound co-ions of the opposite charge. Thus,
porous membrane are sorption operations. In a sorption ion exchange can be cation or anion exchange. Water soften-
process, the sorbed solutes are referred to as sorbate, and the ing by ion exchange involves a cation exchanger, in which a
sorbing agent is the sorbent. reaction replaces calcium ions with sodium ions:
In an adsorption process, molecules, as in Figure 15.1a, or þ
atoms or ions, in a gas or liquid, diffuse to the surface of a ðaqÞ þ 2NaRðsÞ $ CaR2ðsÞ þ 2NaðaqÞ
Ca2þ
solid, where they bond with the solid surface or are held by where R is the ion exchanger. The exchange of ions is revers-
weak intermolecular forces. Adsorbed solutes are referred to ible and does not cause any permanent change to the solid
as adsorbate, whereas the solid material is the adsorbent. To ion-exchanger structure. Thus, it can be used and reused
achieve a large surface area for adsorption per unit volume, unless fouled by organic compounds in the liquid feed that
porous solid particles with small-diameter, interconnected attach to exchange sites on and within the ion exchange resin.

568
C15 09/22/2010 Page 569

Adsorption, Ion Exchange, Chromatography, and Electrophoresis 569

A –

B –
A
Adsorbent –
– – –
Adsorbed layer Fluid phase A
on surfaces in pores – B A
– A
–
4

B
A –
3
2
– Matrix with fixed charges
1
A B Counterions
Figure 15.1 Sorption operations with
– Co-ions solid-particle sorbents. (a) Adsorption.
(a) (b) (b) Ion exchange.

The ion-exchange concept can be extended to the removal the 1960s, following inventions by Milton [2] of synthetic
of essentially all inorganic salts from water by a two-step molecular-sieve zeolites, which provide high adsorptive
demineralization process or deionization. In step 1, a cation selectivity, and by Skarstrom [3] of the pressure-swing cycle,
resin exchanges hydrogen ions for cations such as calcium, which made possible a fixed-bed, cyclic gas-adsorption pro-
magnesium, and sodium. In step 2, an anion resin exchanges cess. The commercial separation of liquid mixtures also
hydroxyl ions for strongly and weakly ionized anions such as began in the 1960s, following the invention by Broughton
sulfate, nitrate, chloride, and bicarbonate. The hydrogen and and Gerhold [4] of the simulated moving bed for adsorption.
hydroxyl ions combine to form water. Regeneration of the Uses of ion exchange date back at least to the time of
cation and anion resins is usually accomplished with sulfuric Moses, who, while leading his followers out of Egypt, sweet-
acid and sodium hydroxide. ened the bitter waters of Marah with a tree [Exodus 15:23–
In chromatography, the sorbent may be a solid adsorbent; 26]. In ancient Greece, Aristotle observed that the salt con-
an insoluble, nonvolatile liquid absorbent contained in the tent of water is reduced when it percolates through certain
pores of a granular solid support; or an ion exchanger. In any sands. Studies of ion exchange were published in 1850 by
case, the solutes to be separated move through the chromato- both Thompson and Way, who experimented with cation
graphic separator, with an inert, eluting fluid, at different exchange in soils before the discovery of ions.
rates because of different sortion affinities during repeated The first major application of ion exchange occurred over
sorption, desorption cycles. 100 years ago for water treatment to remove calcium and
During adsorption and ion exchange, the solid separating other ions responsible for water hardness. Initially, the ion
agent becomes saturated or nearly saturated with the mole- exchanger was a porous, natural, mineral zeolite containing
cules, atoms, or ions transferred from the fluid phase. To silica. In 1935, synthetic, insoluble, polymeric-resin ion
recover the sorbed substances and allow the sorbent to be exchangers were introduced. Today they are dominant for
reused, the asorbent is regenerated by desorbing the sorbed water-softening and deionizing applications, but natural and
substances. Accordingly, these two separation operations are synthetic zeolites still find some use.
carried out in a cyclic manner. In chromatography, regeneration Since the 1903 invention of chromatography by M. S.
occurs continuously, but at changing locations in the separator. Tswett [5], a Russian botanist, it has found widespread use as
Adsorption processes may be classified as purification or an analytical, preparative, and industrial technique. Tswett
bulk separation, depending on the concentration in the feed separated a mixture of structurally similar yellow and green
of the components to be adsorbed. Although there is no sharp chloroplast pigments in leaf extracts by dissolving the
dividing concentration, Keller [1] has suggested 10 wt%. extracts in carbon disulfide and passing the solution through
Early applications of adsorption involved only purification. a column packed with chalk particles. The pigments were
Adsorption with charred wood to improve the taste of water separated by color; hence, the name chromatography, which
has been known for centuries. Decolorization of liquids by was coined by Tswett in 1906 from the Greek words chroma,
adsorption with bone char and other materials has been prac- meaning ‘‘color,’’ and graphe, meaning ‘‘writing.’’ Chroma-
ticed for at least five centuries. Adsorption of gases by a solid tography has revolutionized laboratory chemical analysis of
(charcoal) was first described by C.W. Scheele in 1773. liquid and gas mixtures. Large-scale, commercial applica-
Commercial applications of bulk separation by gas tions described by Bonmati et al. [6] and Bernard et al. [7]
adsorption began in the early 1920s, but did not escalate until began in the 1980s.
C15 09/22/2010 Page 570

570 Chapter 15 Adsorption, Ion Exchange, Chromatography, and Electrophoresis

Also included in this chapter is electrophoresis, which one bed is directed to the other bed as a downward-flowing
involves the size- and charge-based separation of charged purge to regenerate the adsorbent. The purge is exhausted at
solutes that move in response to an electric field applied a pressure of 141.3 kPa. By conducting the purge flow coun-
across an electrophoretic medium. Positively charged solutes tercurrently to the entering air flow, the highest degree of
migrate to the negative electrode; negatively charged water-vapor desorption is achieved.
Other equipment shown in Figure 15.2 includes an air
solutes migrate toward the positive electrode. Typical media
compressor, an aftercooler, piping and valving to switch the
include agarose, polyacrylamide, and starch, which form gels
beds from one step in the cycle to the other, a coalescing filter
with a high H2O content that allows passage of large solutes to remove aerosols from the entering air, and a particulate
through their porous structures. Electrophoresis is widely filter to remove adsorbent fines from the exiting dry air. If
used to separate and purify biomolecules, including proteins the dry air is needed at a lower pressure, an air turbine can be
and nucleic acids. installed to recover energy while reducing air pressure.
During the 5-minute adsorption period of the cycle, the
Industrial Example capacity of the adsorbent for water must not be exceeded. In
this example, the water content of the air is reduced from
Pressure-swing gas adsorption is used for air dehydration and 1.27  103 kg H2O/kg air to the very low value of 9.95 
for separation of air into nitrogen and oxygen. A small unit 107 kg H2O/kg air. To achieve this exiting water-vapor con-
for the dehydration of compressed air is described by White tent, only a small fraction of the adsorbent capacity is utilized
and Barkley [8] and shown in Figure 15.2. The unit consists during the adsorption step, with most of the adsorption occur-
of two fixed-bed adsorbers, each 12.06 cm in diameter and ring in the first 0.2 m of the 1.27-m bed height.
_________________________________________________
packed with 11.15 kg of 3.3-mm-diameter Alcoa F-200 acti-
vated-alumina beads to a height of 1.27 m. The external The bulk separation of gas and liquid mixtures by adsorp-
porosity (void fraction) of the bed is 0.442 and the alumina- tion is an emerging separation operation. Important prog-
bead bulk density is 769 kg/m3. ress is being made in the development of more-selective
The unit operates on a 10-minute cycle, with 5 minutes for
adsorbents and more-efficient operation cycles. In addi-
adsorption of water vapor and 5 minutes for regeneration,
tion, attention is being paid to hybrid systems that include
which consists of depressurization, purging of the water
membrane and other separation steps. The three sorption
vapor, and a 30-s repressurization. While one bed is adsorb-
operations addressed in this chapter have found many
ing, the other bed is being regenerated. The adsorption (dry-
applications, as given in Table 15.1, compiled from list-
ing) step takes place with air entering at 21 C and 653.3 kPa
ings in Rousseau [9]. These cover a wide range of solute
(6.45 atm) with a flow rate of 1.327 kg/minute, passing
molecular weights.
through the bed with a pressure drop of 2.386 kPa. The dew- This chapter discusses: (1) sorbents, including their equili-
point temperature of the air at system pressure is reduced
brium, sieving, transport, and kinetic properties with respect
from 11.2 to 61 C by the adsorption process. During the
to solutes removed from solutions; (2) techniques for con-
270-s purge period, about one-third of the dry air leaving
ducting cyclic operations; and (3) equipment configuration
and design. Both equilibrium-stage and rate-based models
are developed. Although emphasis is on adsorption, basic
Dry air principles of ion exchange, chromatography, and electropho-
resis are also presented. Further descriptions of sorption
operations are given by Rousseau [9] and Ruthven [10].
Particulate
filter

§15.1 SORBENTS
To be suitable for commercial use, a sorbent should have: (1)
high selectivity to enable sharp separations; (2) high capacity
to minimize amount of sorbent; (3) favorable kinetic and
Adsorber Adsorber transport properties for rapid sorption; (4) chemical and ther-
no. 1 no. 2
mal stability, including extremely low solubility in the con-
tacting fluid, to preserve the amount of sorbent and its
properties; (5) hardness and mechanical strength to prevent
Purge Purge
crushing and erosion; (6) a free-flowing tendency for ease of
Coalescing
filter filling or emptying vessels; (7) high resistance to fouling for
Moist long life; (8) no tendency to promote undesirable chemical
air cw reactions; (9) capability of being regenerated when used with
Gas commercial feedstocks containing trace quantities of high-
Aftercooler
compressor MW species that are strongly sorbed and difficult to desorb;
Figure 15.2 Pressure-swing adsorption for the dehydration of air. and (10) low cost.
C15 09/22/2010 Page 571

§15.1 Sorbents 571

 
Table 15.1 Industrial Applications of Sorption Operations 20–500 A, and a macropore is >500 A. Typical commercial
adsorbents, which may be granules, spheres, cylindrical pel-
1. Adsorption
lets, flakes, and/or powders of diameter ranging from 50 mm
Gas purifications:
to 1.2 cm, have specific surface areas from 300 to 1,200 m2/g.
Removal of organics from vent streams
Thus, a few grams of adsorbent can have a surface area equal
Removal of SO2 from vent streams
to that of a football field (120  53.3 yards or 5,350 m2)!
Removal of sulfur compounds from gas streams
This large area is made possible by a particle porosity from
Removal of water vapor from air and other gas streams 
30 to 85 vol% with pore diameters from 10 to 200 A. To
Removal of solvents and odors from air
quantify this, consider a cylindrical pore of diameter dp and
Removal of NOx from N2
length L. The surface area-to-volume ratio is
Removal of CO2 from natural gas
 
Gas bulk separations: S=V ¼ pd p L pd 2p L=4 ¼ 4=d p ð15-1Þ
N2/O2
H2O/ethanol If the fractional particle porosity is ep and the particle density
Acetone/vent streams is rp, the specific surface area, Sg, in area per unit mass of
C2H4/vent streams adsorbent is
Normal paraffins/isoparaffins, aromatics
Sg ¼ 4ep =rp d p ð15-2Þ
CO, CH4, CO2, N2, A, NH3, H2
Liquid purifications:
Thus, if ep is 0.5, rp is 1 g/cm3 ¼ 1  106 g/m3, and dp is

Removal of H2O from organic solutions
20 A (20  1010 m), use of (15-2) gives Sg ¼ 1,000 m2/g.
Removal of organics from H2O
Depending upon the forces between fluid molecules and
Removal of sulfur compounds from organic solutions
solid molecules, adsorption may be physical adsorption (van
Decolorization of solutions
der Waals adsorption) or chemisorption (activated adsorp-
Liquid bulk separations:
tion). Physical adsorption from a gas occurs when inter-
Normal paraffins/isoparaffins
molecular attractive forces between solid and gas molecules
Normal paraffins/olefins
are greater than those between gas molecules. In effect, the
p-xylene/other C8 aromatics
resulting adsorption is like condensation, which is exother-
p- or m-cymene/other cymene isomers
mic and accompanied by a release of heat. The magnitude of
p- or m-cresol/other cresol isomers
the heat of adsorption can be > or < than heat of vaporiza-
Fructose/dextrose, polysaccharides
tion, and changes with amount of adsorption.
2. Ion Exchange
Physical adsorption occurs rapidly, and may be a mono-
Water softening
molecular (unimolecular) layer, or two or more layers thick
Water demineralization
(multimolecular). If unimolecular, it is reversible; if multi-
Water dealkalization
molecular, such that capillary pores are filled, hysteresis may
Decolorization of sugar solutions
occur. The adsorbate density is of the order of magnitude of
Recovery of uranium from acid leach solutions
the liquid rather than the vapor. As physical adsorption takes
Recovery of antibiotics from fermentation broths
place, it begins as a monolayer, becomes multilayered, and
Recovery of vitamins from fermentation broths
then, if the pores are close to the size of the molecules, capil-
3. Chromatography
lary condensation occurs, and pores fill with adsorbate.
Separation of sugars
Accordingly, maximum capacity of a porous adsorbent is
Separation of perfume ingredients
related more to pore volume than to surface area. However,
Separation of C4–C10 normal and isoparaffins
for gases at temperatures above their critical temperature,
adsorption is confined to a monolayer.
Chemisorption involves formation of chemical bonds
§15.1.1 Adsorbents between adsorbent and adsorbate in a monolayer, often with
Most solids adsorb species from gases and liquids, but few a release of heat larger than the heat of vaporization. Chemi-
have a sufficient selectivity and capacity to qualify as serious sorption from a gas generally takes place only at tempera-
candidates for commercial adsorbents. Of importance is a tures greater than 200 C and may be slow and irreversible.
large specific surface area (area per unit volume), which is Commercial adsorbents rely on physical adsorption to
achieved by manufacturing techniques that result in solids achieve separations; solid catalysts rely on chemisorption
with a microporous structure. Pore sizes are usually given in to catalyze chemical reactions.

angstroms, A; nanometers, nm; or micrometers (microns), Adsorption from liquids is difficult to measure or
mm, which are related to meters, m, and millimeters, mm, by: describe. When the fluid is a gas, the amount of gas adsorbed
in a confined space is determined from the measured decrease
1 m ¼ 102 cm ¼ 103 mm ¼ 106 mm ¼ 109 nm ¼ 1010 A8
in total pressure. For a liquid, no simple procedure for deter-

Hydrogen and helium atoms are approximately 1 A in size. mining the extent of adsorption from a pure liquid exists; con-
By the International Union of Pure and Applied Chemistry sequently, experiments are conducted using liquid mixtures.

(IUPAC) definitions, a micropore is <20 A, a mesopore is When porous particles of adsorbent are immersed in a liquid
C15 09/22/2010 Page 572

572 Chapter 15 Adsorption, Ion Exchange, Chromatography, and Electrophoresis

Table 15.2 Representative Properties of Commercial Porous Adsorbents

Capacity for H2O

Pore Particle Vapor at 25 C
Diameter Particle Density Surface Area and 4.6 mmHg,

Adsorbent Nature dp, A Porosity, ep rp, g/cm3 Sg, m2/g wt% (Dry Basis)

Activated alumina Hydrophilic, amorphous 10–75 0.50 1.25 320 7

Silica gel: Hydrophilic/hydrophobic,
Small pore amorphous 22–26 0.47 1.09 750–850 11
Large pore 100–150 0.71 0.62 300–350 —
Activated carbon: Hydrophobic, amorphous
Small pore 10–25 0.4–0.6 0.5–0.9 400–1200 1
Large pore >30 — 0.6–0.8 200–600 —
Molecular-sieve carbon Hydrophobic 2–10 — 0.98 400 —
Molecular-sieve zeolites Polar-hydrophilic, crystalline 3–10 0.2–0.5 1.4 600–700 20–25
Polymeric adsorbents — 40–25 0.4–0.55 — 80–700 —

mixture, the pores, if sufficiently larger in diameter than the method (Brunauer, Emmett, and Teller [11]). Typically, the
liquid molecules, fill with liquid. At equilibrium, because of BET apparatus operates at the normal boiling point of N2
differences in the extent of physical adsorption among liquid (195.8 C) by measuring the equilibrium volume of pure N2
molecules, composition of the liquid in pores differs from that physically adsorbed on several grams of the adsorbent at a
of bulk liquid surrounding adsorbent particles. The observed number of different values of the total pressure in a vacuum
exothermic heat effect is referred to as the heat of wetting, of 5 to at least 250 mmHg. Brunauer, Emmett, and Teller
which is much smaller than the heat of adsorption for a gas. derived an equation to model adsorption by allowing for for-
As with gases, the extent of equilibrium adsorption of a given mation of multimolecular layers. They assumed that the heat
solute increases with concentration and decreases with tem- of adsorption during monolayer formation (DHads) is con-
perature. Chemisorption can also occur with liquids. stant and that the heat effect associated with subsequent lay-
Table 15.2 lists, for six major types of solid adsorbents: ers is equal to the heat of condensation (DHcond). The BET
the nature of the adsorbent and representative values of the equation is
mean pore diameter, dp; particle porosity (internal void frac-  
P 1 ð c  1Þ P
tion), ep ; particle density, rp; and specific surface area, Sg. In ¼ þ ð15-6Þ
yðP0  PÞ ym c ym c P0
addition, for some adsorbents, the capacity for adsorbing
water vapor at a partial pressure of 4.6 mmHg in air at 25 C where P ¼ total pressure, P0 ¼ vapor pressure of adsorbate at
is listed, as taken from Rousseau [9]. Not included is specific test temperature, y ¼ volume of gas adsorbed at STP (0 C,
pore volume, Vp, which is given by 760 mmHg), ym ¼ volume of monomolecular layer of gas
adsorbed at STP, and c ¼ a constant related to the heat of
V p ¼ ep =rp ð15-3Þ adsorption  exp[(DHcond  DHads)=RT].
Experimental data for y as a function of P are plotted,
Also not included in Table 15.2, but of interest when the according to (15-6), as P=[y(P0  P)] versus P=P0, from
adsorbent is used in fixed beds, are bulk density, rb, and bed which ym and c are determined from the slope and intercept
porosity (external void fraction), eb , which are related by of the best straight-line fit of the data. The value of Sg is then
computed from
rb aym N A
eb ¼ 1  ð15-4Þ Sg ¼ ð15-7Þ
rp V

where NA ¼ Avogadro’s number ¼ 6.023  1023 molecules/

In addition, the true solid particle density (also called the
mol, V ¼ volume of gas per mole at STP conditions (0 C,
crystalline density), rs, can be computed from a similar
1 atm) ¼ 22,400 cm3/mol, and a is surface area per adsorbed
expression:
molecule. If spherical molecules arranged in close two-
rp dimensional packing are assumed, the projected surface area is:
ep ¼ 1  ð15-5Þ
rs  
M 2=3
a ¼ 1:091 ð15-8Þ
N A rL
Surface Area and the BET Equation
where M ¼ molecular weight of the adsorbate, and rL ¼ den-
Specific surface area of an adsorbent, Sg, is measured by sity of the adsorbate in g/cm3, taken as the liquid at the test
adsorbing gaseous nitrogen, using the well-accepted BET temperature.
C15 09/22/2010 Page 573

§15.1 Sorbents 573

Although the BET surface area may not always represent where Psp ¼ vapor pressure of liquid in pore, Ps ¼ the normal
the surface area available for adsorption of a particular mole- vapor pressure of liquid on a flat surface, s ¼ surface tension
cule, the BET test is reproducible and widely used to charac- of liquid in pore, and yL ¼ molar volume of liquid in pore.
terize adsorbents. Vapor pressure of the condensed phase in the pores is less
than its normal vapor pressure for a flat surface. The effect of
Pore Volume and Distribution dp on Psp can be significant. For example, for liquid nitrogen
Specific pore volume, typically cm3 of pore volume/g of at 195.8 C, Ps ¼ 760 torr, s ¼ 0.00827 N/m, u ¼ 0, and
adsorbent, is determined for a small mass of adsorbent, mp, yL ¼ 34.7 cm3/mol. Equation (15-14) then becomes
by measuring the volumes of helium, VHe, and mercury, VHg, d p ðA8 Þ ¼ 17:9=lnðPs =Psp Þ ð15-15Þ
displaced by the adsorbent. Helium is not adsorbed, but fills 

the pores. At ambient pressure, mercury cannot enter the From (15-15) for dp ¼ 30 A, Psp
¼ 418 torr, a reduction in


pores because of unfavorable interfacial tension and contact vapor pressure of almost 50%. At 200 A, the reduction is only


angle. Specific pore volume, Vp, is then determined from about 10%. At 418 torr pressure, only pores less than 30 A in
  diameter remain filled with liquid nitrogen. For greater accu-
V p ¼ V Hg  V He =mp ð15-9Þ racy in applying the Kelvin equation, a correction is needed
Particle density is for the thickness of the adsorbed layer. This correction is dis-
mp cussed in detail by Satterfield [12]. For a monolayer, this
rp ¼ ð15-10Þ thickness for nitrogen is about 0.354 nm, corresponding to a
V Hg
P=P0 in (15-6) of between 0.05 and 0.10. At P=P0 ¼ 0.60 and
and true solid density is
0.90, the adsorbed thicknesses are 0.75 and 1.22 nm, respec-
mp
rs ¼ ð15-11Þ tively. The correction is applied by subtracting twice the
V He adsorbed thickness from dp in (15-14) and (15-15).
Particle porosity is then obtained from (15-3) or (15-5).
Distribution of pore volumes over the range of pore size is
of great importance in adsorption. It is measured by mercury
 EXAMPLE 15.1 Particle Porosity.
porosimetry for large-diameter pores (>100 A); by gaseous-

nitrogen desorption for pores of 15–250 A in diameter; and Using data from Table 15.2, determine the volume fraction of pores
by molecular sieving, using molecules of different diameter, in silica gel (small-pore type) filled with adsorbed water vapor when

for pores <15 A in diameter. In mercury porosimetry, the its partial pressure is 4.6 mmHg and the temperature is 25 C. At
extent of mercury penetration into the pores is measured as a these conditions, the partial pressure is considerably below the
function of applied hydrostatic pressure. A force balance vapor pressure of 23.75 mmHg. In addition, determine whether the
along the axis of a straight pore of circular cross section for amount of water adsorbed is equivalent to more than a monolayer,
if the area of an adsorbed water molecule is given by (15-8) and the
the pressure and interfacial tension between mercury and the
specific surface area of the silica gel is 830 m2/g.
adsorbent surface gives the following equation, which is
identical to (14-107) for the bubble test of a sterile filter:
Solution
4sI cosu
dp ¼  ð15-12Þ
P Take 1 g of silica gel particles as a basis. From (15-3) and data in
Table 15.2, Vp ¼ 0.47=1.09 ¼ 0.431 cm3/g. Thus, for 1 g, pore vol-
where for mercury, sI ¼ interfacial tension ¼ 0.48 N/m and
ume is 0.431 cm3. From the capacity value in Table 15.2, amount of
u ¼ contact angle ¼ 140 . With these values, (15-12) becomes adsorbed water ¼ 0.11/(1 þ 0.11) ¼ 0.0991 g. Assume density of
 8  21:6  105 adsorbed water is 1 g/cm3, volume of adsorbed water ¼ 0.0991
dp A ¼ ð15-13Þ cm3, fraction of pores filled with water ¼ 0.0991=0.431 ¼ 0.230,
PðpsiaÞ

and surface area of 1 g ¼ 830 m2. From (15-8):
Thus, forcing mercury into a 100-A-diameter pore requires a " #2=3
very high pressure of 21,600 psia. 18:02
a ¼ 1:091   ¼ 10:51  1016 cm2 /molecule
The nitrogen desorption method for determining pore-size 6:023  1023 ð1:0Þ

distribution in the 15–250-A-diameter range is an extension  
ð0:0991Þ 6:023  1023
of the BET method for measuring specific surface area. By Number of H2 O molecules adsorbed ¼
18:02
increasing nitrogen pressure above 600 mmHg, multilayer ¼ 3:31  1021
adsorbed films reach the point where they bridge the pore, Number of H2 O molecules in a monolayer for 830 m2
resulting in capillary condensation. At P=P0 ¼ 1, the entire 830ð100Þ2
pore volume is filled with nitrogen. Then, by reducing the ¼ ¼ 7:90  1021
10:51  1016
pressure in steps, nitrogen is desorbed selectively, starting
Therefore, only 3.31=7.90 or 42% of one monolayer is adsorbed.
with larger pores. This selectivity occurs because of the
effect of pore diameter on vapor pressure of the condensed
phase in the pore, as given by the Kelvin equation: Activated Alumina
 
4syL cosu
Psp ¼ Ps exp  ð15-14Þ The four most widely used adsorbents in decreasing order
RTd p of commercial usage are carbon (activated and molecular-
C15 09/22/2010 Page 574

574 Chapter 15 Adsorption, Ion Exchange, Chromatography, and Electrophoresis

sieve), molecular-sieve zeolites, silica gel, and activated 60

d carbon
ate
alumina. In Table 15.2, activated alumina, Al2O3, which

Cumulative pore volume, cm3/100 g

ti v
Ac
includes activated bauxite, is made by removing water from 50
hydrated colloidal alumina. It has a moderately high Sg, with ag
el
lic
a capacity for adsorption of water sufficient to dry gases to 40 Si
less than 1 ppm moisture. Because of this, activated alumina
is widely used for removal of water from gases and liquids. 30
Zeolite 5A
20
Silica Gel a
min

C
lu

C
da

MS
SiO2, made from colloidal silica, has a high Sg and high affin-

MS
te
10 t i va
ity for water and other polar compounds. Related silicate Ac
adsorbents include magnesium silicate, calcium silicate, vari- 0
ous clays, Fuller’s earth, and diatomaceous earth. Silica gel is 2 5 10 20 50
Pore diameter,
also desirable for water removal. Small-pore and large-pore
types are available. Figure 15.3 Representative cumulative pore-size distributions of
adsorbents.

Activated Carbon
and alkali or alkali-earth elements, such as Na, K, and Ca,
Partial oxidation of materials like coconut shells, fruit nuts, with the stoichiometric, unit-cell formula Mx/m[(AlO2)x
wood, coal, lignite, peat, petroleum residues, and bones pro- (SiO2)y]z H2O, where M is the cation with valence m, z is the
duces activated carbon. Macropores within the carbon parti- number of water molecules in each unit cell, and x and y are
cles help transfer molecules to the micropores. Two integers such that y=x  1. The cations balance the charge of
commercial grades are available, one with large pores for the AlO2 groups, each having a net charge of 1. To activate
processing liquids and one with small pores for gas adsorp- the zeolite, the water molecules are removed by raising the
tion. As shown in Table 15.2, activated carbon is relatively temperature or pulling a vacuum. This leaves the remaining
hydrophobic and has a large surface area. Accordingly, it is atoms spatially intact in interconnected, cagelike structures
widely used for purification and separation of gas and liquid with six identical window apertures each of from 3.8 to about

mixtures containing nonpolar and weakly polar organic com- 10 A, depending on the cation and crystal structure. These
pounds, which adsorb much more strongly than water. In apertures act as sieves, which permit small molecules to enter
addition, the bonding strength of adsorption on activated car- the crystal cage, but exclude large molecules. Thus, com-
bon is low, resulting in a low heat of adsorption and ease of pared to other types of adsorbents, molecular-sieve zeolites
regeneration. are highly selective because all apertures have the same size.
The properties and applications of five of the most com-
monly used molecular-sieve zeolites are given in Table 15.3,
Molecular-Sieve Carbon
from Ruthven [13]. Zeolites separate not only by molecular
Unlike activated carbon, which typically has pore diameters size and shape, but also by polarity, so they can also separate

starting from 10 A, molecular-sieve carbon (MSC) has pores molecules of similar size. Zeolites have circular or elliptical

ranging from 2 to 10 A, making it possible to separate N2 apertures. Adsorption in zeolites is a selective and reversible
from air. In one process, small pores are made by depositing filling of crystal cages, so cage volume is a pertinent factor.
coke in the pore mouths of activated carbon. Although natural zeolite minerals have been known for more
than 200 years, molecular-sieve zeolites were first synthe-
sized by Milton [2], using reactive materials at temperatures
Molecular-Sieve Zeolites
of 25100 C.
Most adsorbents have a range of pore sizes, as shown in Fig- A type A zeolite is shown in Figure 15.4a as a three-
ure 15.3, where the cumulative pore volume is plotted against dimensional structure of silica and alumina tetrahedra, each
pore diameter. Exceptions are molecular-sieve zeolites, formed by four oxygen atoms surrounding a silicon or alumi-
which are crystalline, inorganic polymers of aluminosilicates num atom. Oxygen and silicon atoms have two negative and

Table 15.3 Properties and Applications of Molecular-Sieve Zeolites


Designation Cation Unit-Cell Formula Aperture Size, A Typical Applications

3A K+ K12[(AlO2)12(SiO2)12] 2.9 Drying of reactive gases

4A Na+ Na12[(AlO2)12(SiO2)12] 3.8 H2O, CO2 removal; air separation
5A Ca2+ Ca5Na2[(AlO2)12(SiO2)12] 4.4 Separation of air; separation of linear paraffins
10X Ca2+ Ca43[(AlO2)86(SiO2)106] 8:0 Separation of air;
13X Na+ Na86[(AlO2)86(SiO2)86] 8:4 removal of mercaptans
C15 09/22/2010 Page 575

§15.1 Sorbents 575

to exchange ammonium ions in fertilizers with calcium ions

in soils. Industrial water softeners using zeolites were intro-
duced about 1910, but they were unstable in the presence of
mineral acids. The instability problem was solved by Adams
and Holmes [16] in 1935, when they synthesized the first
organic-polymer, ion-exchange resins by the polycondensation
of phenol and aldehydes. Depending upon  the phenolic

group,
 the
 resin contains either sulfonic SO 3 or amine
þ
NH3 groups for the reversible exchange of cations or
anions. Today, the most widely used ion exchangers are syn-
Figure 15.4 Structures of molecular-sieve zeolites: (a) Type A unit thetic, organic-polymer resins based on styrene- or acrylic-
cell. (b) Type X unit cell. acid-type monomers, as described by D’Alelio in U.S. Patent
2,366,007 (Dec. 26, 1944).
four positive charges, respectively, causing the tetrahedra to Ion-exchange resins are generally solid gels in spherical or
build uniformly in four directions. Aluminum, with a valence granular form, which consist of (1) a three-dimensional poly-
of 3, causes the alumina tetrahedron to be negatively charged. meric network, (2) ionic functional groups attached to the
The added cation provides the balance. In Figure 15.4a, an network, (3) counterions, and (4) a solvent. Strong-acid, cat-
octahedron of tetrahedra is evident with six faces, with one ion-exchange resins and strong-base, anion-exchange resins
near-circular window aperture at each face. A type X zeolite that are fully ionized over the entire pH range are based on
is shown in Figure 15.4b. This unit-cell structure results in a the copolymerization of styrene and a cross-linking agent,
larger window aperture. Zeolites are treated in monographs divinylbenzene, to produce the three-dimensional, cross-
by Barrer [14] and Breck [15]. linked structure shown in Figure 15.5a. Degree of cross-
linking is governed by the ratio of divinylbenzene to styrene.
Weakly acid, cation exchangers are sometimes based on the
Polymeric Adsorbents
copolymerization of acrylic acid and methacrylic acid, as
Of lesser commercial importance are polymeric adsorbents. shown in Figure 15.5b. These two cross-linked copolymers
Typically, they are spherical beads, 0.5 mm in diameter, swell in the presence of organic solvents and have no ion-
made from microspheres about 104 mm in diameter. They exchange properties.
are produced by polymerizing styrene and divinylbenzene To convert the copolymers to water-swellable gels with
for use in adsorbing nonpolar organics from aqueous solu- ion-exchange properties, ionic functional groups are added to
tions, and by polymerizing acrylic esters for adsorbing polar the polymeric network by reacting copolymers with various
solutes. They are regenerated by leaching with organic chemicals. For example, if the styrene–divinylbenzene co-
solvents. polymer is sulfonated, as shown in Figure 15.6a, the cation-
exchange
  resin, shown in Figure 15.6b, is obtained with
SO 3 groups permanently attached to the polymeric net-
§15.1.2 Ion Exchangers
work to give a negatively charged matrix and exchangeable,
The first ion exchangers were naturally occurring inorganic mobile, positive hydrogen ions (cations). The hydrogen ion
aluminosilicates (zeolites) used in experiments in the 1850s can be exchanged on an equivalent basis with other cation

Figure 15.5 Ion-exchange resins: (a) Resin from

styrene and divinylbenzene; (b) Resin from
acrylic and methacrylic acid.
C15 09/22/2010 Page 576

576 Chapter 15 Adsorption, Ion Exchange, Chromatography, and Electrophoresis

H+–O3S
SO3–H+

SO3–H+ SO3–H+
CH CH2 CH CH2
SO3–H+
SO3–H+
+ H2SO4
SO3–H+
SO3–H+
SO3H
SO3–H+
(a)
H+–O3S
SO3–H+

(b)

CH CH2 CH CH2

+ CH3OCH2Cl

CH2Cl

CH CH2 CH CH2 Figure 15.6 Introducing ionic

functional groups into resins.
(a) Sulfonation to a cation
+ (CH3)3N
exchanger. (b) Fixed and mobile
ions in a cation exchanger.
CH2Cl CH2–N(CH3)3Cl (c) Chloromethylation and
(c) amination to an anion exchanger.

counterions, such as Na+, Ca2+, K+, or Mg2+, to maintain on the number of equivalents of mobile charge in the resin.
charge neutrality of the polymer. For example, two H+ ions Thus, 1 mol H+ is one equivalent, whereas 1 mol Ca2+ is two
are exchanged for one Ca2+ ion. The liquid whose ions are equivalents. Exchanger capacity is usually quoted as eq/kg of
being exchanged also contains other ions of unlike charge, dry resin or eq/L of wet resin. Wet capacity depends on resin
such as Cl for a solution of NaCl, where Na+ is exchanged. water content and degree of swelling, whereas dry capacity is
These other ions are called co-ions. Often the liquid treated is fixed. For copolymers of styrene and divinylbenzene, maxi-
H2O, which dissolves to some extent in the resin and causes mum capacity is based on the assumption that each benzene
it to swell. Other solvents, such as methanol, are also soluble ring in the resin contains one sulfonic-acid group.
in the resin. If the styrene–divinylbenzene copolymer is
chloromethylated and aminated, a strong-base, anion-
exchange resin is formed, as shown in Figure 15.6c, which EXAMPLE 15.2 Ion-Exchange Capacity.
can exchange Cl ions for other anions, such as OH, A commercial ion-exchange resin is made from 88 wt% styrene and
HCO 
3 ; SO4 ; and NO3 .
2
12 wt% divinylbenzene. Estimate the maximum ion-exchange ca-
Commercial ion exchangers in the H, Na, and Cl form are pacity in eq/kg resin (same as meq/g resin).
available under the trade names of AmberliteTM, DuoliteTM,
DowexTM, Ionac1, and Purolite1, typically in the form of Solution
spherical beads from 40 mm to 1.2 mm in diameter. When
saturated with water, the beads have typical moisture con- Basis: 100 g of resin before sulfonation.
tents from 40 to 65 wt%. When water-swollen, rp ¼ 1.1–
1.5 g/cm3. When packed into a vessel, rb ¼ 0.56–0.96 g/cm3 M g gmol
with eb of 0.35–0.40. Styrene 104.14 88 0.845
Before water is demineralized by ion exchange, poten- Divinylbenzene 130.18 12 0.092
tial organic foulants must be removed. As discussed by 100 0.937
McWilliams [17], this can be accomplished by coagulation,
clarification, prechlorination, and use of ion-exchanger traps Sulfonation at one location on each benzene ring requires 0.937 mol
that exchange inorganic anions for anionic organic molecules. of H2SO4 to attach a sulfonic acid group (M ¼ 81.07) and split
The maximum ion-exchange capacity of a strong-acid cat- out one water molecule. This is 0.937 equivalent, with a weight
ion or strong-base anion exchanger is stoichiometric, based addition of 0.937(81.07) ¼ 76 g. Total dry weight of sulfonated
C15 09/22/2010 Page 577

§15.1 Sorbents 577

resin ¼ 100 þ 76 ¼ 176 g maximum ion-exchange capacity, or exchange mechanism is desired, a synthetic, polymer ion
0:937 exchanger is used. With a polymer gel or a microporous
¼ 5:3 eq=kgðdryÞ solid, a separation based on sieving, called exclusion, can be
ð176=1;000Þ
operative. Unique to chromatography are liquid-supported or
Depending on the extent of cross-linking, resins from copolymers of
-bonded solids, where the mechanism is absorption into the
styrene and divinylbenzene are listed as having actual capacities of
from 3.9 (high degree of cross-linking) to 5.5 (low degree of cross-
liquid, also referred to as a partition mode of separation or
linking). Although a low degree of cross-linking favors dry capacity, partition chromatography. With mobile liquid phases, the
almost every other ion-exchanger property, including wet capacity stationary liquid phase may be stripped or dissolved. Accord-
and selectivity, is improved by cross-linking, as discussed by ingly, methods of chemically bonding the stationary liquid
Dorfner [18]. phase to the solid support have been developed.
In packed columns >1 mm inside diameter, the sorbents
are in the form of particles. In capillary columns <0.5 mm
§15.1.3 Sorbents for Chromatography inside diameter, the sorbent is the inside wall or a coating on
that wall. If coated, the capillary column is referred to as a
Sorbents (called stationary phases) for chromatographic sep- wall-coated, open-tubular (WCOT) column. If the coating is
arations come in many forms and chemical compositions a layer of fine particulate support material to which a liquid
because of the diverse ways that chromatography is applied. adsorbent is added, the column is a support-coated, open-
Figure 15.7 shows a classification of analytical chromato- tubular (SCOT) column. If the wall is coated with a porous
graphic systems, taken from Sewell and Clarke [19]. The adsorbent only, the column is a porous-layer, open-tubular
mixture to be separated, after injection into the carrier fluid (PLOT) column.
to form the mobile phase, may be a liquid (liquid chromatog- Each type of sorbent can be applied to sheets of glass, plas-
raphy) or a gas (gas chromatography). Often, the mixture is tic, or aluminum for use in thin-layer (or planar) chromatogra-
initially a liquid, but is vaporized by the carrier gas, giving a phy or to a sheet of cellulose material for use in paper
gas mixture as the mobile phase. Gas carriers are inert and do chromatography. If a pump, rather than gravity, is used to
not interact with the sorbent or feed. Liquid carriers (sol- pass a liquid mobile phase through a packed column, the name
vents) can interact and must be selected carefully. high-performance liquid chromatography (HPLC) is used.
The stationary sorbent phase is a solid, a liquid supported The two most common adsorbents used in chromatogra-
on or bonded to a solid, or a gel. With a porous-solid adsorb- phy are porous alumina and porous silica gel. Of lesser
ent, the mechanism of separation is adsorption. If an ion- importance are carbon, magnesium oxide, and carbonates.

Figure 15.7 Classification of analytical chromatographic systems.

[From P.A. Sewell and B. Clarke, Chromatographic Separations, John Wiley & Sons, New York (1987) with permission.]
C15 09/22/2010 Page 578

578 Chapter 15 Adsorption, Ion Exchange, Chromatography, and Electrophoresis

(b) Use partition or gas–liquid chromatography, that is, with a gas

mobile phase and a bonded liquid coating on a solid for the sta-
tionary phase.
(c) Use ion-exchange chromatography, that is, with a liquid as the
mobile phase and polymer resin beads as the stationary phase.

Figure 15.8 Bonded phases from the reaction of surface silanol

§15.2 EQUILIBRIUM CONSIDERATIONS
groups with (a) monofunctional and (b) bifunctional chlorosilanes.
In adsorption, a dynamic equilibrium is established for solute
distribution between the fluid and solid surface. This is
Alumina is a polar adsorbent and is preferred for the separa- expressed in terms of: (1) concentration (if the fluid is a liq-
tion of components that are weakly or moderately polar, with uid) or partial pressure (if the fluid is a gas) of the adsorbate in
more polar compounds retained more selectively by the the fluid and (2) solute loading on the adsorbent, expressed as
adsorbents and therefore eluted from the column last. mass, moles, or volume of adsorbate per unit mass or per unit
Alumina is a basic adsorbent, preferentially retaining acidic BET adsorbent surface area. Unlike vapor–liquid and liquid–
compounds. Silica gel is less polar than alumina and is an liquid equilibria, where theory is often applied to estimate
acidic adsorbent, preferentially retaining basic compounds, phase distribution in the form of K-values (§2.2.2), no accept-
such as amines. Carbon is a nonpolar (apolar) stationary able theory has been developed to estimate fluid–solid adsorp-
phase with the highest attraction for nonpolar molecules. tion equilibria. It is thus necessary to obtain equilibrium data
Adsorbent-type sorbents are better suited for separation of for a particular solute, or mixture of solutes and/or solvent,
a mixture on the basis of chemical type (e.g., olefins, esters, and the solid-adsorbent material of interest. If the data are
acids, aldehydes, alcohols) than for separation of individual taken over a range of fluid concentrations at a constant tem-
members of a homologous series. For the latter, partition perature, a plot of adsorbent solute loading versus solute con-
chromatography—wherein an inert-solid support, often sil- centration or partial pressure in the fluid, called an adsorption
ica gel, is coated with a liquid phase—is preferred. For gas isotherm, is made. This equilibrium isotherm places a limit on
chromatography, that liquid must be nonvolatile. For liquid the extent to which a solute is adsorbed from a specific fluid
chromatography, the stationary liquid phase must be mixture on a given adsorbent for one set of conditions. The
insoluble in the mobile phase, but since this is difficult to rate at which solute is adsorbed is discussed in §15.3.
achieve, the stationary liquid phase is usually bonded to the
solid support. An example of a bonded phase is the result of
§15.2.1 Pure-Gas Adsorption
reacting silica with a chlorosilane. Both monofunctional and
bifunctional silanes are used, as shown in Figure 15.8, where The five experimental physical-adsorption isotherms for pure
R is a methyl (CH3) group and R0 is a hydrocarbon chain (C6, gases shown in Figure 15.9 are due to Brunauer, as described
C8, or C18) where the terminal CH3 group is replaced with a in [20, 21]. The simplest isotherm is Type I, which corre-
polar group, such as CN or NH2. If the resulting station- sponds to unimolecular adsorption, as characterized by a max-
ary phase is more polar than the mobile phase, it is normal- imum limit in the amount adsorbed. This type describes gases
phase chromatography; otherwise, it is reverse-phase at temperatures above their critical temperature. The more
chromatography. complex Type II isotherm is associated with multimolecular
In liquid chromatography, the order of elution of solutes in BET adsorption and is observed for gases at temperatures
the mobile phase can be influenced by the solvent carrier of below their critical temperature and for pressures below, but
the mobile phase by matching the solvent polarity with the approaching, the saturation pressure. The heat of adsorption
solutes and using more-polar adsorbents for less-polar sol- for the first adsorbed layer is greater than that for the succeed-
utes and less-polar adsorbents for more-polar solutes. ing layers, each of which is assumed to have a heat of adsorp-
tion equal to the heat of condensation. Both Types I and II are
desirable isotherms, exhibiting strong adsorption.
EXAMPLE 15.3 Chromatography Mode. The Type III isotherm in Figure 15.9 is convex and
For the separation of each of the following mixtures, select an
undesirable because the extent of adsorption is low except at
appropriate mode of chromatography from Figure 15.7: (a) gas high pressures. According to BET theory, it corresponds to
mixture of O2, CO, CO2, and SO2; (b) vaporized mixture of anthra- multimolecular adsorption where heat of adsorption of the
cene, phenanthrene, pyrene, and chrysene; and (c) aqueous solution first layer is less than that of succeeding layers. Fortunately,
containing Ca2+ and Ba2+. this type of isotherm is rare, an example being adsorption of
iodine vapor on silica gel. In the limit, as heat of adsorption
Solution of the first layer approaches zero, adsorption is delayed until
the saturation pressure is approached.
(a) Use gas–solid chromatography, that is, with a gas mobile phase
and a solid-adsorbent stationary phase.
Derivation of the BET equation (15-6) assumes that an
infinite number of molecular layers can be adsorbed, thus
C15 09/22/2010 Page 579

§15.2 Equilibrium Considerations 579

I Figure 15.10. The five adsorption isotherms of Figure 15.10a

Amount adsorbed
cover pressures from vacuum to almost 800 mmHg and tem-
peratures from 23.5 to 151.5 C. For ammonia, the normal
boiling point is 33.3 C and the critical temperature is
132.4 C. For the lowest-temperature isotherm, up to 160 cm3
(STP) of ammonia per gram of charcoal is adsorbed, which is
equivalent to 0.12 g NH3/g charcoal. All five isotherms are of
Type I. When the amount adsorbed is low (<25 cm3/g), iso-
0 1.0
Relative pressure, P/P0 therms are almost linear and a form of Henry’s law, called the
linear isotherm, is obeyed:
II III
q ¼ kp ð15-16Þ
Amount adsorbed

Amount adsorbed
where q is equilibrium loading or amount adsorbed of a given
species/unit mass of adsorbent; k is an empirical, temperature-
B dependent constant for the component; and p is the partial
pressure of the species. As temperature increases, the amount
adsorbed decreases because of Le Chatelier’s principle for an
0 1.0 0 1.0 exothermic process. This is shown clearly in the crossplot of
Relative pressure, P/P0 Relative pressure, P/P0 adsorption isobars in Figure 15.10b, where absolute tempera-
ture is employed. Another crossplot of the data yields adsorp-
IV V tion isosteres in Figure 15.10c. These curves for constant
Amount adsorbed

Amount adsorbed

amounts adsorbed, resemble vapor-pressure plots, for which

the adsorption form of the Clausius–Clapeyron equation,

d ln p DH ads
¼ ð15-17Þ
B
dT RT 2

dlog p DH ads

0 1.0 0 1.0 or ¼ ð15-18Þ
Relative pressure, P/P0 Relative pressure, P/P0 d ð1=T Þ 2:303RT
Figure 15.9 Brunauer’s five types of adsorption isotherms. (P=P0 ¼
total pressure/vapor pressure.)
is used to determine the exothermic heat of adsorption, shown in
Figure 15.10d, where DHads is initially 7,300 cal/mol, but
decreases as the amount adsorbed increases, reaching 6,100 cal/
precluding the possibility of capillary condensation. In a mol at 100 cm3/g. These values can be compared to the heat of
development by Brunauer et al. [20] before the development vaporization of NH3, which at 30 C is 4,600 cal/mol.
of the BET equation, the number of layers is restricted by Adsorption-isotherm data for 18 different pure gases and a
pore size, and capillary condensation is assumed to occur at a variety of solid adsorbents are analyzed by Valenzuela and
reduced vapor pressure in accordance with the Kelvin equa- Myers [23]. The data show that adsorption isotherms for a
tion (15-14). The resulting equation is complex, but also pre- given pure gas at fixed temperature vary with the adsorbent.
dicts adsorption isotherms of Types IV and V in Figure 15.9, For propane vapor at 25–30 C, as shown in Figure 15.11, for
where the maximum extent of adsorption occurs before the pressures up to 101.3 kPa, the highest specific adsorption is
saturation pressure is reached. Types IV and V are the capil- with Columbia G-grade activated carbon, while the lowest is
lary-condensation versions of Types II and III, respectively. with Norton Z-900H, a zeolite molecular sieve. Columbia G-
As shown in Figure 15.9, hysteresis occurs in multimolec- grade activated carbon has about twice the adsorbate capacity
ular adsorption regions for isotherms of Types IV and V. The of Cabot Black Pearls activated carbon.
upward adsorption branch of the hysteresis loop is due to Literature data compiled by Valenzuela and Myers [23]
simultaneous, multimolecular adsorption and capillary con- also show that for a given adsorbent, loading depends
densation. Only the latter occurs during the downward strongly on the gas. This is illustrated in Table 15.4 for a tem-
desorption branch. Hysteresis can also occur in any isotherm perature of 38 C and a pressure range of 97.9 to 100 kPa
when strongly adsorbed impurities are present. Measure- from the data of Ray and Box [24] for Columbia L activated
ments of pure-gas adsorption require adsorbents with clean carbon. Included in the table are normal boiling points and
pore surfaces, which is achieved by pre-evacuation. critical temperatures. As might be expected, the species are
adsorbed in approximately the inverse order of volatility.
Correlation of experimental gas adsorption isotherms is
Linear Isotherm
the subject of numerous articles and books. As summarized
Physical-adsorption data of Titoff [22] for ammonia gas on by Yang [25], approaches have ranged from empirical to the-
charcoal, as discussed by Brunauer [21], are shown in oretical. In practice, the classic equations of Freundlich and
C15 09/22/2010 Page 580

580 Chapter 15 Adsorption, Ion Exchange, Chromatography, and Electrophoresis

Volume adsorbed per gram of adsorbent,

Volume adsorbed per g of adsorbent,

–23.5°C 700 mm
150 150
0°C 400 mm

100 mm

cm3 at STP
30°C
cm3 at STP

100 100

40 mm

50 50
80°C

151.5°C

100 200 300 400 500 600 700 800 150 200 250 300 350 400 450
Pressure, mm Hg Temperature, K

(a) (b)

800
100cm 3

35cm 3

10cm 3

700

600 3.0 v=
v= 10
v = 75 0c
Pressure, mm Hg

v = 50 cm 3 m3
75cm 3

500 ,–
v 3 c , ⌬H
v = 2 5cm m 3, –⌬H
=
20cm 3

0 ad
10 cm 3 3, –⌬H ads s =
cm –⌬ a = 61
400 2.0 3 ,– H ds = 62 00
, – ⌬H ad 0 Figure 15.10 Different displays
log p

64 0 ca
⌬H a s =
00 c al/ l/m
ds 6 5 ol
ad = 0 ca mo of adsorption-equilibrium
50cm 3

300 s = 6 80 0 l/m le e
73 c al
0 /m ole
v is per gram. v is per gram. 0 0 ca
ol
data for NH3 on charcoal.
ca l/m e
200 1.0 l/m ole (a) Adsorption isotherms.
ol
e (b) Adsorption isobars.
100 (c) Adsorption isosteres.
(d) Isosteric heats of adsorption.
150 200 250 300 350 400 450 2.2 2.6 3.0 3.4 3.8 4.2 [From S. Brunauer, The Adsorption
Temperature, K I of Gases and Vapors, Vol. I,
× 103
T Princeton University Press (1943)
(c) with permission.]
(d)

Langmuir, discussed next, are dominant because of their sim- and van Bemmelen, is empirical and nonlinear in pressure:
plicity and ability to correlate Type I isotherms.
q ¼ kp1=n ð15-19Þ
Freundlich Isotherm
where k and n are temperature-dependent constants for a par-
The equation attributed to Freundlich [26], but which, ticular component and adsorbent. The constant, n, lies in the
according to Mantell [27], was devised earlier by Boedecker range of 1 to 5, and for n ¼ 1, (15-19) reduces to the linear

6.0 d carbon Table 15.4 Comparison of Equilibrium Adsorption of Pure

G acti vate
Columbia Gases on 20–40 mesh Columbia L Activated Carbon Particles
5.0 (Sg ¼ 1,152 m2/g) at 38 C and 1 atm

Pure gas q, mol/kg Tb,  F Tc,  F

Loading, gmol/kg

4.0
H2 0.0241 423.0 399.8
bo n
3.0
d car N2 0.292 320.4 232.4
tivate
ac
Pearls MSC-5A carbon molecular sieve CO 0.374 313.6 220.0
2.0 Black
l
silic
a ge CH4 0.870 258.7 116.6
i son Z-900H zeolite
D a v Norton CO2 1.64 109.3 87.9
1.0
C2H2 2.67 119 95.3
0.0 C2H4 2.88 154.6 48.6
0 50 100 150 200 250 C2H6 3.41 127.5 90.1
Pressure, kPa
C3H6 4.54 53.9 196.9
Figure 15.11 Adsorption isotherms for pure propane vapor at 298– C3H8 4.34 43.7 216.0
303 K.
C15 09/22/2010 Page 581

§15.2 Equilibrium Considerations 581

isotherm (15-16). Experimental q–p isothermal data can be Using (15-25), the best straight line is drawn through a plot
fitted to (15-19) by a nonlinear curve fit or by converting (15- of points p=q versus p, giving a slope of (1=qm) and an inter-
19) to the following linear form, and using a graphical cept of 1=(qmK). Theoretically, K should change rapidly with
method or a linear-regression program to obtain k and n; temperature but qm should not, because it is related through
ym by (15-7) to Sg. The Langmuir isotherm predicts an
log q ¼ log k þ ð1=nÞlog p ð15-20Þ asymptotic limit for q at high pressure, whereas the Freundlich
In the graphical method, data are plotted as log q versus log p; isotherm does not, as shown e.g. by the curve for Columbia G
the best straight line through the data has a slope of (1=n) and activated carbon in Figure 15.11.
an intercept of log k. In general, k decreases, while n increases
with increasing temperature, approaching a value of 1 at high
temperatures. Although (15-19) is empirical, it can be derived Other Adsorption Isotherms
by assuming a heterogeneous surface with a nonuniform dis- Valenzuela and Myers [23] fit isothermal, pure-gas adsorp-
tribution of heat of adsorption (Brunauer [21]). tion data to the three-parameter isotherms of Toth:
mp
Langmuir Isotherm q¼ ð15-26Þ
ðb þ pt Þ1=t
The Langmuir equation [28] is restricted to Type I iso-
where m, b, and t are constants for a given adsorbate–
therms. It is derived from mass-action kinetics, assuming
adsorbent system and temperature.
that chemisorption is the reaction. Let the surface of the
Honig and Reyerson devisied the three-constant UNILAN
pores of the adsorbent be homogeneous (DHads ¼ constant),
equation:
with negligible interaction forces between adsorbed mole-


cules. If u is the fraction of surface covered by adsorbed n c þ pes
molecules, (1  u) is the fraction of bare surface, and the q ¼ ln ð15-27Þ
2s c þ pes
net rate of adsorption is the difference between the rate of
adsorption on the bare surface and the rate of desorption on where n, s, and c are the constants for a given system and
the covered surface: temperature. The Toth and UNILAN isotherms reduce to the
Langmuir isotherm for t ¼ 1 and s ¼ 0, respectively.
dq=dt ¼ ka pð1  uÞ  kd u ð15-21Þ

where ka and kd are the adsorption and desorption kinetic

constants. At equilibrium, dq=dt ¼ 0 and (15-21) reduces to EXAMPLE 15.4 Freundlich and Langmuir Isotherms.

Kp Data for the equilibrium adsorption of pure methane gas on acti-

u¼ ð15-22Þ vated carbon (PCB from Calgon Corp.) at 296 K were obtained by
1 þ Kp
Ritter and Yang [Ind. Eng. Chem. Res., 26, 1679–1686 (1987)]:
where K ¼ ka=kd is the adsorption-equilibrium constant and
u ¼ q=qm ð15-23Þ q, cm3 (STP) of 45.5 91.5 113 121 125 126 126
CH4/g carbon
where qm is the maximum loading corresponding to complete P ¼ p, psia 40 165 350 545 760 910 970
surface coverage. Thus, the Langmuir adsorption isotherm is
restricted to a monomolecular layer. Combining (15-23) with
Fit the data to: (a) the Freundlich isotherm, and (b) the Langmuir
(15-22) results in the Langmuir isotherm:
isotherm. Which isotherm provides a better fit? Do the data give a
reasonable fit to the linear isotherm?
Kqm p
q¼ ð15-24Þ
1 þ Kp
Solution
At low pressures, if Kp 1, (15-24) reduces to the linear iso- Using the linearized forms of the isotherm equations, a spreadsheet
therm, (15-16), while at high pressures where Kp
1, q ¼ or other program can be used to do a linear regression:
qm. At intermediate pressures, (15-24) is nonlinear in pressure.
(a) Using (15-20), log k ¼ 1.213, k ¼ 16.34, 1=n ¼ 0.3101, and
Although originally devised by Langmuir for chemisorption, n ¼ 3.225. Thus, the Freundlich equation is:
(15-24) is widely applied to physical-adsorption data.
In (15-24), K and qm are treated as constants obtained by q ¼ 16:34p0:3101
fitting the nonlinear equation to experimental data or by
employing the following linearized form, numerically or (b) Using (15-25), 1=qm ¼ 0.007301, qm ¼ 137.0, 1=(qmK) ¼
graphically: 0.5682, and K ¼ 0.01285. Thus, the Langmuir equation is

p 1 p 1:760p
¼ þ ð15-25Þ q¼
q qm K qm 1 þ 0:01285p
C15 09/22/2010 Page 582

582 Chapter 15 Adsorption, Ion Exchange, Chromatography, and Electrophoresis

The predicted values of q from the two isotherms are: In a similar fashion, as shown by Yon and Turnock [30], the
Freundlich and Langmuir equations can be combined to give
q, cm3 (STP) of CH4/g carbon the following extended relation for gas mixtures:
p, psia Experimental Freundlich Langmuir 1=n
ð qi Þ 0 k i pi i
qi ¼ P 1=n ð15-33Þ
40 45.5 51.3 46.5 1 þ k j pj j
165 91.5 79.6 93.1 j
350 113 101 112
where (qi)0 is the maximum loading, which may differ from
545 121 115 120
(qi)m for a monolayer. Equation (15-33) represents data for
760 125 128 124
nonpolar, multicomponent mixtures in molecular sieves rea-
910 126 135 126
sonably well. Broughton [31] has shown that both the
970 126 138 127
extended-Langmuir and Langmuir–Freundlich equations
The Langmuir isotherm fits the data significantly better than the
lack thermodynamic consistency. Therefore, (15-32) and
Freundlich. Average percent deviations, in q, are 1.01% and 8.64%, (15-33) are frequently referred to as nonstoichiometric iso-
respectively. One reason for the better Langmuir fit is the trend to an therms. Nevertheless, for practical purposes, their simplicity
asymptotic value for q at the highest pressures. Clearly, the data do often makes them the isotherms of choice.
not fit a linear isotherm well at all. Both (15-32) and (15-33) are also referred to as constant-
selectivity equilibrium equations because they predict a sepa-
ration factor (selectivity), ai,j, for each pair of components, i,
j, in a mixture that is constant for a given temperature
§15.2.2 Gas Mixtures and Extended Isotherms and independent of mixture composition. For example, (15-
32) gives
Commercial applications of physical adsorption involve mix-
tures rather than pure gases. If adsorption of all components qi =qj ðqi Þm K i
except one (A) is negligible, then adsorption of A is esti- ai;j ¼ ¼
pi =pj ðqj Þm K j
mated from its pure-gas-adsorption isotherm using the partial
pressure of A in the mixture. If adsorption of two or more As with vapor–liquid and liquid–liquid phase equilibria
components in the mixture is significant, the situation is com- for three or more components, data for binary and multi-
plicated. Experimental data show that one component can component gas–solid adsorbent equilibria are scarce and
increase, decrease, or have no influence on adsorption of less accurate than corresponding pure-gas data. Valen-
another, depending on interactions of adsorbed molecules. A zuela and Myers [23] include experimental data on
simple theoretical treatment is the extension of the Langmuir adsorption of gas mixtures from 9 published studies on 29
equation by Markham and Benton [29], who neglect interac- binary systems, for which pure-gas-adsorption isotherms
tions and assume that the only effect is the reduction of the were also obtained. They also describe procedures for
vacant surface area for the adsorption of A because of applying the Toth and UNILAN equations to multi-
adsorption of other components. For a binary gas mixture component mixtures based on the ideal-adsorbed-solution
of A and B, let uA ¼ fraction of surface covered by A and uB ¼ (IAS) theory of Myers and Prausnitz [32]. Unlike the
fraction of surface covered by B. Then, ð1  uA  uB Þ ¼ extended-Langmuir equation (15-32), which is explicit in
fraction of vacant surface. At equilibrium: the amount adsorbed, the IAS theory, though more accurate,
ðkA Þa pA ð1  uA  uB Þ ¼ ðkA Þd uA ð15-28Þ is not explicit and requires an iterative solution procedure.
Additional experimental data for higher-order (ternary and/or
ðkB Þa pB ð1  uA  uB Þ ¼ ðkB Þd uB ð15-29Þ higher) gas mixtures are given by Miller, Knaebel, and Ikels
[33] for 5A molecular sieves and by Ritter and Yang [34] for
Solving these equations simultaneously, and combining re-
activated carbon. Yang [25] presents a discussion of mixture
sults with (15-23), gives
adsorption theories, together with comparisons of these theo-
ð qA Þ m K A pA ries with mixture data for activated carbon and zeolites. The
qA ¼ ð15-30Þ
1 þ K A pA þ K B pB data on zeolites are the most difficult to correlate, with the sta-
tistical thermodynamic model (SSTM) of Ruthven and Wong
ðqB Þm K B pB [35] giving the best results.
qB ¼ ð15-31Þ
1 þ K A pA þ K B pB

where (qi)m is the maximum amount of adsorption of species

i for coverage of the entire surface. Equations (15-30) and EXAMPLE 15.5 Extended-Langmuir Isotherm.
(15-31) are readily extended to a mixture of j components:
The experimental work of Ritter and Yang, cited in Example 15.4,
ð qi Þ m K i pi also includes adsorption isotherms for pure CO and CH4, and a
qi ¼ P ð15-32Þ binary mixture of CH4 (A) and CO (B). Ritter and Yang give the
1 þ K j pj
j following Langmuir constants for pure A and B at 294 K:
C15 09/22/2010 Page 583

§15.2 Equilibrium Considerations 583

obtained over the entire concentration range, the distinction

qm, cm3(STP)/g K, psi1
between solute and solvent is arbitrary and the resulting
CH4 133.4 0.01370 adsorption isotherms, as discussed by Kipling [36], can
CO 126.1 0.00624 exhibit curious shapes, unlike those obtained for pure gases
or gas mixtures. To illustrate this, let n0 ¼ total moles of bi-
Use these constants with the extended-Langmuir equation to predict nary liquid contacting the adsorbent, m ¼ mass of adsorbent,
the specific adsorption volumes (STP) of CH4 and CO for a vapor x01 ¼ mole fraction of solute before contact with adsorbent,
mixture of 69.6 mol% CH4 and 30.4 mol% CO at 294 K and a total x1 ¼ mole fraction of solute in the bulk solution after adsorp-
pressure of 364.3 psia. Compare the results with the following
tion equilibrium is achieved, and qe1 ¼ apparent moles of sol-
experimental data of Ritter and Yang:
ute adsorbed per unit mass of adsorbent.
A solute material balance, assuming no adsorption of sol-
Total volume adsorbed, cm3/(STP)/g 114.1
vent and a negligible change in the total moles of liquid mix-
Mole fractions in adsorbate:
ture, gives
CH4 0.867  
CO 0.133 e n0 x01  x1
q1 ¼ ð15-34Þ
m
Solution If isothermal data are obtained over the entire concentration
pA ¼ yA P ¼ 0:696ð364:3Þ ¼ 253:5 psia range, processed with (15-34), and plotted as adsorption iso-
pB ¼ yB P ¼ 0:304ð364:3Þ ¼ 110:8 psia therms, the resulting curves are not as shown in Figure
From (15-30): 15.12a. Instead, curves of the type in Figures 15.12b and c
133:4ð0:0137Þð253:5Þ are obtained, where negative adsorption appears to occur in
qA ¼ ¼ 89:7 cm3 ðSTPÞ/g Figure 15.12c. Such isotherms are best referred to as compos-
1 þ ð0:0137Þð253:5Þ þ ð0:00624Þð110:8Þ
ite isotherms or isotherms of concentration change, as sug-
126:1ð0:00624Þð110:8Þ
qB ¼ ¼ 16:9 cm3 ðSTPÞ/g gested by Kipling [36]. Likewise, adsorption loading, qe1 , of
1 þ ð0:0137Þð253:5Þ þ ð0:00624Þð110:8Þ (15-34) is more correctly referred to as surface excess.
The total amount adsorbed ¼ q ¼ qA þ qB ¼ 89.7 þ 16.9 ¼ 106.6 Under what conditions are composite isotherms of the
cm3 (STP)/g, which is 6.6% lower than the experimental value. shapes in Figures 15.12b and c obtained? This is shown by
Mole fractions in the adsorbate are xA ¼ qA=q ¼ 89.7/106.6 ¼ examples in Figure 15.13, where various combinations of
0.841 and xB ¼ 1  0.841 ¼ 0.159. These adsorbate mole fractions hypothetical adsorption isotherms for solute (A) and solvent
deviate from the experimental values by 0.026. For this example, the (B) are shown together with resulting composite isotherms.
extended-Langmuir isotherm gives reasonable results.
When the solvent is not adsorbed, as seen in Figure 15.13a, a
composite curve without negative adsorption is obtained. In
all other cases of Figure 15.13, negative values of surface
excess appear.
§15.2.3 Liquid Adsorption Valenzuela and Myers [23] tabulate literature values for
When porous adsorbent particles are immersed in a confined equilibrium adsorption of 25 different binary-liquid mixtures.
pure gas, the pores fill with gas, and the amount of adsorbed With one exception, all 25 mixtures give composite iso-
gas is determined by the decrease in total pressure. With a therms of the shapes shown in Figures 15.12b and c. The
liquid, the pressure does not change, and no simple experi- exception is a mixture of cyclohexane and n-heptane with sil-
mental procedure has been devised for determining the extent ica gel, for which surface excess is almost negligible (0 
of adsorption of a pure liquid. If the liquid is a homogeneous 0.05 mmol/g) from x1 ¼ 0.041 to 0.911. They also include
binary mixture, it is customary to designate one component literature references to 354 sets of binary, 25 sets of ternary,
the solute (1) and the other the solvent (2). The assumption is and 3 sets of data for higher-order liquid mixtures.
then made that the change in composition of the bulk liquid When data for the binary mixture are available only in the
in contact with the porous solid is due entirely to adsorption dilute region, solvent adsorption, if any, may be constant, and
of the solute; solvent adsorption is thus tacitly assumed not to all changes in total amount adsorbed are due to the solute. In
occur. If the liquid mixture is dilute in solute, the conse- that case, the adsorption isotherms are of the form of Figure
quences are not serious. If, however, experimental data are 15.12a, which resembles the shape obtained with pure gases.
Adsorption
Adsorption
Adsorption

Concentration Concentration Concentration Figure 15.12 Representative isotherms of

(a) (b) (c) concentration change for liquid adsorption.
C15 09/22/2010 Page 584

584 Chapter 15 Adsorption, Ion Exchange, Chromatography, and Electrophoresis

A A
Radke and Prausnitz [37] have been fitted to the Freundlich and
Langmuir isotherms, (15-35) and (15-36), with the average devia-
tions indicated, for solute concentrations up to 50 mmol/L:
Co Co
mp mp
osi osi
te te Absolute Average
B
B Acetone in Water (25 C): Deviation of q, %
(a) (b) q1 ¼ 0:141c0:597
1 (1) 14.2
A A 0:190c1
q1 ¼ (2) 27.3
Com 1 þ 0:146c1
pos Com
ite B
Adsorption

B pos Propionitrile in water

ite
(25 C):
(c) (d)
q2 ¼ 0:138c0:658
2 (3) 10.2
B
0:173c2
q2 ¼ (4) 26.2
B 1 þ 0:0961c2
A
where qi ¼ amount of solute adsorbed, mmol/g, and ci ¼ solute con-
A centration in aqueous solution, mmol/L.
Use these single-solute results with an extended Langmuir-type
Co Com isotherm to predict the equilibrium adsorption in a binary-solute,
m pos
po ite
sit aqueous system containing 40 and 34.4 mmol/L, respectively, of
e acetone and propionitrile at 25 C with the same adsorbent. Compare
(e) (f) the results with the following experimental values from Radke and
Concentration Prausnitz [37]:
Figure 15.13 Origin of various types of composite isotherms for q1 ¼ 0:715 mmol/g; q2 ¼ 0:822 mmol/g; and qtotal ¼ 1:537 mmol/g
binary-liquid adsorption.
[From J.J. Kipling, Adsorption from Solutions of Non-electrolytes, Aca-
Solution
demic Press, London (1965) with permission.]
From (15-32), the extended liquid-phase Langmuir isotherm is
It is then common to fit the data with concentration forms of ðqi Þm K i ci
the Freundlich (15-19) or Langmuir (15-24) equations: qi ¼ P ð5Þ
1 þ K j cj
j

q ¼ kc1=n ð15-35Þ From (2), (q1)m ¼ 0.190=0.146 ¼ 1.301 mmol/g.

From (4), (q2)m ¼ 0.173=0.0961 ¼ 1.800 mmol/g.
From (5):
Kqm c
q¼ ð15-36Þ 1:301ð0:146Þð40Þ
1 þ Kc q1 ¼ ¼ 0:749 mmol/g
1 þ ð0:146Þð40Þ þ ð0:0961Þð34:4Þ
Candidate systems for this case are small amounts of organ- 1:800ð0:0961Þð34:4Þ
q2 ¼ ¼ 0:587 mmol/g
ics dissolved in water and small amounts of water dissolved 1 þ ð0:146Þð40Þ þ ð0:0961Þð34:4Þ
in hydrocarbons. For liquid mixtures dilute in two or more
solutes, multicomponent adsorption may be estimated from a Summing, qtotal ¼ 1.336 mmol/g.
concentration form of the extended-Langmuir equation (15- Compared to experimental data, the percent deviations for q1, q2, and
32) based on constants, qm and K, obtained from experiments qtotal, respectively, are 4.8%, 28.6%, and 13.1%. Better agreement
on single solutes. However, when solute–solute interactions is obtained by Radke and Prausnitz using an IAS theory. It is expected
are suspected, it may be necessary to determine constants that a concentration form of (15-33) would also give better agreement,
from multicomponent data. As with gas mixtures, the con- but that requires that the single-solute data be refitted for each solute
centration form of (15-32) also predicts constant selectivity to a Langmuir–Freundlich isotherm of the form
for each pair of components in a mixture. q0 kc1=n
q¼ ð6Þ
1 þ kc1=n

EXAMPLE 15.6 Adsorption of VOCs.

Small amounts of VOCs in water can be removed by adsorption.
§15.2.4 Ion-Exchange Equilibria
Generally, two or more VOCs are present. An aqueous stream con- Ion exchange differs from adsorption in that one sorbate
taining small amounts of acetone (1) and propionitrile (2) is to be (a counterion) is exchanged for a solute ion, the process being
treated with activated carbon. Single-solute equilibrium data from governed by a reversible, stoichiometric, chemical-reaction
C15 09/22/2010 Page 585

§15.2 Equilibrium Considerations 585

equation. Thus, selectivity of the ion exchanger for one coun- Table 15.5 Relative Molar Selectivities, K, for
terion over another may be just as important as the ion- Cations with 8% Cross-Linked Strong-Acid Resin
exchanger capacity. Accordingly, the law of mass action is
Li+ 1.0 Zn2+ 3.5
used to obtain an equilibrium ratio rather than to fit data to a
H+ 1.3 Co2+ 3.7
sorption isotherm such as the Langmuir or Freundlich
Na+ 2.0 Cu2+ 3.8
equation.
NHþ 4 2.6 Cd2+ 3.9
As discussed by Anderson [38], two cases are important. In
K+ 2.9 Be2+ 4.0
the first, the counterion initially in the ion exchanger is
Rb+ 3.2 Mn2+ 4.1
exchanged with a counterion from an acid or base solution, e.g.,
Cs+ 3.3 Ni+ 3.9
Naþ 
ðaqÞ þ OHðaqÞ þ HRðsÞ $ NaRðsÞ þ H2 Oðl Þ Ag+ 8.5 Ca2+ 5.2
Note that hydrogen ions leaving the exchanger immediately UO2þ2 2.5 Sr2+ 6.5
react with hydroxyl ions to form water, leaving no counterion Mg2+ 3.3 Pb2+ 9.9
on the right-hand side of the reaction. Accordingly, ion Ba2+ 11.5
exchange continues until the aqueous solution is depleted of
sodium ions or the exchanger is depleted of hydrogen ions.
At equilibrium, xA and yA are independent of total equiv-
In the second, more-common, case, the counterion being
alent concentrations C and Q. Such is not the case when the
transferred from exchanger to fluid remains as an ion. For
two counterions are of unequal charge, as in the exchange of
example, exchange of counterions A and B is expressed by:
Ca2+ and Na+. A derivation for this general case gives
An 
ðl Þ þ nBRðsÞ $ ARnðsÞ þ nBðl Þ ð15-37Þ  n1
C yA ð1  xA Þn
where A and B must be either cations (positive charge) or K A;B ¼ ð15-44Þ
Q xA ð1  yA Þn
anions (negative charge). For this case, at equilibrium, a
chemical-equilibrium constant based on the law of mass Thus, for unequal counterion charges, KA,B depends on the
action can be defined: ratio C=Q and on the ratio of charges, n, as defined in (15-37).
qAR cn  When KA,B data for a system of counterions with a partic-
K A;B ¼ n n B ð15-38Þ ular ion exchanger are not available, the method of Bonner
qBR cAn
and Smith [39], as modified by Anderson [38], is used for
where molar concentrations ci and qi refer to the liquid and screening purposes or preliminary calculations. In this
ion-exchanger phases, respectively. The constant, KA,B is not method, KA,B is
a rigorous equilibrium constant because (15-38) is in terms of K ij ¼ K i =K j ð15-45Þ
concentrations instead of activities. Although it could be cor-
rected by including activity coefficients, it is used in the form where values for relative molar selectivities Ki and Kj are
shown, with KA,B referred to as a molar selectivity coefficient given in Table 15.5 for cations with an 8% cross-linked,
for A displacing B. For the resin phase, concentrations are in strong-acid resin, and in Table 15.6 for anions with strong-
equivalents per unit mass or unit bed volume of ion base resins. For values of K in these tables, the units of C and
exchanger. For the liquid, concentrations are in equivalents Q are, respectively, eq/L of solution and eq/L of bulk bed vol-
per unit volume of solution. For dilute solutions, KA,B is con- ume of water-swelled resin.
stant for a given pair of counterions and a given resin. A typical cation-exchange resin of the sulfonated styrene–
When exchange is between two counterions of equal divinylbenzene type, such as Dowex 50, as described by
charge, (15-38) reduces to a simple equation in terms of equi- Bauman and Eichhorn [40] and Bauman, Skidmore, and
librium concentrations of A in the liquid solution and in the Osmun [41], has an exchangeable ion capacity of 5 
ion-exchange resin. Because of (15-37), the total concentra- 0.1 meq=g of dry resin. As shipped, water-wet resin might
tions, C and Q, in equivalents of counterions in the solution contain 41.4 wt% water. Thus, wet capacity is 5(58.6/100) ¼
and the resin, remain constant during the exchange. Accord- 2.9 meq/g of wet resin. If bulk density of a drained bed of wet
ingly: resin is 0.83 g/cm3, bed capacity is 2.4 eq/L of resin bed.
ci ¼ Cxi =zi ð15-39Þ
qi ¼ Qyi =zi ð15-40Þ Table 15.6 Approximate Relative Molar
where xi and yi are equivalent fractions, rather than mole Selectivities, K, for Anions with Strong-Base Resins
fractions, of A and B, such that I 8 OH (Type II) 0.65
xA þ xB ¼ 1 ð15-41Þ NO3 4 HCO 3 0.4
yA þ yB ¼ 1 ð15-42Þ Br 3 CH3COO 0.2
HSO 4 1.6 F 0.1
and zi ¼ valence of counterion i. Combining (15-38) with NO 1.3 OH (Type I) 0.05–0.07
2
(15-42) gives for counterions A and B of equal charge, CN 1.3 SO2 0.15
4
y ð1  xA Þ Cl 1.0 CO2 0.03
K A;B ¼ A ð15-43Þ 3
xA ð 1  y A Þ BrO3 1.0 HPO24 0.01
C15 09/22/2010 Page 586

586 Chapter 15 Adsorption, Ion Exchange, Chromatography, and Electrophoresis

1.0 similar trend was observed by Selke and Bliss [43, 44] for
1N
Equivalent ion fraction of Cu2+ in resin
0.0 exchange between Ca2+ and H+ using a similar resin, Amber-
0.1 lite IR-120. Selectivity sensitivity is shown dramatically in
Figure 15.15, from Myers and Byington [45], where the natu-
ral logarithm of the separation factor, SCu2þ ; Naþ , as computed
5
0. from data of Figure 15.14 with (15-46), is plotted as a func-
1.
0 tion of equivalent ionic fraction, xCu2þ . For dilute solutions of
0.5 Cu2+, SCu2þ ; Naþ ranges from 0.5 at a total concentration of
0 4 N to 60 at 0.01 N. In terms of K Cu2þ ; Naþ of (15-44), with n
2.
¼ 2, the corresponding variation is only from 0.6 to 2.2.
0
3.
. 0
4
EXAMPLE 15.7 Ion-Exchange Equilibrium.
An Amberlite IR-120 ion-exchange resin similar to that of Example
0
0 0.5 1.0 15.2, but with a maximum ion-exchange capacity of 4.90 meq/g of
Equivalent ion fraction of Cu2+ in solution dry resin, is used to remove cupric ion from a waste stream contain-
Figure 15.14 Isotherms for ion exchange of Cu2+ and Na+ on ing 0.00975-M CuSO4 (19.5 meq Cu2+/L solution). The spherical
Dowex 50-X8 as a function of total normality in the bulk solution. resin particles range in diameter from 0.2 to just over 1.2 mm. The
equilibrium ion-exchange reaction is of the divalent–monovalent
[From A.L. Myers and S. Byington, Ion Exchange Science and Technology,
M. Nijhoff, Boston (1986) with permission.]
type:
þ
ðaqÞ þ 2HRðsÞ $ CuR2 ; þ2HðaqÞ
Cu2þ

As with other separation processes, a separation factor, As ion exchange takes place, the meq of cations in the aqueous
SP ¼ SA,B (§1.8), which ignores the valence of the exchang- solution and in the resin remain constant.
Experimental measurements by Selke and Bliss [43, 44] show an
ing ions, can be defined for an equilibrium stage. For binaries
equilibrium curve like Figure 15.14 at ambient temperature that is
in terms of equivalent ionic fractions: markedly dependent on total equivalent concentration of the aque-
y ð1  xA Þ ous solution, with the following equilibrium data for cupric ions
SA;B ¼ A ð15-46Þ
xA ð 1  y A Þ with a 19.5 meq/L solution:
which is identical to (15-43). Data for an exchange between
c, meq Cu2+/L Solution 0.022 0.786 4.49 10.3
Cu2+ (A) and Na+ (B) (counterions of unequal charge) with
q, meq Cu2+/g Resin 0.66 3.26 4.55 4.65
Dowex 50 cation resin over a wide range of total-solution
normality at ambient temperature are shown in terms of yA
These data follow a highly nonlinear isotherm.
and xA in Figure 15.14, from Subba Rao and David [42]. At
low total-solution concentration, the resin is highly selective (a) From the data, compute the molar selectivity coefficient, K, at
for copper ion, whereas at high total-solution concentration, each value of c for Cu2+ and compare it to the value estimated from
the selectivity is reversed to slightly favor sodium ions. A (15-45) using Table 15.5. (b) Predict the milliequivalents of Cu2+
exchanged at equilibrium from 10 L of 20 meq Cu2+/L, using 50 g
of dry resin with 4.9 meq of H+/g.
5
Natural log of relative separation factor

Solution
4
0.01 N (a) Selke and Bliss do not give a value for the resin capacity, Q, in
eq/L of bed volume. Assume a value of 2.3. From (15-44):
3  
C yCu2þ ð1  xCu2þ Þ2
0.1 K Cu2þ ; Hþ ¼
Q xCu2þ ð1  yCu2þ Þ2
2

where (C=Q) ¼ 0.0195=2.3 ¼ 0.0085.

1 1.0 xCu2þ ¼ cCu2þ =19:5 and yCu2þ ¼ qCu2þ =4:9
2.0
0 Using the above values of c and q from Selke and Bliss:
4.0

–1
0 0.2 0.4 0.6 0.8 1.0 q, meq Cu2+/g xCu2þ yCu2þ K Cu2þ ; Hþ
Equivalent ion fraction of Cu2+
in solution 0.66 0.00113 0.135 1.35
Figure 15.15 Relative separation factor of Cu2+ and Na+ for ion 3.26 0.0403 0.665 1.15
exchange on Dowex 50-X8 as a function of total normality in the 4.55 0.230 0.929 4.04
bulk solution. 4.65 0.528 0.949 1.30
[From A.L. Myers and S. Byington, Ion Exchange Science and Technology,
M. Nijhoff, Boston (1986) with permission.]
C15 09/22/2010 Page 587

§15.3 Kinetic and Transport Considerations 587

Table 15.7 Recombinant Pharmaceuticals

The average value of K is 2.0. Values in Table 15.5 when substituted Purified by Adsorption
into (15-45) predict K Cu2þ ; Hþ ¼ 3:8=1:3 ¼ 2:9, which is somewhat
higher. Annual Sales,
Product Billions of $
(b) Assume a value of 2.0 for K Cu2þ ; Hþ with Q ¼ 2.3 eq/L. The
total-solution concentration, C, is 0.02 eq/L. Equation (15-44) Insulin 1.0
becomes Growth hormone 0.9
 
0:02 yCu2þ ð1  xCu2þ Þ2 Interferons 0.5
2:0 ¼ ð1Þ Erythropoietin 0.4
2:3 xCu2þ ð1  yCu2þ Þ2
Tissue plasminogen activator 0.2
Initially, the solution contains (0.02)(10) ¼ 0.2 equivalent of cupric
ion with xCu2þ ¼ 1:0. Let a ¼ equivalents of Cu exchanged. Then, at
equilibrium, by material balance:
pressure-swing adsorption, ion exchange, chromatography,
0:02  ða=10Þ and simulated-moving-bed systems.
xCu2þ ¼ ð2Þ
0:02 Examples of industrial-scale, packed-bed adsorption for
ða=50Þ purification and bulk separation of mixtures of small mole-
yCu2þ ¼ ð3Þ cules are included in Table 15.1. Large molecules produced
0:0049
in biochemical processes are also separated by adsorption.
Substitution of (2) and (3) into (1) gives Important examples are found in biopharmaceutical separa-
h ih i2
ða=50Þ 0:02ða=10Þ tions that include resolution of synthetic chiral mixtures to
0:0049 1  0:02
2:0 ¼ 0:0087 h ih i2 ð4Þ produce pure enantiomeric drugs such as the antidepressant
0:02ða=10Þ
0:02 1  ð0:0049
a=50Þ
fluoxetine (trade names: Prozac1, Sarafem1); the separation
of sweeteners fructose and glucose by high-throughput simu-
Solving (4), a nonlinear equation, for a gives 0.1887 eq of Cu lated moving beds; and anion-exchange adsorption of infec-
exchanged. Thus, 0.1887/[(0.020)(10)] ¼ 0.944 or 94.4% of the tious, nonreplicative adenovirus type 5 for purification as a
cupric ion is exchanged.
viral vector for gene therapy. Some biopharmaceuticals pro-
duced by recombinant methods that rely on adsorptive sepa-
rations are listed in Table 15.7.
Equilibria in Chromatography To accomplish separation by adsorption in a packed bed,
solutes dispersed in a mobile fluid solvent are transported by
As discussed in §15.1.3, separation by chromatography
bulk flow (i.e., convection) through interstices between
involves many sorption mechanisms, including adsorption on
packed particles. Solutes diffuse between the moving fluid
porous solids, absorption or extraction (partitioning) in liq-
phase and a stationary fluid phase within pores of the solid
uid-supported or bonded solids, and ion transfer in ion-
adsorbent. Instantaneous equilibrium partitioning of solute
exchange in resins. Thus, at equilibrium, depending upon the
between fluid and adsorptive surfaces in the stationary phase,
sorption mechanism, equations such as (15-19), (15-24), (15-
due to a solute-specific thermodynamic driving force, is
32), and (15-33) for gas adsorption; (15-35) and (15-36) for
resisted by consecutive mass-transfer processes illustrated in
liquid adsorption; (6-17) to (6-20) for gas absorption; (8-1)
Figure 15.16, which is adapted from Athalye et al. [46]:
for liquid extraction; and (15-38), (15-43), and (15-44) for
ion exchange apply.
At equilibrium, the distribution (partition) constant for
solute, i, is
K i ¼ qi =ci ð15-47Þ
where q is concentration in the stationary phase and c is con-
centration in the mobile phase. Solutes with the highest
equilibrium constants will elute from the chromatographic
column at a slower rate than solutes with smaller constants.

§15.3 KINETIC AND TRANSPORT

CONSIDERATIONS
Multicomponent mixtures may often be separated efficiently
in a single unit by selective partitioning of targeted specie(s)
between a flowing fluid phase and stationary sorptive phase.
The latter is typically porous or resin particles packed into a
cylindrical vessel. Equipment configurations and operating
procedures to accomplish adsorptive separations take various
forms, including slurry adsorption, temperature- and Figure 15.16 Transport-rate processes in adsorption.
C15 09/22/2010 Page 588

588 Chapter 15 Adsorption, Ion Exchange, Chromatography, and Electrophoresis

1. Solute transport by bulk flow (convection) and disper- Adsorbent Adsorbent

particle particle
sion through interstices of the bed of adsorptive
particles Bulk Bulk
fluid fluid
2. External (interphase) solute transport from the bulk
flow to the outer perimeter of the adsorbent particle Tb
through a thin film or boundary layer Ts
3. Internal (intraphase) solute transport by diffusion in Tb Ts
quiescent fluid-filled pores in the adsorbent particle
Cb
4. Surface diffusion along the internal porous surface of
the adsorbent particle. Cs Cs Cb

At the adsorptive surface, kinetic interaction occurs

between solute and sorptive sites within the adsorbent
particle. (a) (b)
External transport includes convective dispersion of the
Figure 15.17 Solute concentration and temperature profiles for a
solute within the bulk fluid, and diffusion through a boundary porous adsorbent particle surrounded by a fluid: (a) adsorption,
layer surrounding adsorbent particles. Axial dispersion of (b) desorption.
individual solute molecules in the bulk fluid occurs primarily
by microscale, fluid-phase phenomena such as mixing via
solid obstructions to flow, eddies, and recirculation from Costs associated with adsorbent usage and handling and
regional pressure gradients. Boundary-layer transport, internal consumption of solvent and regenerant [47, 48] often domi-
transport, and surface diffusion occur via random Brownian nate manufacturing expenses [49]. This is especially true in
motion. Kinetic interaction, which depends on solute orienta- purification of high-value-added pharmaceuticals [50] or bio-
tion and frequency of surface collisions, is virtually instanta- technology products such as antibiotics or recombinant DNA
neous for physical adsorption of gases and small solutes, but [51]. Thus, understanding how transport resistances affect
can become controlling in bioprocesses for orientation- separation is important to improve packed-bed adsorption
specific sorption such as antibody interaction with fixed efficiency and enhance process economics. Applicable
antigen, or chemisorption (bond formation). descriptions of mass transport in chromatographic separa-
During regeneration of the adsorbent, the reverse of the tions are also valuable to predict scale-up and to control oper-
four steps occurs, following desorption. Adsorption and ating performance [52]. Therefore, a general model that
desorption may be accompanied by heat transfer arising from describes all steps in the adsorption process is presented first
exothermic heat of adsorption and endothermic heat of to illustrate the important features of the process. This is fol-
desorption. Although external mass transfer is limited to con- lowed by correlations and methods for predicting the individ-
vection, external heat transfer from the particle outer surface ual resistances in the general model and by specialized cases
occurs not only by convection through a thermal boundary of the general model.
layer surrounding each particle, but also by thermal radiation
between particles when the fluid is a gas, and by conduction
at points of contact by adjacent particles. Conduction and §15.3.1 Convection-Dispersion Model
radiation mechanisms for heat transfer also exist within Solute convected by dispersed, plug flow of fluid at average
particles. interstitial (actual) velocity u in the axial direction z, within a
In a fixed bed of adsorbent particles, solute concentration uniform bed of particles, has a position- and time-dependent,
and temperature change continuously with time and location bulk, fluid-phase concentration cf{z, t} given by Ruthven
during adsorption and desorption processes, when significant [10] and others as
thermal effects due to sorption occur. For a given particle at a
@cf @cf @ 2 cf 1  eb @
cb
particular time, profiles illustrating temperature and solute þu E 2 ¼ ð15-48Þ
concentration in the fluid are as shown in Figures 15.17a and @t @z @z eb @t
b for adsorption and desorption, respectively, where sub- where eb is the fractional, interstitial void volume (bed poros-
scripts b and s refer to bulk fluid and particle outer surface, ity); E is a Fickian, convective, axial-dispersion coefficient
respectively. The fluid concentration gradient is usually that accounts for both axial molecular diffusion and the inter-
steepest within the particle, whereas the temperature gradient action of lateral molecular diffusion with local nonuniform-
is usually steepest in the boundary layer surrounding the par- ities in fluid velocity caused by complicated flow paths
ticle. Thus, although resistance to heat transfer is mainly around the stationary particles; and
cb is the volume-averaged
external to the adsorbent particles, the major resistance to stationary-phase concentration per unit mass, which for a
mass transfer resides in the adsorbent particle. However, spherical particle of radius Rp is given by
dilute aqueous solute solutions typical of bioseparations Z
rarely result in temperature gradients external to, or within, 3 Rp 2

cb ¼ 3 r cb dr ð15-49Þ
the particles. Rp 0
C15 09/22/2010 Page 589

§15.3 Kinetic and Transport Considerations 589

The superficial velocity of the fluid through the bed is by an appropriate mechanism to obtain a relatively
us ¼ eb u. At a particular time t and axial location in the bed pure product. Desorption mechanisms include: increas-
z, the first term on the LHS of (15-48) accounts for the ing bed temperature or reducing bed pressure in the
change in solute concentration in the bulk fluid with time. case of a gas, or introducing an isocratic or gradient
The second term on the LHS accounts for the change in cf change in solvent composition (e.g., I, pH, hydropho-
with axial location arising from bulk convection. The third bicity) to reduce sorbate equilibrium partitioning to the
term accounts for axial dispersion, and the term on the RHS solid phase. The adsorption-desorption cycle may be
accounts for the change in solute loading on the stationary repeated, sometimes after an intermediate, validated
phase. cleaning step, to increase stationary-phase utilization.
The equilibrium loading, q, measures the mass of solute During the adsorption part of the cycle, solute concen-
adsorbed on surfaces of the stationary phase per unit mass of tration and loading fronts move with time through the
adsorbent, whereas
cb includes both adsorbed solute and bed. This method can be used to partially separate one
unadsorbed solute diffusing in the pore volume of the station- of a mixture of solutes, such as oxygen or nitrogen
ary phase. For example, at equilibrium in nonadsorbing gel- from air, as illustrated in §15.3.5.
filtration chromatography, q ¼ 0, while
cb ¼ ep cf . Occasion- 2. Displacement. This mode is widely used (e.g., dis-
ally q and
cb are interchanged in the literature. The change in placement chromatography) to separate and concen-

cb with time for a rapidly equilibrating particle may be trate target protein(s) from a mixture. Relative to
related to flux at its external surface using a linear driving frontal mode, mobile-phase feeds in displacement
force (LDF) approximation introduced by Glueckauf [53], mode nearly always contain two or more solutes to be
4 3 @
cb   separated. Following near-saturation of the bed with
pRp ¼ kc;tot 4pR2p
cb a  cf ð15-50Þ
3 @t solute(s), desorption (elution) in consecutive zones of
that uses an overall mass-transfer coefficient, kc,tot, defined pure substances is effected by substitution with a more
below in (15-58), which consists of a series of transport-rate strongly adsorbed solute (displacer) that is fed into the
resistance terms for concentration-independent equilibrium bed. The cycle may be repeated after removing the dis-
and transport properties illustrated in Figure 15.16. At equili- placer from the bed.
brium, solute is partitioned between the bulk fluid and aver- 3. Differential. This mode is used to recover a variety of
age stationary-phase loading according to bioproducts. A small pulse of solute dissolved in the
cf 1 mobile phase is loaded onto the bed. Rather than satu-
a¼ ¼  ð15-51Þ

cb ep ð1 þ K d Þ rating the bed as in frontal or displacement modes, the
with K d ¼ ka =kd ¼ cs =cp being the constant equilibrium dis- solute pulse is carried through the bed (eluted) at a rate
tribution coefficient given by the kinetic rate constant for lower than pure solvent. This reduced rate is due to sol-
adsorption relative to desorption, with subscripts s and p ute-specific interactions with the stationary phase that
referring to surface and pore volume of the porous stationary are modulated by solvent composition (e.g., I, pH,
phase, respectively. An inclusion porosity, ep , defines the hydrophobicity), which may be unchanged (isocratic
effective stationary-phase volume fraction accessible to a elution) or changed in a linear or nonlinear fashion
specific solute. Particle porosity ep is the internal void frac- (gradient elution). Solute product(s) emerge(s) from
tion of a particle, whereas ep includes only voids penetrable the column diluted by the solvent in the form of expo-
by a particular solute due to size or steric hindrance. For nentially modified Gaussian concentration peaks.
example, Source 15QTM anion-exchange resin has ep  0:4, Mass transfer in each of the three modes determines effi-
whereas for binding of adenovirus type 5 (MW 165 MDa), ciency of separation, utilitization of adsorbent and solvent,
its ep ¼ 0:0. and recovery of solute.

Modes of Time-Dependent Sorption

Solute Concentration Distribution
The mobile phase consists of fluid solvent (also called carrier
or eluent) and solutes, which are components that may be To identify transport-rate effects on a separation, consider an
adsorbed and desorbed, dissolved, or suspended in eluent. A asymptotic solution to (15-48)–(15-51) obtained by Reis
solution of eluent and desorbed solute(s) is often referred to et al. [55] for differential, isocratic elution of a sharp-pulse
as eluate. Time-dependent (de)sorption described in (15-48) input of
to (15-51) may be carried out in any of the following three m o dfzg
cfz; 0g ¼ ð15-52Þ
major adsorption operating modes [54] to separate a mixture Aeb
of solutes:
in a long, packed bed, where mo is solute mass, A is bed
1. Frontal. This mode is widely used for purification of cross-sectional area, and d{z} is the Dirac delta function,
small molecules and biomolecules in fixed-bed adsorb- which is zero everywhere except at z ¼ 0, where its magni-
ers. The mobile phase (gas or liquid) is fed continu- tude is infinity. The resulting fluid-phase solute concentration
ously to the stationary phase until the bed approaches illustrated in Figure 15.18 is distributed normally around
saturation with the solute. The solute is then desorbed the mean solute position, zo = vut, along the axis, z, when
C15 09/22/2010 Page 590

590 Chapter 15 Adsorption, Ion Exchange, Chromatography, and Electrophoresis

Figure 15.18 Elution of 0.1 mL injection of 10 gm/mL hemoglobin

eluted at 0.6 mL/min from an 11.5  2.5 cm i.d. glass column
packed with Toyopearl HW65C (- - -). The corresponding Gaussian
expression (-) yields ep ¼ 0.498 and H ¼ 0.312 mm [46].

Figure 15.19 Gaussian peak with the properties: maximum peak

time, t, is treated as a parameter, height cmax; standard deviation s; width at inflection points, wi;
" #
width at half height, wh; width at base intercept, w; and average
mo v  ð z  zo Þ 2
cf fz; tg ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp ð15-53Þ retention time, tR.
Aeb 2pHzo 2Hzo [From Horvath and Melander [57] with permission.]
where v is the fraction of solute in the moving fluid phase at
equilibrium, given by height, H, and the overall mass-transfer coefficient in linear
1 differential chromatography (LDC), LDC kc,tot:
v¼ ð15-54Þ
1  eb 1  
1þ E vð1  vÞRb u
eb a H¼2 þ ð15-57Þ
u 3akc;tot
and H is the height of a theoretical chromatographic plate,
and N is the number of theoretical chromatographic plates as  2
1 1 Rp 3 Kd
defined by Glueckauf [56]. Equation (15-53) is a Gaussian ¼ þ  þ ð15-58Þ
expression with solute mean residence time kc;tot kc 5ep De Rp ka ep 1 þ K d
NH

t ¼ ð15-55Þ
vu
and variance s, where

tH
s2 ¼ ð15-56Þ
vu
which reflects random solute partitioning in an adsorptive
bed. These features are illustrated in Figure 15.19. This solu-
tion is comparable to those of Giddings [58], Aris and
Amundson [59], Ruthven [10], and Guiochon [60], and may
be extended to describe gradient elution [61] and membrane
chromatography [62] or to evaluate countercurrent adsorp-
tion [63].

Separation Efficiency (Plate Height or Bandwidth)

Figure 15.20 van Deemter plot comparing reduced plate height,
Each transport-rate resistance to equilibrium partitioning
h ¼ H=2Rp, of 0.1-mL bovine hemoglobin at 10 (), 50 (D), and 100
broadens the adsorptive band in (15-53), as shown in Figures
(&) mg/mL eluted from 11.5  2.5 cm i.d. glass column packed
15.18 to 15.20, and lowers separation efficiency, which is with Toyopearl HW65C. Contributions to h (dotted lines) are due to
proportional to H1. This is measured by noting that peak (A) boundary-layer transport; (B) axial dispersion by [174, 175];
width measured at 60.6% of maximum peak height corre- (C) axial dispersion by [70]; (D) intraparticle diffusion. Total h
sponds to the square root of the variance of the solute distri- values (lines) arise from A þ B þ D (lower) and A þ C þ D
bution, s, in (15-56), as shown by van Deemter et al. [64] and (upper). NReNSc ¼ 100 at u ¼ 1.5 cm/minute.
Karol [65], and by examining the relation between plate [From Athalye et al. [46] with permission.]
C15 09/22/2010 Page 591

§15.3 Kinetic and Transport Considerations 591

§15.3.2 Correlations for Transport-Rate

Coefficients
Equation (15-59) shows that decreasing s [i.e., decreasing H
in (15-56)] improves resolution, R, in a separation. Equation
(15-57) shows that H is decreased by reducing axial convec-
tive dispersion (i.e., coefficient E). Equation (15-58) shows
that H is also decreased by increasing fluid-phase mass trans-
fer, represented by kc; effective pore diffusivity, De; and/or
kinetic-adsorption rate, ka. Identifying correlations for these
four transport-rate coefficients allows the user to select geo-
metric and operating parameters that maximize separation
efficiency, R.

Convective Dispersion
For low values of the diffusion Peclet number for axial con-
vection dispersion, N Pe ¼ N Re N Sc ¼ 2Rp ueb =Di 1, the
Figure 15.21 Fluid concentrations c1 and c2 plotted for convective dispersion coefficient is given by E ¼ Di =tf ,
chromatographic separation of 1 and 2 where t ¼ 0.95t1, d12 ¼ 0.05,
where Di is the molecular diffusivity of the solute and tf is
and R12 ¼ 1 at z=L ¼ 1.
the interstitial tortuosity factor, 1.4 for a packed bed of
spheres [66, 67]. For very high NReNSc values, the dispersion
Peclet number, N PeE ¼ 2Rp ueb =E, asymptotically approaches
where kc is the fluid-phase mass-transfer coefficient, and the limit of pure hydrodynamic dispersion [67–70]:
De represents effective solute diffusivity in the pore liq-
2p
uid. The first term on the RHS of (15-57) represents N PeE ¼ ð15-61Þ
transport-rate resistance due to convective dispersion. The 1p
three terms on the RHS of (15-58) represent series resist- where p ¼ 0.17 þ 0.33exp(24=NRe). For intermediate val-
ances associated with the particle, arising from external ues of NReNSc, convective axial-dispersion coefficients can
boundary-layer transport, internal intraparticle diffusion, be predicted reasonably well over a wide range of conditions
and kinetic sorption rate, respectively. Contributions of for both gases and liquids by [174, 175]:
these individual transport-rate resistances to H are illus-
trated in van Deemter plots [64], like that in Figure 15.20 1 1  p     eb
¼ Y þ Y 2 exp Y 1  1 þ
for a size-exclusion resin. N PeE p tN Re N Sc
ð15-62Þ

Resolving Solute Mixtures where Y ¼ pð1  pÞN Re N Sc /½23:16ð1  eb Þ. These correla-
tions may be used to select values of u and/or Rp for a partic-
Resolution, R, of similar components 1 and 2, illustrated in ular set of Di and eb to minimize E and convective-dispersion
Figure 15.21, occurs due to solute-specific partitioning band broadening.
between moving fluid and stationary bulk phases and is
defined by the ratio of peak separation relative to the average
peak width: Boundary-Layer (External) Transport
pffiffiffiffi
j
t1 
t2 j d N Wakao and Funazkri [71] proposed a correlation for mass
R ¼ ð15-59Þ transport of species i from the bulk fluid flowing through the
2ðs1 þ s2 Þ 4
bed to the outer surface of the bed particles of diameter Dp ¼
where d is the fractional difference in migration velocities 2Rp through the boundary layer or film around the particles.
in the moving fluid phase of species 1 and 2 or, in general, It is given by
i and j:    
kc;i Dp rueb Dp 0:6 m 1=3
jvi  vj j N Shi ¼ ¼ 2 þ 1:1 ð15-63Þ
di;j ¼ 2 ð15-60Þ Di m rDi
vi þ vj
and was the result of reanalyzing 37 sets of previously pub-
The differential fluid-phase/stationary-phase partitioning fac- lished mass-transfer data for particles packed in a bed, with
tor, di,j, equals twice the product of relative selectivity and Sherwood-number corrections for axial dispersion. This cor-
retention factors [58], and contains only measurable geomet- relation is compared to 12 sets of gas-phase and 11 sets of
ric and thermodynamic parameters peculiar to any species liquid-phase data in Figure 15.22. The data cover a Schmidt
solid-phase pair i and j. number range of 0.6 to 70,600, a Reynolds number range of 3
C15 09/22/2010 Page 592

592 Chapter 15 Adsorption, Ion Exchange, Chromatography, and Electrophoresis

104

103
Sherwood number, NSh

102

10

Figure 15.22 Correlation of experimental data

1 for Sherwood number in a packed bed.
1 10 102 103 104 105 106 [From N. Wakao and T. Funazkri, Chem. Eng.
(NRe 0.6
NSc )
1/3 2
Sci., 33, 1375 (1978) with permission [71].]

to 10,000, and a particle diameter from 0.6 to 17.1 mm. Parti- may be compared by considering a short cylinder with diam-
cle shapes include spheres, short cylinders, flakes, and eter, D, equal to the length, L.
granules.
1. Dp ¼ diameter of a sphere with the same external sur-
The value of 2 for the first term on the RHS of (15-63)
face area:
corresponds to N Sh ¼ kc Dp =Di ¼ 2, the Sherwood (Nusselt) pD2p ¼ pDL þ pD2 =2
number value for steady-state mass (heat) transport to a  0:5
spherical particle surrounded by an infinite, quiescent fluid, and Dp ¼ DL þ D2 =2 ¼ 1:225D
which is obtained by solving (3-74) using boundary condi- 2. Dp ¼ diameter of a sphere with the same volume:
tions for constant concentrations (temperatures) at the parti-  1=3
cle surface and far away, respectively. (See Exercise 3.31.) A pD3p =6 ¼ pD2 L=4 and Dp ¼ 3D2 L=2 ¼ 1:145D
fluid flowing with
 momentum-to-mass
 (heat) diffusivity ratio 3. Dp ¼ 4 times the hydraulic radius, rH, where for a
N Sc ¼ m=rDi N Pr ¼ Cp m=k past the particle at a dimen- packed bed, 4rH ¼ 6=ay , where ay ¼ external particle
sionless rate N Re ¼ Dp G=m, where G is the mass velocity surface area/volume of the particle.
equal to rueb in (15-63), adds the second term on the RHS of
(15-63), which raises the Sherwood (Nusselt) number to val- Thus,
ues as high as 160 (30), as shown by Ranz and Marshall pDL þ pD2 =2 6 6D
ay ¼ ¼ and Dp ¼ 4rH ¼ ¼ 1:0D
[72, 73], who proposed an earlier correlation. Other correla- pD2 L=4 D 6
tions for packed beds have the form of Chilton and Colburn The hydraulic radius concept is equivalent to replacing Dp in
[74] j-factors, as first proposed by Gamson et al. [75]: 0
the Reynolds number by cD p, where C is sphericity, given
j D ¼ ðN StM ÞðN Sc Þ2=3 ¼ f fN Re g ð15-64Þ by Suggestion 2. The sphericity is defined by:
Surface area of a sphere of same volume as particle
j H ¼ ðN St ÞðN Pr Þ2=3 ¼ f fN Re g ð15-65Þ c¼
Surface area of particle
with N Re ¼ Dp G=m, For a cylinder of D ¼ L,

and N StM ¼ kc r=G and N St ¼ h=CP G pD2p pð1:145DÞ2

c¼  2 ¼ ¼ 0:874
pD 3 2
Some Chilton–Colburn-type correlations that use N Re ¼ pDL þ 2 pD
4 2
Dp G=eb m to account for bed void fraction, eb , are reported
0
by Sen Gupta and Thodos [76], Petrovic and Thodos [77], and cDp ¼ ð0:874Þð1:145DÞ ¼ D, the diameter of the
and Dwivedi and Upadhay [78]. cylinder.
When (15-63) for mass transfer and its analog for heat Suggestions 2 and 3 are widely used. Suggestion 3 is con-
transfer, given in the following example, are used with beds veniently applied to crushed particles of irregular surface, but
packed with nonspherical particles, Dp is the equivalent with no obvious longer or shorter dimension, and isotropic in
0
diameter of a spherical particle. The following suggestions shape. In that case, D p is taken as the size of the particle and
have been proposed for computing the equivalent diameter the sphericity is approximately 0.65, as discussed by Kunii
from geometric properties of the particle. These suggestions and Levenspiel [79].
C15 09/22/2010 Page 593

§15.3 Kinetic and Transport Considerations 593

EXAMPLE 15.8 Transport Coefficients. adsorption:

Acetone vapor in a nitrogen stream is removed by adsorption in a
" #
ep  1 
1 1
rp K d
fixed bed of activated carbon. At a location in the bed where the Dei ¼ Di þ DK þ Di;surf  ð15-67Þ
pressure is 136 kPa, the bulk gas temperature is 297 K, and the bulk t ep
mole fraction of acetone is 0.05, estimate the external gas-to-parti-
cle mass-transfer coefficient for acetone and the external particle-to- Because intraparticle tortuosity, t, is not necessarily the
gas heat-transfer coefficient. Additional data are as follows: Average same for pore-volume diffusion as for surface diffusion,
particle diameter ¼ 0.0040 m, and gas superficial molar velocity ¼ (15-67) must be used with caution, as discussed by
0.00352 kmol/m2-s. Riekert [81]. Sladek, Gilliland, and Baddour [82] reported
surface diffusivity values of physical adsorption of light
Solution gases in the range 5  103 to 106 cm2/s, with larger
Because the temperature and composition are known only for the values associated with low differential heat of adsorption.
bulk gas and not at the particle external surface, use gas properties They proposed correlating surface diffusivity for nonpolar
at bulk gas conditions. Relevant fluid properties for use in (15-63) adsorbates in cm2/s using
and its heat-transfer analog,
    Di;surf ¼ 1:6  102 exp½0:45ðDH ads Þ=mRT  ð15-68Þ
hDp Dp G 0:6 Cp m 1=3
N Nu ¼ ¼ 2 þ 1:1 ð15-66Þ where m ¼ 1 for insulating adsorbents and m ¼ 2 for
k m k
conducting adsorbents (e.g., carbon). A detailed review of
are as follows: m ¼ 0.0000165 Pa-s (kg/m-s); r ¼ 1.627 kg/m3; k ¼ surface diffusion is given by Kapoor, Yang, and Wong [83].
0.0240 W/m-K ¼ 0.024  103 kJ/m-K-s; heat capacity at constant
pressure ¼ 31.45 kJ/kmol-K; molecular weight ¼ M ¼ 29.52; Thus,
specific heat CP ¼ 31.45/29.52 ¼ 1.065 kJ/kg-K. Also, gas mass EXAMPLE 15.9 Effective Diffusivity in Porous
velocity G ¼ 0.00352(29.52) ¼ 0.1039 kg/m2-s. Silica Gel.
Assume c ¼ 0:65; therefore, Dp ¼ 0.65(0.004) ¼ 0.0026 m. The
diffusivity, Di, of acetone in nitrogen at 297 K and 136 kPa is inde- Porous silica gel of 1.0 mm particle diameter, with a particle den-
pendent of composition and is 0.085  104 m2/s. sity of 1.13 g/cm3, an inclusion porosity of 0.486, an average

pore radius of 11 A, and a tortuosity of 3.35 is to be used to ad-
N Re ¼ Dp G=m ¼ 0:0026ð0:1039Þ=ð0:0000165Þ ¼ 16:4 sorb propane from helium. At 373 K, diffusion in the pores is
N Sc ¼ m=rDi ¼ 0:0000165=ð1:627Þð0:0000085Þ ¼ 1:19 controlled by Knudsen and surface diffusion. Estimate the effec-
tive diffusivity. The differential heat of adsorption is 5,900 cal/
N Pr ¼ CP m=k ¼ ð1:065Þð0:0000165Þ=ð0:000024Þ ¼ 0:73 mol. At 100 C, the adsorption-equilibrium constant, Kd (for a
linear isotherm), is 19 cm3/g.
From (15-63):
N Sh ¼ 2 þ 1:1ð16:4Þ0:6 ð1:19Þ1=3 ¼ 8:24
Solution
which from Figure 15.22 is well within the data range of the correla- 8 ¼ 22 1010 m ¼ 22  108 cm.
Pore diameter, d p ; ¼ 22 A
tion. Thus, the mass-transfer coefficient for acetone is Molecular weight of propane, Mi, ¼ 44.06.
 
kci ¼ N Sh Di =Dp ¼ 8:24ð0:0000085=0:0026Þ From
 (14-21),
 propane Knudsen diffusivity is DK ¼ 4; 850
22  108 ð373=44:06Þ1=2 ¼ 3:7  103 cm2 /s. From (15-68), using
¼ 0:027 m/s ¼ 0:088 ft/s
m ¼ 1, Ds ¼ 1:6  102 expfð0:45Þð5; 900Þ=½ð1Þð1:987Þð373Þg
From (15-66): ¼ 4:45  104 cm2 =s. Equation (15-67) reduces to De ¼ ðep =tÞDK þ
 
N Nu ¼ 2 þ 1:1ð16:4Þ0:6 ð0:73Þ1=3 ¼ 7:31; ðrp Kd =ep ÞDi;surf ¼ ð0:486=3:35Þ 3:17  103 þ ð1:13Þð19Þ ð4:45
  104 =3:35 ¼ 0:46  103 þ 2:85  103 ¼ 3:31 103 cm2 /s.
h ¼ N Nu k=Dp ¼ 7:31ð0:0240=0:0026Þ
¼ 67:5 W=m2 -K or 11:9 Btu/h-ft2 - F Experiments by Schneider and Smith [80] give a value of 1.22 
103 cm2/s for De, with a value of 0.88  103 for the contribution
of surface diffusion. Thus, the estimated contribution from surface
diffusion is high, by a factor of about 3. In either case, the fractional
Internal Transport and Effective Pore Diffusivity contribution due to surface diffusion is large.

The largest transport-rate resistance to equilibrium solute

partitioning arises from diffusion within tortuous, fluid-con- Measurement and analysis of De in adsorbent media is
taining pores inside adsorbent particles or surface diffusion often undertaken to examine ways to reduce transport-rate
along surfaces lining the pores. Effective values of liquid dif- resistance due to pore diffusivity. For example, Helfferich
fusivity (14-14) or gas diffusivity (14-18) introduced for pore [84] reports that pore diffusion is controlling in ion-exchange
diffusion, Dp, in membranes may be used directly in (15-58). beds at ion concentrations above 1.0 N, whereas resistance
However, they neglect the possibility of an additional internal external to ion-exchange resins dominates below 0.01 N.
transport mechanism, surface diffusion. Schneider and Smith Figure 15.23 shows concentration-independent self-diffusion
[80] suggested a modification to account for surface diffu- coefficients of Na+, Zn2+, and Y3+ in a sulfonated styrene–
sion, Di,surf, arising from the radial concentration gradient divinylbenzene cation-exchange resin measured using iso-
inside an adsorbent particle characterized by linear tope ions at 0.2 and 25 C. Values of self-diffusivity increase
C15 09/22/2010 Page 594

594 Chapter 15 Adsorption, Ion Exchange, Chromatography, and Electrophoresis

Solution
10–6
Self-diffusion coefficient, cm2/s The resolving power, R, for a unit value of d may be estimated from
+
Na 25°C N ¼ LH using (15-57) to obtain the height equivalent to a theoretical
10–7 plate, H. Calculation of H requires estimating coefficients associated
Na+ 0.2°C with each transport-rate process, i.e., convective dispersion, E;
boundary-layer transport, kc; intraparticle effective diffusivity, De;
10–8
Zn2+ 25°C
and kinetic sorption, ka, Kd. The necessary parameters, their sour-
ces, and corresponding values are tabulated below. Superscripts in
Zn2+ 0.2°C the table correspond to appended notes that provide detailed
10–9
descriptions and illustrative calculations for each step of the solution.
Y3+ 25°C

10–10 Y3+ 0.2°C Parameter

Parameter Source Value

Inclusion porosity of Data (Fig. 0.498

5 10 15 20 25
hemoglobin, ep 15.18)
Nominal wt% divinylbenzene
Partition coefficient, a (15-51)1 2.01
Figure 15.23 Self-diffusion coefficients for cations in a resin as a Void volume, eb Data2 0.38
function of cross-linking with divinylbenzene. Fractional migration, velocity, v (15-54)3 0.552
[From B.A. Soldano, Ann. NY Acad. Sci., 57, 116 (1953) with permission Convective-dispersion (15-61)4 f{NReNSc}5
[85].]
coefficient, E
Boundary-layer coefficient, kc (15-67) f{NReNSc}6
for smaller, monovalent cations at higher temperature in res- Height equivalent to a (15-57) f{NReNSc}7
ins with lower-percentage cross-linking. theoretical plate, H
Resolution, R (15-59) 8

Kinetic Adsorption Rate

Notes:
As shown by (15-58), chemisorption (bond formation) or ori-
1
Hemoglobin that does not adsorb in SEC has Kd  0. So the parti-
entation-controlled affinity interactions may be so slow as to tion coefficient may be calculated using (15-51) as
contribute significantly to band broadening. Biological affin- 1
a¼ ¼ 2:01
ity processes, e.g., antibody–antigen complexation, often 0:498
become the rate-limiting mass-transport process [86]. On the 2
A well-packed column has a void volume eb  0:38.
other hand, transport-rate resistance due to kinetic solute– 3
The fractional migration velocity, v, may be calculated using
adsorbent interaction is typically neglected for physical (15-54) as  1
adsorption of gases and small solutes. Adsorption onto ion- 1  0:38
v¼ 1þ 0:498 ¼ 0:552
exchange and hydrophobic-interaction media, for example, is 0:38
considered instantaneous [87]. For fast adsorption kinetics, 4
The value of the convective-dispersion coefficient, E ¼ 2Rb u=
i.e., ka
kc , (15-58) reduces to
N PeE , may be calculated using (15-61) to obtain NPe.
5
For the conditions of this problem, values of E=uRp that correspond
1 1 Rp
¼ þ  ð15-69Þ to u ¼ 0.5 cm/minute (at NReNSc ¼ 33.33), u ¼ 1.0 cm/minute
kc;tot kc 5ep De (NReNSc ¼ 66.67), and u ¼ 1.5 cm/minute (NReNSc ¼ 100) have been
calculated and plotted as line B in Figure 15.20. It is not necessary to
graphically obtain these values to find H using (15-57), since H/2Rp is
EXAMPLE 15.10 Resolving Power. also shown in Figure 15.20 for the requested values of u.
6
kc for spherical, stationary-phase particles in the range 102 <
In nonadsorbing, size-exclusion chromatography (SEC), solutes are N Re N Sc < 103 may be estimated using [46]
retained in a packed bed to the degree in which they penetrate "  #1=3
porous stationary-phase media. Retention time varies approximately 2Rp kc 1:09 3
N Sh ¼ ¼ N Re N Sc þ 43
ð1Þ
inversely with log(MW). SEC is often used to desalt, or exchange Di eb
buffers, of protein solutions. A 10 mg/mL aqueous solution of
bovine hemoglobin is to be purified by size-exclusion chromatogra- Generally, a value of kc obtained from (1) would be substituted into
phy using a 46.5 (L)  2.5 (D) cm i.d. glass column packed with 75- (15-68) to obtain kc,tot, which would then be substituted into (15-57)
mm-diameter Toyopearl HW65C gel matrix. Intraparticle effective to obtain H. However, it is not necessary to calculate kc directly for
diffusivity, De, of hemoglobin is 3.0  107 cm2/s. Estimate the this problem, since H=2Rp is also shown in Figure 15.20 for requested
resolving power per unit thermodynamic driving force, R/d, for this values of u.
protein in this column at u ¼ 0.5, 1.0, and 1.5 cm/min. What
7
The magnitude of H=2Rp varies with NReNSc, as illustrated in Figure
resolution, R, is expected from a protein of comparable diffusi- 15.20 (lower solid line). Reduced values of h ¼ H=2Rp from Figure
vity whose inclusion porosity, ep , is 111.4% of bovine hemo- 15.20 for u ¼ 0.5 cm/minute (at NReNSc ¼ 33.33), u ¼ 1.0 cm/minute
globin? What size column would be required to obtain R ¼ 1, as in (NReNSc ¼ 66.67), and u ¼ 1.5 cm/minute (NReNSc ¼ 100) are 4.3,
Figure 15.21? 7.1, and 9.6, respectively. Therefore the corresponding values of
C15 09/22/2010 Page 595

§15.3 Kinetic and Transport Considerations 595

H (in mm) are 322.5, 532.5 and 720, respectively. From these values Table 15.9 Biochromatographic Purification Factors
of H, using the definition, N ¼ L=H, values of N at 0.5, 1.0, and 1.5
cm/minute are calculated, using L ¼ 46.5 cm, to be 1440, 870, and Type Purification factor1 Examples
645, respectively. As an example: Biospecific affinity 50–10,000 Protein immunoglobulins
L 465ð1000Þ Dye affinity 10–100 Blue dextran/protein
N¼  ¼ ¼ 1442 ð2Þ
2Rp h 75ð4:3Þ Cation exchange 2–40 Cytochrome C
8
Using (15-59), resolution per unit driving force, R/d, at 0.5, 1.0, and Size exclusion 2–20 Hemoglobin
1.5 cm/minute is then calculated to be 9.5, 7.4, and 6.4, respectively. 1
Purification factor was defined in §1.9.4
As an example, pffiffiffiffi pffiffiffiffiffiffiffiffiffiffi
R N 1442
¼ ¼ ¼ 9:5 ð3Þ
d 4 4 high internal surface areas for adsorption (100 to 1,500 m2/g).
Resolving hemoglobin from a protein with an inclusion porosity ep ¼ But silica may irreversibly bind or denature some proteins and
1:114ð0:498Þ ¼ 0:555 gives a driving force d ¼ 0.05. Using (15-59), is unstable in common basic regenerants like 1.0-M NaOH
this yields R ¼ 0.47, 0.37, and 0.32 at 0.5, 1.0, and 1.5 cm/min, re- (pH 14). Selective, reversible adsorption of hydrophilic spe-
spectively. As an example, cies to uncoated silica is called normal-phase chromatogra-
pffiffiffiffi
d N phy. More commonly, long-chain alkanes are bound to silica
R¼ ¼ 9:5ð0:05Þ ¼ 0:47 ð4Þ
4 to selectively adsorb hydrophobic species (e.g., small mole-
From (15-59), to obtain R ¼ 1 with d ¼ 0.05, the number of theoreti- cules, peptides, proteins, and DNA) in reversed-phase chro-
cal stages would be N ¼ 6,400. For H ¼ 320 mm at 0.5 cm/minute, a matography. The bonded alkanes can form monolayers and be
column 206 cm long would be required. polymerized or end-capped. End-capping adds an organic
layer after the bonding step to coat all bare silica surfaces.
Excellent resolution makes reversed-phase chromatography
§15.3.3 Biochromatography Adsorbents attractive for species that do not lose activity upon interaction
Rapid advances in biochromatography for recovery of biopoly- with bonded silica.
mers rely on increased understanding of biomolecules and
intermolecular forces to select or formulate suitable adsorb- Polymer Resin
ent stationary phases. Of major importance are resin particles
of silica or polymer that are conjugated with chemistries that Low-cost, pH-stable polymer resins (10–100 mm) are made
allow separation of bioproducts using ion-exchange and by adding a cross-linking agent (e.g., bis-acrylamide for
hydrophobic-interaction, affinity, reversed-phase, or size- polyacrylamide resins) to an emulsion of polymer in an
exclusion chromatographies [88, 89]. Table 15.8 summarizes immiscible solvent. Styrene divinylbenzene (STDVB) forms
characteristics of these five classes of adsorptive phases. The a rigid, pH-stable, mildly hydrophobic backbone, primarily
thermodynamic basis for the physicochemical interaction derivatized for ion exchange by reaction. Polyacrylamide (PA)
associated with each adsorbent phase is provided in §2.9.2. A forms a hydrogel useful in size-exclusion chromatography.
range of purification factors is achieved using four stationary Natural, hydrophilic, hydrogel-forming polymers like agarose,
phases common to biopharmaceutical fluid–solid separations. large-pore dextran, and microcrystalline cellulose can sepa-
Table 15.9 shows examples of these factors. rate enzymes, antibodies, and virus by size exclusion as well
With few exceptions (e.g., capillary columns), adsorbents as by interaction with derivatized phenol, antibodies, dyes,
used in protein chromatography are porous. Use of non- heavy metals, nonspecific ion-exchange groups, and biospe-
porous pellicular adsorbents in biochromatography is rare. cific epitopes. Substituted cross-linked agarose gels like
Each of the five types of chromatographic adsorbents sum- Sepharose resist shrinking with changes in pH or ionic
marized in Table 15.8 is widely used as a resin in a packed strength, I.
bed, as illustrated in Figures 15.16 and 15.21.
Ion-Exchange Chromatography
Silica Resin
Table 15.10 identifies the type, group, approximate pKa, and
Silica resins (1–25 mm) are incompressible at typical high- formula at physiological pH (§2.9.1) of the most common
pressure liquid chromatography (HPLC) pressures, and have ion-exchange groups used to electrostatically bind small

Table 15.8 Large-Scale Protein Chromatography

Type Basis for Separation Resolution/Speed/Capacity Application

Ion exchange Charge High/high/high Protein, whole virus

Hydrophobic interaction Surface hydrophobicity Good/good/high Polypeptide
Affinity Bioaffinity Excels/high/high Antibody–antigen, dye–ligand
Reversed phase Surface hydrophobicity Excels/high/high DNA, plasmid
Size exclusion Size Moderate/good/low Protein, plasmid, DNA
C15 09/22/2010 Page 596

596 Chapter 15 Adsorption, Ion Exchange, Chromatography, and Electrophoresis

Table 15.10 Ion-Exchange Resins

Na+ Type Group Formula pKa

Strong acid Sulfopropyl, SP SO

3 2
Weak acid Carboxymethyl, CM COO 4–4.5
Weak base Diethylamino-ethyl, 2C2H5N+HC2H5 9–9.5
DEAE
Strong base Quaternary ethyl 4C2H5N+ 12
amine, QEA

molecules (e.g., peptides, antibiotics), biopolymers (proteins,

nucleic acids), and particles (virus). Derivatized ion-
exchange groups exhibit typical charge capacities of
0.5 mmol/cm3 (0.5 M). These charges are balanced by coun-
Figure 15.24 Capacity of CM-cellulose at a ¼ 0 for (1) hen egg
terions that are displaced by a binding species. Adjacent
white lysozyme; (2) AMP kinase; (3) phosphoglycerate kinase;
DEAE substituents electrostatically repel protons. This low- (4) phosphoglycerate mutase; (5) creatine kinase; (6) enolase;
ers the local pKa, which allows substantial titration around (7) lactate dehydrogenase; (8) glyceraldehydes phosphate
pH 6–8, and increases the pH in the micro-environment of dehydrogenase; (9) aldolase; (10) pyruvate kinase.
the matrix relative to the surrounding buffer, which affects [From Scopes [90] with permission.]
species solubility (see Donnan effects in §2.9.1). The effect
occurs in the opposite direction for CM-cellulose. Protein
stability, typically higher slightly above physiological pH technique called chromatofocusing [91]. Regeneration of
than below, suggests using anion, rather than cation, ion-exchange resins by 1.0-M NaOH (pH 14) is common.
exchangers for protein recovery. Dissolved nucleic acids may
also interfere with cation exchange, unless first removed by
Reversed-Phase Chromatography
polycation (protamine) precipitation.
To minimize interactions between buffer and ion- Hydrophobicity in reversed-phase change varies with the
exchange groups, aqueous buffers used with an ion-exchange chain length and density of alkanes that are typically bonded
resin should possess the same sign in their charged form and to silica: octyl (C8), octadecyl (C18), phenyl (C6H6), and
contain only simple counterions (e.g., Na+, K+, Cl, methyl (C1). Steric effects yield exposed silica after bonding
CH3COO). Anion exchange DEAE-cellulose, for example, that can be covered by polymerizing alkyl chains at their
could be used with Tris HCl buffer at pH 8.0 (see Table 2.13) attachment point or end-capping the exposed silica with
since it contains HTris+, Cl counterion, and Tris (neutral) methyl or ethyl groups. Anionic, strong-acid counterions like
species. A dissolved, anionic protein neutralized by HTris+ trifluoroacetate, acetate, or chloride that selectively partition
would displace Cl associated with DEAE while discharging with the targeted co-ion and alter its hydrophobicity are typi-
HTris-Cl, the acid salt of Tris. Every 1 mg/mL protein cally used to separate species in a biological mixture. Tar-
adsorbed releases 1 mM (millimolar) of buffer salt. From geted nonpolar species are dissolved in aqueous mobile
(2-115) and (2-120), buffer salt discharge can decrease pH phase with minimal organic content added to promote
and increase I of the mobile phase, reducing anticipated phase interaction, and introduced to reversed-phase resins
adsorption. Effects of buffer salt discharge are mitigated by where they partition based on hydrophobic content, or
using  10 mM of a suitable buffer [88] within 0.3 unit of its mixed-mode interactions. The organic (e.g., acetonitrile,
pKa and at most 5 mg/mL of protein adsorbate. methanol, isopropanol) content is gradually increased in the
Ion-exchange adsorptive capacity varies inversely with mobile phase to selectively elute adsorbed species.
log(MW) until biological species are excluded from resin
pores, as shown in Figure 15.24 for a (given by (15-51) ¼ 0.
Hydrophobic-Interaction Chromatography
Exclusion occurs at MW  106 for cellulose and 107 for
Trisacryl. Increasing ionic strength, typically Na+ or K+ and Proteins with low water solubility—like globulins, mem-
Cl, up to I  0.5 to 1.0 M, is usually used to elute adsorbed brane-associated proteins, or enzymes that precipitate at 20–
species by shielding electrostatic interactions between target 40% (NH4)2SO4—adhere to polymer resins derivatized with
and resin (2-135). Adjustment to pH is rarely used due to C4, C6, C8, and C10; linear aliphatic chains; or phenyl moie-
high matrix buffering capacities. However, continuously add- ties. Other candidates may bind hydrophobic resins at pH
ing the acid form of a high-buffering ampholyte buffer, values close to their pKa and/or at high I. In contrast to
which has low ionic strength, to a column containing polye- reversed-phase-type two-phase partitioning in hydrobic inter-
thyleneimine–agarose ion exchange, which allows continu- actions, lyophilic salts strip proteins of solvated water, pre-
ous titration, yields a very steady pH gradient in which cipitating aggregation or nucleation of the protein onto the
proteins emerge at or above their isolectric point (§2.9.1), a surface (see §2.9). Reducing salt content redissolves the
C15 09/22/2010 Page 597

§15.3 Kinetic and Transport Considerations 597

adsorbed proteins. Reproducibility is sensitive to tempera- or glycine buffer at pH 9. Figure 2.23 shows the fundamental
ture, buffer type, salt used, and pH. effect of separation distance on electrostatic interactions
between adjacent particles like biomolecule and adsorbent.
Affinity Chromatography
Size-Exclusion Chromatography
Biospecific interactions introduced in §2.9.3, like enzyme–
ligand, enzyme–cofactor, receptor–agonist (antagonist), or Large molecules (MW  1–2  106) that are excluded from
antibody–antigen, are the basis for affinity chromatography. the largest pores of underivatized polymer gels (like hydro-
One member of the interacting pair is conjugated to a poly- philic agarose and cross-linked dextran or hydrophobic poly-
mer resin to selectively bind the other from a biological mix- acrylamide) elute from the column in the void volume. This
ture. Affinity ligands include starch for binding amylases and volume, Vo, is 30 to 35% of total column volume, Vt. Smaller
glycogen-metabolizing enzymes, cellulose for binding cellu- molecules, down to MW  1  104, exhibit size- and shape-
lases, and phosphocellulose for binding nucleic-acid binding dependent permeability and elute in order of decreasing
proteins. Pseudo-affinity dye adsorbents like Cibacron apparent size. Resins are available with pores that provide
Blue F3GA, an analog of ADP-ribose that binds purine- 90% exclusion of molecules whose volume is 5–6 times
nucleotide-binding enzymes, or Procion Red H-E3B, which larger than those excluded from 10% of the bead volume.
binds NADP-binding proteins, are also used. To avoid steric Size-exclusion chromatography, also called gel permeation
hindrance of affinity interaction, spacer arms and nonob- or molecular sieving, is limited in capacity by lack of bind-
structing attachment methods are used. Interaction energies ing. Elution is typically isocratic, unless mixed-mode adsorp-
> 35 kJ/mol required for affinity binding typically require tion requires increasing I. Figures 15.18 and 15.20 illustrate
supplemental nonspecific hydrophobic interactions (see isocratic elution of protein in size exclusion.
§2.9), usually provided by a hydrophobic spacer arm of hex-
amethylene, or equivalent. Elution by displacement with a §15.3.4 Reducing Transport-Rate Resistances
compound that shows higher avidity to the binding ligand is
in the Bed: Scale-Up and Process Alternatives
superior to elution via nonspecific changes in I or pH. Col-
umn regeneration by changing I or pH is common. Though Individual contributions from transport-rate resistances to
selectivity of affinity chromatography is excellent, the theoretical plate height, H, illustrated in Figures 15.16 and
expense of procuring and derivatizing the conjugated epitope 15.20, show that H in packed beds generally increases as
restricts its use to analytical applications such as affinity operating velocity, u, (and throughput) rise. Examination of
recognition of cloned epitopes and high-throughput screen- (15-57) and (15-58) reveals that smaller-diameter adsorbent
ing, or to preparation of high-value-added bioproducts. particles decrease H and increase separation efficiency, pri-
Affinity chromatography is often portrayed as a simple marily by reducing the resistance due to pore-volume diffu-
‘‘lock-and-key’’ mechanism, e.g., between a receptor, -k, and sion. However, 2.5 mm is generally regarded as a practical
a complementary target, y. However, the actual mechanistic lower limit for adsorbent radius, Rp, in high-pressure liquid
interaction is much more complex, consisting of several steps chromatography (HPLC) systems, since pressure drop rises
that include electrostatic interactions, solvent displacement, (1) in inverse proportion to decreases in Rp for highly turbu-
steric selection, and charge and conformational rearrange- lent packed-bed flow; and (2) in inverse proportion to
ment, as described in §2.9.3. decreases in R2p for laminar packed-bed flow, as found in
Ergun’s equation (14-10).
Immobilized Metal Affinity Chromatography
Scale-Up
Electron-donor amino acid residues in proteins like histidine,
tryptophan, and cysteine that are surface-accessible form Sopher and Nystrom [92], Janson and Hedman [93], and
metal coordination complexes with divalent transition metal Pharmacia [94] recommend scaling up chromatographic sep-
ions like Ni2+, Cu2+, and Zn2+. This complexation is the basis arations by maintaining H, u, and L while increasing volu-
for immobilized metal affinity chromatography (IMAC) (see metric throughput, mass loading, and gradient slope in
Example 2.14 and Table 2.18). Metalloproteins, which proportion to an increase in column cross-sectional area.
require metal centers for activity, are another target of metal However, frictional forces from the column wall that support
ions immobilized with spacer arms to the resin. Complexa- packed particles disappear below a length-to-diameter ratio
tion is typically enabled by conjugating iminodiacetic acid of roughly 2.5 [95]. This causes settling and cracking in
(IDA) or tris (carboxymethyl) ethylene diamine (TED) to a scaled-up packed beds. These phenomena were linked by
polymer gel via a spacer arm. The chelating IDA or TED is Janson and Hedman [93, 96] and Love [97] to channeling
charged with a small, concentrated (50 mM) pulse of metal and backmixing. Deterioration in large-scale packed beds
salt up to half the length of the column, to allow for metal- may be counteracted using dynamic compression to increase
ion migration. The mobile phase contains 1-M salt to mini- packing homogeneity and long-term bed stability, while
mize nonspecific ion-exchange interactions and high pH to gradually increasing bed density, as reported by Guiochon
de-protonate donor groups on targeted proteins. Elution typi- and co-workers [60]. Alternatively, Grushka [98] and Wankat
cally employs a stronger complexing agent such as imidazole [99] suggest increasing the length-to-diameter aspect ratio
C15 09/22/2010 Page 598

598 Chapter 15 Adsorption, Ion Exchange, Chromatography, and Electrophoresis

during scale-up, which would require increasing Rp to lessen yielding up to 100-fold smaller back-pressures [110–113],
the rise in pressure resulting from lengthening the packed 10-fold smaller process volumes, and shorter response times
bed. relative to packed beds [114]. This can increase throughput
10-fold or more [115, 116], and allow as much as 10-fold
decreases in processing times, solvent and tankage require-
Process Alternatives to Chromatography
ments, and solute residence times [117]. Spiral-wound, hol-
Transport-rate limitations in process-scale adsorption moti- low-fiber, cross-linked polymer rod, single-sheet, and radial-
vate examining alternatives to fluid–solid partitioning in flow configurations are also used, with similar benefits.
packed beds that increase separation efficiency as well as Of the organic and inorganic membrane materials intro-
throughput, particularly for preparative adsorption of high- duced in Chapter 14, adsorptive membranes are often com-
value, high-molecular-weight biomolecules from liquid solu- posed of hydrophilic native or regenerated cellulose, reduced
tions. Growing demand for chromatography as a preparative with borohydride to neutralize ion-exchange activity of resid-
tool in biotechnology [100–102] and pharmaceutical [103] ual carboxylic and aldehyde side groups, or with acrylic
applications has motivated development of perfusive copolymers synthesized by free-radical polymerization of a
[104, 105], ‘‘hyperdiffusive’’ [106], chromarod [107, 108] mixture of monovinyl monomer, such as styrene or meth-
and adsorptive-membrane [109] medias to reduce or virtually acrylate, and divinyl monomer, such as divinylbenzene, in a
eliminate mass-transfer resistance by intraparticle diffusion. heated mold [107, 108, 111, 112, 115]. Macroporous poly
Operating strategies have advanced to include counterflow, (glycidyl methacrylate-coethylene dimethacrylate) (GMA-
recycle [60], and displacement to more efficiently utilize EDMA) is a commonly used copolymer. Epoxy groups are
chromatographic columns. Adsorptive-membrane separation modified to furnish functional-group sites for hydrophobic-
and countercurrent contacting of bulk liquid and adsorptive- interaction (HIC), ion-exchange (IEC), or affinity membrane
solid phases are two promising alternatives that reduce costs adsorption [113].
associated with adsorbent, regenerant, and solvent, and In adsorptive membranes, the length scale for solute diffu-
increase throughput. Adsorptive stacked membranes essen- sion to an adsorptive site [Rp in (15-57) and (15-58)] is
tially eliminate transport-rate resistance due to intraparticle reduced to much less than the size of flow-through membrane
diffusion by derivatizing adsorptive sites on the surfaces of pores (1 mm) [118]. This allows adsorptive-membrane
micron-scale, flow-through pores. Countercurrent contacting, capacity to be maintained at substantially higher throughputs.
which increases the local average thermodynamic driving Eliminating diffusional resistance reduces the expression for
force for equilibrium partitioning, is nearly achieved by theoretical plate height in (15-57) to H ¼ 2E=u, which is
timed-valve delivery to a modest number of packed-bed sec- evaluated for N Re N Sc > 1 using an N PeE correlation obtained
tions in simulated-moving-bed (SMB) operations, described by analysis of creeping flow in high-void-fraction, random
in §15.4. configurations of fixed spheres [119]:
 
Adsorptive Membranes 1 3 p2 N Re N Sc eb
¼ eb þ eb ð1  eb Þln þ ð15-70Þ
N Pe 8 12 2 N Re N Sc
Membrane adsorption typically utilizes a rigid cylindrical
column, illustrated in Figure 15.25. Microporous, 200-mm-
For use in (15-70), an equivalent mean particle diameter, Dp,
thick, hydrophilic, polymeric membrane rounds derivatized
for the membrane bed is estimated from its average pore size,
with interactive moieties are layered one on top of another
dp, and bulk porosity, eb , using Dp ¼ 3d p ð1  eb Þ=3eb .
and compression-gasketed at the periphery to prevent bypass-
Measured values of adsorptive-membrane plate heights
ing. The adsorptive-membrane cross section perpendicular to
and capacities are shown in Tables 15.11 and 15.12, respec-
the flow direction is considerably longer than the flow path,
tively. van Deemter equations like (15-57) and van Deemter
plots like Figure 15.20 show that values of plate heights first
decrease and then increase as velocity increases. Variations
in measured plate height from different sources, or from

Table 15.11 Experimental Adsorptive-Membrane

Theoretical Plate Heights (H); from [118]

H Range, micron Velocity, u, Range, cm/min

0.59–3.3 0.52–3.8
3–7 0.07–0.22
25 0.1–2
50–110 1.5–4
80–160 1–45
400 0.04–1
Figure 15.25 Cut-away view of a ChromaSorbTM 0.08 mL Small- 250–800 0.035–6.5
Scale Screening and Development Membrane Adsorber.
C15 09/22/2010 Page 599

§15.3 Kinetic and Transport Considerations 599

Table 15.12 Reported Capacity Values of Adsorptive Membranes for Several Biological Macromolecules:
Monoclonal Antibody (MAb), Malate Dehydrogenase (Md), Human Serum Albumin (HSA), Ribonuclease
(Rib), Lysozyme (Lys), Ovalbumin (Ova), Bovine Serum Albumin (BSA), Gamma-Globulin (G-G),
Immunglobulin G (IgG), and a Mixture of IgG and IgA (BGG); from [118]

Membrane Static Capacity (mg/mL) Dynamic Capacity (mg/mL)

C-4 200–400 50
Cation exchange 50 (MAb)
Dye affinity 50.8 (Md) 45.7 (Md)
Anion exchange 5.8 (HSA)
Copolymer 20 (Rib), 26 (Lys), 47 (Ova) 5 (Rib), 0 (Lys), 5 (Ova)
Copolymer 40 (Ova)
L-Phe affinity hollow fiber 50 mg BGG/g fiber
Anion exchange 30–40 g BSA/g membrane
Anion exchange 20
Dye affinity 8.6 (Lys), 5.6 (BSA) 7.8 (Lys), 7.6 (BSA)
Protein-A/IgG affinity 4.74 (IgG-rabbit), 0.51 (protein A)
Protein-A affinity 3.3 (G-G) 2.9 (G-G)
Ion exchange 8 IgG/cartridge

theoretical prediction, have been quantitatively shown to Substituting Rv ¼ 0.06  104 cm and Rp ¼ 7.5  104 into
arise from differences in the parameters in (15-57), as well as the above equation gives a static capacity of 1.2  1013 virions
from external contributions to plate height such as band per mL. This value is 1/24th of the reported capacity value, sug-
broadening due to mixing in extra-column peripheral vol- gesting negligible effective virus penetration into pores of the
umes such as injectors, detectors, tubing, and valves and non- resin. Adenovirus is comprised of 87% protein and 13% nucleic
acid with a total viral mass of 1.65  108 Da. Estimated static
uniform flow in adsorptive-membrane beds. Capacity may be
virus capacity is therefore
measured experimentally and predicted using (15-51). Exam-
ple 15.11 illustrates prediction of capacity for virus adsorp-

1:2  1013 virions 1:65  108 grams
tion. Static capacity, measured under nonflowing (e.g., batch) ¼ 3:3 mg/mL
mL resin 6:02  1023 virions
conditions, typically exceeds dynamic capacity, measured
under flowing conditions, since slow, diffusive, mass- This value is comparable to capacity of protein A-affinity interac-
transfer limits complete utilitization of surface area of the tion in Table 15.12. The experimentally reported capacity is 0.137
stationary phase. mg virus/mL resin, which is about an order of magnitude smaller
than measured values for protein chromatography.
Static capacities reported for protein adsorption on several mem-
EXAMPLE 15.11 Capacity of an Anion-Exchange brane monoliths measured 3.3 to 50 mg/mL, whereas typical chro-
Resin. matographic protein capacities are 25–60 mg/mL for PorosTM
and MonoQTM media, 110–115 mg/mL for HyperD1 resin, and
Adenovirus type 5 is a candidate viral vector for gene therapy. Esti- 300 mg/mL for soft Sephadex1.
mate the static capacity of 15-mm-diameter SourceQTM anion-
exchange resin for binding 120-nm-diameter adenovirus type 5.
Compare the estimate with an experimental static capacity of 5 
1011 virus/mL reported for adenovirus on this resin, and with protein
Counterflow
capacities of anion-exchange resins and membrane monoliths.
Reductions in counterflow solvent and adsorbent usage rela-
tive to packed-bed adsorption at comparable purities may be
Solution
evaluated using an equilibrium-stage description of steady-
Using a packing factor of 0.547 for random sequential adsorption, state counterflow introduced by Kremser [120], as described
an estimate for static capacity of 120-nm virions with a projected in §9.2, and Souders and Brown [121]. Klinkenberg et al.
surface area of Av adsorbed on total outer surface, Ap, of 15 mm [123] specialized this description for continuous counter-
SourceQTM beads packed to a void volume of 0.38 is given by current adsorption of solute 1 and 2 from small feed mass
virions Ap 1  eb 3ð0:62Þð0:547Þ flow rate, F, relative to pure fluid (U) and solid-phase (S)
¼ ¼ mass flow rates into an M-stage enricher and an N-stage strip-
mL resin 1 Vp R2v Rp
Av per, respectively, separated by a feed stage, e, in a model col-
0:547
umn, shown in Figure 15.26. The fluid-phase mass fraction of
where A, V, and R are surface area, volume, and radius; and sub-
solute i exiting the stripper, yN,i, and the solid-phase mass
scripts p and v represent resin particle and virus.
fraction exiting the enricher, x1,i, are related recursively to
C15 09/22/2010 Page 600

600 Chapter 15 Adsorption, Ion Exchange, Chromatography, and Electrophoresis

which may be obtained from statistical tables, e.g., Hogg and

Ledolter [124]. Comparing (15-73) and (15-59) reveals that
the number of counterflow stages required for a selected sep-
aration increases proportionally to d1, whereas the number
of chromatographic stages required in differential operation
increases proportionally to d2. For close separations, i.e.,
d 1, the higher local driving force in counterflow results
in (requires) significantly fewer theoretical stages.

Relative Solvent and Adsorbent Usage

Using (15-52)–(15-56), (15-59)–(15-60), and (15-71)–
(15-73), the relative solid phase required for steady counter-
Figure 15.26 Equilibrium-stage representation of ideal steady flow (SC) relative to differential chromatography to achieve
counterflow. The enricher and stripper consist of M and N stages,
comparable purities can be evaluated using [63]:
respectively. Feed enters stage e. Pure solid (S) and fluid (U) streams
enter the stripper and enricher, respectively. Mass flow rates of

N SC;tot ASC H SC 3 U d3 Ff2Rg
component i exiting the column from the stripper, MU,N,i, and the ¼ ln ð15-74Þ
enricher, MS,N,i are shown. In the column, component i enters and
NAH 32 F R4 Ff2Rg
exits a representative stage j ¼ N  1 of height HSC with mass
Solid-phase savings using counterflow increases for high
fraction values in the lower solid phase, xj,i and upper fluid phase,
resolution of closely related solutes, e.g., racemic mixtures.
yj,i, as shown.
The physical basis for this decreased adsorbent usage appears
when Figure 15.21 is compared with Figure 15.27: 5% of the
the mass fraction of i at a central feed stage, ye,i, by chromatographic bed actively separates 1 from 2, whereas
ye;i 100% of the counterflow column actively separated 1 from 2.
yN;i ¼ N ð15-71Þ
P j In Figure 15.27, the profiles for the easier and the more diffi-
Gi cult separation appear consistent because (1) unity resolution
j¼0
is specified in both cases and (2) scaled axes are used to rep-
ye;i resent the data. Equation (15-73) shows that to achieve the
x1;i ¼ ð15-72Þ
PM same final purity, a 100-fold increase of NSC is required for
ai Gji
j¼0
the more difficult (d ¼ 0.00209) separation.
The counterflow solvent volume, Vsolv,SC, required relative
where the extraction ratio, Gi ¼ ðai rs =ru ÞjU=Sj, summarizes to differential chromatography (DC) may be estimated as
the phase-partitioning of solute between steady mass flow
rates of fluid eluent, U ¼ ð1  rÞeb VT, and solid adsorbent, V solv;SC 2U
¼ pffiffiffiffi ð15-75Þ
S ¼ rð1  eb ÞVT. Equation (15-51) defines the solute parti- V solv;DC F N
tion coefficient, ai. Each stage, j, has volume V, which is
transferred at a rate T, and r is the fractional relative motion
of the solid phase, i.e., the modulus of the ratio of solid-
phase to fluid-phase velocities. Fluid mass flow rate U is re-
lated to the countercurrent interstitial liquid velocity, uSC,
by U ¼ ASC eb uSC ru, where ASC is the counterflow-column
cross-sectional area. Equilibrated fluid- and solid-phase mass
fractions exiting the jth stage are related by yj;i ¼ ai xj;i rs =ru.
The total number of counterflow stages, N SC;tot ¼
N þ M þ 1, required to attain a fractional purity, PU,N,i, of
species i exiting the column at stage N in fluid-phase U in an
optimal binary split is given by [63].
 
4 PU;N;i
N SC;tot ¼ ln 1 ð15-73Þ
d 1  PU;N;i

where  d is given by (15-60) and, in general, Pp;j;i ¼

M p;j;i = M p;j;i þ M p;j;not i , for mass flow rate Mp,j,i of solute i
Figure 15.27 Relative concentrations in counterflow for two
leaving stage j in phase p, is the product of solute mass frac- separations. In the easier (d ¼ 0.209) separation, solute 1 (D)
tion and its corresponding phase flow rate i.e., MU,j,i ¼ Uyj,i. partitions more to the upper fluid phase while 2 (^) tends to the
Fractional purity, P, obtained in differential adsorption may lower solid phase. In the more difficult (d ¼ 0.00209) separation,
be evaluated in terms of resolution, R, in (15-59) as P ¼ solute 1 (---) and solute 2 (-) partition likewise. Stage numbers M þ
F{2R}, where F is the cumulative distribution function, N are selected to resolve 1 and 2 to unity (PU,1 ¼ 0.9772).
C15 09/22/2010 Page 601

§15.3 Kinetic and Transport Considerations 601

Solvent savings also increases for high purity separations, by continuous addition of a mobile feed at a volumetric flow
but less dramatically than adsorbent savings. While adsorb- rate QF containing dilute solute at concentration cF in frontal
ent- and solvent-usage requirements are lower in steady loading mode, also referred to as percolation or simply fixed-
counterflow for high-resolution purification of closely related bed adsorption. Frontal loading concentrates dilute solute,
solutes, separation is limited to binary separations. Like since cs > cb
cf  cF usually characterizes loading, and
distillation, counterflow separations effect a binary split reduces dilution caused by transport-rate resistances, which
between key components. Additional components in a multi- spread an initially sharp solute pulse cf{z,0} in (15-52) across
component feed that partition more strongly to the solid or a bed volume of 3.3 s ¼ 3.3N1/2AH=fsu given by (15-56).
liquid phases are separated more efficiently, but may present Subsequent application of a fluid eluent at a thermodynamic
additional complexity in terms of solid- or fluid-phase regen- state (temperature, pressure, composition) that favors
eration in real systems. In packed-bed adsorption, the number desorption can selectively recover concentrated solute at cf > cF.
of species that can be resolved in batch or semi-batch opera-
tion is limited only by the magnitude of the thermodynamic
Ideal Fixed-Bed Adsorption
driving force that distinguishes partitioning of the respective
solutes and the separation efficiency of the system. Ideal (local-equilibrium) fixed-bed adsorption represents the
limiting case of: (1) neglible external and internal transport-
rate resistances; (2) ideal plug flow; and (3) adsorption iso-
EXAMPLE 15.12 Steady Counterflow Separation of therm beginning at the origin. Local equilibrium between
Albumins. fluid and adsorbent is thus achieved instantaneously, result-
Bovine serum albumin ðep ¼ 0:30Þ and ovalbumin ðep ¼ 0:34Þ in an
ing in a shock-like stoichiometric front, shown in Figure
equimolar mixture are to be purified by size exclusion to a resolu- 15.28, that moves at a constant velocity throughout the bed.
tion of R ¼ 1.0 in a packed bed of Toyopearl1 (TSK gel) HW55F, The bed is divided into two zones or sections: (1)
which has a bed porosity of eb ¼ 0:34. The differential migration Upstream of the stoichiometric front, fluid-phase solute con-
velocity, d, for these two solutes is 0.048. Assume a volumetric feed centration, cf, equals the feed concentration, cF, and spent ad-
stream/fluid-phase dilution in counterflow of 10. Determine the rela- sorbent is saturated with adsorbate at a loading cb in
tive number of equilibrium stages, the relative solvent requirement, equilibrium with cF. The length (height) and weight of this
and the relative solid-phase requirement for steady counterflow sep- section are LES and WES, respectively, where ES refers to
aration relative to differential chromatography. the equilibrium section, called the equilibrium zone. (2)
Downstream of the stoichiometric front and in the exit fluid,
Solution cf ¼ 0, the adsorbent is adsorbate-free. The length and
Using tables for a cumulative normal distribution, P ¼ F{2R} for weight of this section are LUB and WUB, respectively,
R ¼ 1 corresponds to mutual fractional purity, PS;1;OA ¼ PU;N;BSA ¼ where UB refers to unused bed.
0:9772. Substituting R ¼ 1 and PU,N,BSA into (15-59) and (15-73), After a stoichiometric time ts, the stoichiometric wave
respectively, and using d ¼ 0.048 yields N SC;tot =N ¼ 313=6; 975. front reaches the end of the bed; the value of cf,out abruptly
Using (15-74) with U=F ¼ 10 and N ¼ 6,975 stripping stages shows rises to the inlet value, cF; no further adsorption is possible;
that only 24% as much solvent is needed for steady counterflow rel- and the adsorption step is terminated. This point is referred to
ative to differential chromatography. Substituting values for U=F, as the breakpoint and the stoichiometric wave front becomes
PU,N,BSA, and d into (15-73) indicates that 0.04% as much adsorbent the ideal breakthrough curve. For ideal adsorption in a
might be expected for steady counterflow relative to differential
packed bed of length LB, the location of the concentration
chromatography.
wave front Lideal  LB in Figure 15.28, as a function of time,
is obtained by equating the solute entering in the feed to that
in the adsorbate:
§15.3.5 Mitigating Transport-Rate Resistances:
Frontal Loading QF cF tideal ¼
cb Að1  eb ÞLideal ð15-76Þ

Selective partitioning of a solute from mobile fluid to the sta- where cb is the loading in equilibrium with cF, and A is the
tionary adsorbent phases may be used to saturate a packed bed bed cross-sectional area. Defining the total mass of adsorbent
Concentration front at some

cF
c, solute concentration

Stoichiometric
front
Spent Unused
time t

Feed Effluent adsorbent adsorbent

(in equilibrium with

entering fluid)

WES WUB
LES LUB Figure 15.28 Stoichiometric
0 L LB (equilibrium) concentration
z, distance through the bed front for ideal fixed-bed adsorption.
C15 09/22/2010 Page 602

602 Chapter 15 Adsorption, Ion Exchange, Chromatography, and Electrophoresis

in the bed by S ¼ Að1  eb ÞLB and rearranging (15-76) gives, at t1, no part of the bed is saturated. At t2, the bed is almost
for ideal fixed-bed adsorption corresponding to Figure 5.28, saturated for a distance Ls. At Lf, the bed is almost clean.
QF cF tideal Beyond Lf, little mass transfer occurs at t2 and the adsorbent
Lideal ¼ LES ¼ LB ð15-77Þ is still unused. The region between Ls and Lf is called the

cb S
mass-transfer zone, MTZ, at t2, where adsorption takes place.
LUB ¼ LB  LES ð15-78Þ Because it is difficult to determine where the MTZ zone
begins and ends, Lf can be taken where cf=cF ¼ 0.05, with Ls
LES at cf=cF ¼ 0.95. From time t2 to time tb, the S-shaped front
WES ¼ S ð15-79Þ
LB moves through the bed.
At the breakthrough point, tb, the leading point of the
WUB ¼ S  WES ð15-80Þ MTZ just reaches the end of the bed. Feeding is discontinued
at tb to prevent loss of unadsorbed, dilute solute, whose
outlet concentration begins to rise rapidly. Rather than using
Solute Concentration Distributions in Frontal Loading cf=cF ¼ 0.05, the breakthrough concentration can instead be
Actual solute concentration distributions during frontal load- taken as the minimum detectable or maximum allowable
ing are not ideal, but may be obtained from (15-48)–(15-51) solute concentration in the effluent fluid. When feeding inad-
by superimposing solutions of the form (15-53) using Green’s vertently continues after tb, the time to reach cf,out=cF ¼ 0.95
functions [125]. This produces concentration profiles for cf, is designated te.
illustrated in Figure 15.29a, that are broadened by transport-
rate resistances summarized in (15-57) and (15-58). Analytical Solution
The cf profiles in 15.29a are normalized relative to feed
concentration, cF, and plotted as a function of axial distance, For a single solute and an initially clean bed free of solute
z, within the column at successive times t1, t2, and tb after adsorbate, Anzelius [126] obtained an analytical solution
loading begins. A corresponding S-shaped breakthrough from (15-48)–(15-51) and (15-69) for frontal loading, neglec-
curve for cf=cF, shown in Figure 15.29b, is plotted as a func- ting axial dispersion, that is summarized by Ruthven [10] and
tion of time, t, at the column outlet, z ¼ LB. In Figure 15.29a, discussed by Klinkenberg [127], who provided this useful
approximation for solute concentration distribution with
respect to axial distance and time [126]:

 
1.0 cf 1 pffiffiffi pffiffiffi 1 1
 1 þ erf t  j þ pffiffiffi þ pffiffiffi ð15-81Þ
Equilibrium MTZ Unused bed cF 2 8 t 8 j
zone at t2 at t2 at t2
where erf{x} is the error function defined in (3-76) and j and
t1 t2 tb t are dimensionless distance and displacement-corrected
cf /cF
time coordinates, respectively, given by
 
3kc;tot z 1  eb
j¼ ð15-82Þ
Rp u eb
0
0 Ls Lf LB 3akc;tot  z
t¼ t ð15-83Þ
Distance through bed, z Rp u
(a)
where 3=Rp ¼ av is the surface area per unit volume for a
1.0 sphere, and resistances due to external transport, pore diffu-
0.95 –
sivity, and kinetics (if present) are included in kc,tot, as shown
in (15-58). For gas separations at low loadings, avkc,tot may
be represented by the experimentally obtained product kK,
cf ,out/cF Breakthrough where K is an equilibrium constant defined by (15-16) equiv-
curve alent to a1 in (15-51), and k is an overall mass-transfer
coefficient obtained from experimental data. This approxi-
mation is accurate to < 0.6% error for j > 2.0. Klinkenberg
[128] also provided an approximate solution for profiles of
– 0.05
0 solute concentration in equilibrium with the average sorbent
0 tb te
loading:
Time, t

 
(b) cf
cb 1 pffiffiffi pffiffiffi 1 1
¼   1 þ erf t  j  pffiffiffi  pffiffiffi ð15-84Þ
Figure 15.29 Solute wave fronts in a fixed-bed adsorber with cF
cb 2 8 t 8 j
mass-transfer effects. (a) Concentration-distance profiles.
(b) Breakthrough curve. where cf ¼
cb a and
cb is the loading in equilibrium with cF.
C15 09/22/2010 Page 603

§15.3 Kinetic and Transport Considerations 603

1
EXAMPLE 15.13 Breakthrough Curves Using the
0.9
Klinkenberg Equations. 0.8
s s
Air at 70 F and 1 atm, containing 0.9 mol% benzene, enters a fixed- 0.7 le
on e

2
i

5
0.6 s c
bed adsorption tower at 23.6 lb/minute. The tower is 2 ft in inside en n

10
cf /cF 0.5 m ista
diameter and packed to a height of 6 ft with 735 lb of 4  6 mesh i

15
ξ ,d d

20
0.4
silica gel (SG) particles with a 0.26-cm effective diameter and an

25

30
0.3 )
external void fraction of 0.5. The adsorption isotherm for benzene .2 b ed
0.2 32 of
has been determined to be linear for the conditions of interest: d
0.1 en
q ¼ Kc ¼ 5;120c ð1Þ 0
(
0 5 10 15 20 25 30 35 40
where q ¼ lb benzene adsorbed per ft3 of silica gel particles, and τ , dimensionless time
c ¼ equilibrium concentration of benzene in the gas, in lb benzene Figure 15.30 Gas concentration breakthrough curves for Example
per ft3 of gas. 15.13.
Mass-transfer experiments simulating conditions in the 2-ft-
diameter bed have been fitted to a linear-driving-force (LDF) model: When z ¼ bed height ¼ 6 ft, j ¼ 32.2 and
@
q  z 
¼ 0:206K ðc  c Þ ð2Þ t ¼ 0:206 t  ð4Þ
@t 197
where time is in minutes and 0.206 is the constant k in minute1, For t ¼ 155 minutes (the ideal time), and z ¼ 6 ft, using (4), t ¼ 32.
which includes resistances both in the gas film and in the adsorbent Thus, breakthrough curves should be computed from (15-81) for
pores, with the latter resistance dominant. values of t and j no greater than about 32. For example, when j ¼
Using the approximate concentration-profile equations of Klin- 32.2 (exit end of the bed) and t ¼ 30, which corresponds to a time
kenberg [127], compute a set of breakthrough curves and the time t ¼ 145.7 minutes, the concentration of benzene in the exiting gas,
when the benzene concentration in the exiting air rises to 5% of the from (15-81), is
inlet. Assume isothermal, isobaric operation. Compare break- " !#
through time with time predicted by the equilibrium model. c 1 1 1
¼ 1 þ erf 30  32:2 þ
0:5 0:5
þ
cF 2 8ð30Þ0:5 8ð32:2Þ0:5
Solution
1 1
¼ ½1 þ erf ð0:1524Þ ¼ erfcð0:1524Þ
For the equilibrium model, the breakthrough curve is vertical, and 2 2
the bed becomes completely saturated with benzene at cF. ¼ 0:4147 or 41:47%
MW of entering gas ¼ 0:009ð78Þ þ 0:991ð29Þ ¼ 29:44:
This far exceeds the specification of c=cF ¼ 0.05, or 5%, at the exit.
Density of entering gas ¼ ð1Þð29:44Þ=ð0:730Þð530Þ ¼ 0:076/lb/ft3 : Thus, the time of operation of the bed is considerably less than the
Gas flow rate ¼ 23:6=0:0761 ¼ 310 ft3 /minute: ideal time of 155 minutes. Figure 15.30 shows breakthrough curves
computed from (15-84) over a range of the dimensionless time, t,
ð23:6Þ for values of the dimensionless distance, j, of 2, 5, 10, 15, 20, 25,
Benzene flow rate in entering gas ¼ ð0:009Þð78Þ
29:44 30, and 32.2, where the last value corresponds to the bed exit. For
¼ 0:562 lb/minute and c=cF ¼ 0.05 and j ¼ 32.2, t is seen to be nearly 20.
0:562 From (4), with z ¼ 6 ft, the time to breakthrough is t ¼ 0:206
20
þ
cF ¼ ¼ 0:00181 lb benzene/ft3 of gas
197 ¼ 97:1 minutes, which is 62.3% of the ideal time.
6
310
Figure 15.29 or (15-84) can be used to compute the bulk concen-
From (1), tration of benzene at various locations in the bed for t ¼ 20. The
lb benzene
q ¼ 5;120ð0:00181Þ ¼ 9:27 results are as follows:
ft3 SG
The total adsorption of benzene at equilibrium j z, ft c/cF
9:27ð3:14Þð2Þ2 ð6Þð0:5Þ
¼ ¼ 87:3 lb 2 0.373 1.00000
4
87:3 5 0.932 0.99948
Time of operation ¼ ¼ 155 minutes 10 1.863 0.97428
0:562
For the actual operation, taking into account external and internal 15 2.795 0.82446
mass-transfer resistances, and replacing avkc,tot in (15-82) and (15- 20 3.727 0.53151
83) with kK obtained from experimental data, 25 4.658 0.25091
  30 5.590 0.08857
ð0:206Þð5;120Þz 1  0:5 32.2 6.000 0.05158
j¼ ¼ 1;055 z=u
u 0:5
At t ¼ 20, the adsorbent loading, at various positions in the bed, can
310 be computed from (15-84), using q ¼ 5,120c. The maximum load-
u ¼ interstitial velocity ¼   ¼ 197 ft/min ð3Þ
3:14  22 ing corresponds to cF. Thus, qmax ¼ 9.28 lb benzene/ft3 of SG.
0:5
4 Breakthrough curves for the solid loading are plotted in Figure
1;055 15.31. As expected, those curves are displaced to the right from the
j¼ z ¼ 5:36z, where z is in ft.
197 curves of Figure 15.30. At t ¼ 20:
C15 09/22/2010 Page 604

604 Chapter 15 Adsorption, Ion Exchange, Chromatography, and Electrophoresis

1 4.0

MTZ, width of wave front, ft

0.9 3.5
0.8 s 3.0
es
0.7 nl
2
io e 2.5

5
0.6 s c
en n

10
q/qF* 0.5 2.0
i m ista

15
ξ ,d d 1.5

20
0.4

25
0.3 d) 1.0

30
.2 be
0.2 32 of 0.5
0.1 n d 0.0
(e 0 2 4 6 8 10 12 14 16 18 20
0
0 5 10 15 20 25 30 35 40 τ , dimensionless time
τ , dimensionless time
Figure 15.33 Broadening of wave front in Example 15.13.
Figure 15.31 Adsorbent loading breakthrough curves for Example
15.13. Favorable Adsorption Isotherms Sharpen Breakthrough
Broadening of the wave front in Example 15.13 due to
c
q lb benzene transport-rate resistance is summarized in Figure 15.33 by
z, ft ¼

q;
j cF qF ft3 SG plotting the MTZ width for 0:95  cf =cF  0:05 versus
dimensionless time t up to a value of 20, where the front breaks
2 0.373 0.99998 9.28 through the 6-ft-long bed. MTZ broadening increases from
5 0.932 0.99883 9.27 2 feet at t ¼ 6 to 4 feet at t ¼ 20. The rate of broadening
10 1.863 0.96054 8.91 slows as t increases; however, broadening in a deeper bed per-
15 2.795 0.77702 7.21 sisted even at t ¼ 100. This is typical of frontal loading per-
20 3.727 0.46849 4.35 formed with a linear adsorption isotherm (curve A in Figure
25 4.658 0.20571 1.909 15.34a) or with an unfavorable Type III isotherm (curve C in
30 5.590 0.06769 0.628 Figure 15.34a). On the other hand, a favorable Type I Lang-
32.2 6.000 0.03827 0.355 muir or Freundlich isotherm (curve B in Figure 15.34a)
rapidly diminishes wave-front broadening to produce a
Values of q
are plotted in Figure 15.32 and integrated over the ‘‘self-sharpening’’ wave front, as illustrated in Figure 15.29.
bed length to obtain the average bed loading: This has been evaluated by DeVault [129] and others. Solute
Z 6 velocity at the concentration wave front, uc, within a packed

qavg ¼
qdz=6 bed of significant capacity, i.e., K d
1, is obtained by substi-
0
tuting (15-51) into (15-54) and multiplying by u:
3
The result is 5.72 lb benzene/ft of SG, which is 61.6% of the maxi- u
mum loading based on inlet benzene concentration. uc ¼ ð15-85Þ
1  eb 
If the bed height were increased by a factor of 5, to 30 ft, j ¼ 1þ ep K d
161. The ideal time of operation would be 780 minutes or 13 h. eb
With mass-transfer effects taken into account as before, the dimen- Solute velocity in (15-85) is relatively small at lower values of
sionless operating time to breakthrough is computed to be t ¼ 132, cf (i.e., higher Kd) in Figure 15.34a, curve B, but increases as
or breakthrough time is
cf increases. Thus, lagging wave-front regions at higher solute
132 30 concentration move faster than leading wave-front regions at
t¼ þ ¼ 641 minutes
0:206 197 lower solute concentrations, as shown in Figure 15.34b. Self-
sharpening of breakthrough curves via nonlinear Type I
which is 82.2% of the ideal time. This represents a substantial adsorption isotherms mitigates broadening due to consecutive
increase in bed utilization. transport-rate resistances and thus allows more adsorptive bed
capacity to be efficiently utilized.

10
9
8
q, lb benzene/ft3 SG

7
6
5
4
3 τ = 2.0
2 t = 97.1 minutes
1
0
0 1 2 3 4 5 6 Figure 15.32 Adsorbent loading profile for
z, distance through the bed, ft Example 15.13.
C15 09/22/2010 Page 605

§15.3 Kinetic and Transport Considerations 605

q0*

B
le
ab
v or
q* Fa A
r
ea
Lin C e
bl
o ra c
f av
Un Figure 15.34 Effect of shape of isotherm
on sharpness of concentration wave front.
0 c0 z (a) Isotherm shapes. (b) Self-sharpening wave
c
(b)
(a) front caused by a favorable adsorption isotherm.

Scale-Up Using Constant-Pattern Front

Persistent transport-rate resistance eventually limits self-
sharpening, and an asymptotic or constant-pattern front LES

(CPF) is developed. For such a front, MTZ becomes constant

Loading = qF*
LES
and curves of cf=cF and
cb =
cb become coincident. The bed

Loading = 0
depth at which CPF is approached depends upon the non-
LES
linearity of the adsorption isotherm and the importance of
B LES
adsorption kinetics. Cooney and Lightfoot [130] proved the A
existence of an asymptotic wave-front solution, including LUB
LES
effects of axial dispersion. Initially, the wave front broadens
because of mass-transfer resistance and/or axial dispersion.
Analytical solutions for CPF from Sircar and Kumar [131]
and a rapid approximate method based on Freundlich and 1
B
Langmuir isotherms from Cooney [132] are available to esti-
mate CPF concentration profiles and breakthrough curves
using mass-transfer and equilibrium parameters. q/qF
When the constant-pattern-front assumption is valid, it
A
can be used to determine the length of a full-scale adsorbent 0
tb ts te
bed from breakthrough curves obtained in small-scale labora- t, time for adsorption step
tory experiments. This widely used technique is described by
Figure 15.35 Determination of bed length from laboratory
Collins [133] for purification applications. measurements.
Total bed length is taken to be
LB ¼ LES þ LUB ð15-86Þ
the sum of the length of an ideal, equilibrium-adsorption sec- EXAMPLE 15.14 Scale-Up for Fixed-Bed Adsorption.
tion, LES, unaffected by mass-transfer resistance Collins [133] reports the experimental data below for water-vapor
cF Q F t b adsorption from nitrogen in a fixed bed of 4A molecular sieves for
LES ¼ ð15-87Þ
qF rb A bed depth ¼ 0.88 ft, temperature ¼ 83 F, pressure ¼ 86 psia, G ¼
entering gas molar velocity ¼ 29.6 lbmol/h-ft2, entering water con-
where QF is the volumetric feed flow rate, plus a length of tent ¼ 1,440 ppm (by volume), initial adsorbent loading ¼ 1 lb=100
unused bed, LUB, determined by the MTZ width and the lb sieves, and bulk density of bed ¼ 44.5 lb/ft3. For the entering gas
cf=cF profile within that zone, using moisture content, cF, the equilibrium loading, qF, ¼ 0.186 lb H2O/
Le lb solid.
LUB ¼ ðts  tb Þ ð15-88Þ
ts
cexit, ppm cexit, ppm
where Le=ts ¼ the ideal wave-front velocity. The stoichio- (by volume) Time, h (by volume) Time, h
metric time ts divides the MTZ (e.g., CPF zone) into equal
areas A and B as shown in Figure 15.35, and Le=ts corre- <1 09.0 650 10.8
sponds to the ideal wave-front velocity. Alternatively, ts, may 1 9.0 808 11.0
be determined using 4 9.2 980 11.25
Z te   9 9.4 1,115 11.5
cf 33 9.6 1,235 11.75
ts ¼ 1 dt ð15-89Þ
0 cF 80 9.8 1,330 12.0
142 10.0 1,410 12.5
For example, if ts equalizes areas A and B when it is equi- 238 10.2 1,440 12.8
distant between tb and te, then LUB ¼ MTZ/2. A conserva- 365 10.4 1,440 13.0
tive estimate of MTZ ¼ 4 ft may be used in the absence of 498 10.6
experimental data.
C15 09/22/2010 Page 606

606 Chapter 15 Adsorption, Ion Exchange, Chromatography, and Electrophoresis

Determine the bed height required for a commercial unit to be oper- Let ts ¼ the midpoint or (9.76 þ 12.25)=2 ¼ 11 h. The ideal wave-
ated at the same temperature, pressure, and entering gas mass veloc- front velocity ¼ Le=ts ¼ 0.88=11 ¼ 0.08 ft/h. From (15-87), LUB ¼
ity and water content to obtain an exiting gas with no more than 9 0.08(11  9.76) ¼ 0.1 ft. MTZ ¼ 0.2 ft and LB ¼ 1:96 þ
ppm (by volume) of water vapor with a breakthrough time of 20 h. 0:1 ¼ 2:06 ft.

Solution

1;440ð18Þ §15.3.6 Multicomponent Differential

cF ¼ ¼ 0:02592 lb H2 O/lbmol N2
106 Chromatography
QF
G¼ ¼ 29:6 lbmol N2 /h-ft2 of bed cross-section
pD2 =4 Most separation systems discussed thus far perform a binary
Initial moisture content of bed ¼ 0:01 lb H2 O/lb solid split between key components (e.g., LK and HK in distilla-
tion). Chromatography can separate multicomponent mix-
From (15-87), revised for a gas flow rate based on the lbmol of N2 tures into 2+ products, whose elution time, t ¼ zL =f s u,
instead of the volume in ft3 of N2,
increases proportionally to relative thermodynamic partition-
ð0:02592Þð29:6Þð20Þ ing of each species from moving fluid to stationary adsorp-
LES ¼ ¼ 1:96 ft tive phase (15-51). Transport-rate processes (e.g., 15-57 and
ð0:186  0:01Þð44:5Þ
15-58) dilute the product concentration and mix adjacent sol-
Use the integration method to obtain LUB. From the data:
utes. Figure 15.36 illustrates differential chromatography,
Take te ¼ 12.8 h (1,440 ppm) and tb ¼ 9.4 h (9 ppm). also called batch or elution chromatography, in which a feed
By numerical integration of breakthrough-curve data, using (15-88): mixture insufficient to load the sorbent is pulsed into the feed
ts ¼ 10.93 h. end of the column.
From (15-88), An eluant carrier gas or solvent moving at a constant plug-
  flow interstitial velocity u that has little or no affinity for the
10:93  9:40
LUB ¼ ð0:88Þ ¼ 0:12 ft sorbent moves the fraction of solute desorbed into the fluid
10:93
phase (15-54) along the length of the column at solute wave
From (15-86), LB ¼ 1.96 þ 0.12 ¼ 2.08 ft, or a bed utilization of (migration) velocity uc ¼ vu, given in (15-85), as solute read-
1:96 sorbs and desorbs in succession via mass action. Transport-
 100% ¼ 94:2%. rate processes broaden each solute peak as it proceeds along
2:08
Alternatively, an approximate calculation can be made. Let tb, the column. The solute migration velocity, given by (15-85),
the beginning of breakthrough, be 5% of the final ppm, or 0.05 is smaller for solutes with higher affinity, i.e., smaller a cor-
(1,440) ¼ 72 ppm. Using the experimental data, this corresponds to responding to a larger Kd in (15-51). Initially, overlapping
tb ¼ 9.76 h. Let te, the end of breakthrough, be 95% of the final solute peaks are gradually separated by the fractional differ-
ppm, or 0.95(1,440) ¼ 1,370 ppm, corresponding to te ¼ 12.25 h. ence in migration velocities of adjacent, noninteracting

(a) (b) (c) (d) (e)

Feed
end
Solute A
A

B B
C
C

A
A
Distance
into
B
bed

Figure 15.36 Movement of

concentration waves during
Eluant separation in a chromatographic
end
Concentration in bulk fluid column.