Ch.

MASS TRANSFER
BY MIGRATION
AND DIFFUSION
4.1 DERIVATION OF A GENERAL MASS TRANSFER EQUATION

▪ The difference of electrochemical potential over a distance

due to a concentration gradient or potential gradient

▪ A flux of species J occurs to alleviate any difference of electrochemical potential

∂µ i
J i ( x) = −Ci ui Ci Di ∂ µ i
∂x J i ( x) = −
D RT ∂x
ui = i
RT
: Nernst-Einstein relation
4.1 DERIVATION OF A GENERAL MASS TRANSFER EQUATION

▪ Consider the Nernst-Planck equations

Ci Di ∂ µ i
J i ( x) = − + Ci v : if solution is moving
RT ∂x

∂ ∂ 
=−
Ci Di
 ∂x ( RT ln a ) + ( z Fφ ) + Ci v
∂x
i i
RT  When ai = Ci

Ci Di  ∂ ∂ 
=−  ( RT ln C ) + ( z Fφ ) + Ci v
RT  ∂x ∂x
i i


∂Ci ( x) zi F ∂φ ( x)
J i ( x) = − Di − Di Ci + Ci v
∂x RT ∂x
4.1 DERIVATION OF A GENERAL MASS TRANSFER EQUATION

▪ Consider the Nernst-Planck equations

For linear mass transfer

diffusion migration convection

▪ In this chapter, we are concerned with systems in which convection is absent.

Convective mass transfer (Chapter 9)
▪ In an unstirred or stagnant solution with no density gradients
solution velocity (v) = 0
the general flux equation for species j becomes
4.1 DERIVATION OF A GENERAL MASS TRANSFER EQUATION

▪ If species j is charged,
then the flux, Jj, is equivalent to a current density.

▪ If we consider a linear system with a cross-sectional area, A, normal to the axis of mass
flow.
Jj (mol s-1 cm-2) is equal to -ij/zjFA [C/s per (C mol-1 cm2)]
where ij is the current component at any value of x arising from a flow of species j

▪ Then,
4.1 DERIVATION OF A GENERAL MASS TRANSFER EQUATION

id,j : diffusion currents of species j

im,j : migration currents of species j
4.2 MIGRATION

1) In the bulk solution (away from the electrode)

concentration gradients are generally small
the total current is carried mainly by migration

▪ For species j in the bulk region of a linear mass-transfer system having a cross-sectional
Area A,
ij = im,j

▪ Einstein-Smoluchowski equation
relationship between the mobility of species j and the diffusion coefficient
4.2 MIGRATION

▪ For a linear electric field

: where ΔE/l is the gradient (V/cm) arising from the change in potential ΔE over distance l.

▪ The total current in bulk solution is given by

4.2 MIGRATION

▪ The conductance of the solution, L (Ω-1),

the reciprocal of the resistance, R (Ω),
given by Ohm's law

: where к, the conductivity (Ω-1 cm-1) is given by

(discussed in Ch. 2)

▪ The fraction of the total current that a given ion j carries

tj, the transference number of j
4.3 MIXED MIGRATION AND DIFFUSION NEAR AN ACTIVE ELECTRODE

2) Near the electrode,

: an electroactive substance is transported by both diffusion and migration processes.

▪ That current can be separated into diffusion and migration currents

reflect the diffusive and migrational components to the flux of the electroactive species
at the surface:

▪ Note that im and id may be in the same or opposite directions

depending on the direction of the electric field and the charge on the electroactive
species.
4.3 MIXED MIGRATION AND DIFFUSION NEAR AN ACTIVE ELECTRODE

▪ Examples of three reductions

: a positively charged, a negatively charged, and an uncharged substance

▪ The migrational component is always in the same direction as id for cationic

species reacting at cathodes and for anionic species reacting at anodes.

▪ It opposes id when anions are reduced at cathodes and when cations are oxidized at
anodes.
4.3.1 Balance Sheets for Mass Transfer During Electrolysis

▪ For many electrochemical systems, the mathematical treatments can be simplified if the
migrational component to the flux of the electroactive substance is made negligible.
discuss the conditions under which that approximation holds.

▪ Although migration carries the current in the bulk solution during electrolysis,
diffusional transport also occurs in the vicinity of the electrodes, because concentration
gradients of the electroactive species arise there.
Under some circumstances, the flux of electroactive species to the electrode is due
almost completely to diffusion.
To illustrate these effects, let us apply the "balance sheet" approach to transport in
several examples.
4.3.1 Balance Sheets for Mass Transfer During Electrolysis

Example 4.1
▪ Consider the electrolysis of a solution of hydrochloric acid at platinum electrodes

▪ Since the equivalent ionic conductance of H+, λ+, and of Cl-, λ-, relate as λ+ ≈ 4λ-

tH+ = 0.8 and tCl- = 0.2

λH + 4
tH + = = = 0 .8
λH + λCl
+ − 4 +1
4.3.1 Balance Sheets for Mass Transfer During Electrolysis

▪ Assume that a total current equivalent to 10e per unit time is passed through the cell,
producing five H2 molecules at the cathode and five Cl2 molecules at the anode.
▪ The total current is carried in the bulk solution by the movement of 8H+ toward the
cathode and 2C- toward the anode
t+ = 0.8 and t- = 0.2

Migration
4.3.1 Balance Sheets for Mass Transfer During Electrolysis

▪ To maintain a steady current,

10H+ must be supplied to the cathode per unit time,
so an additional 2H+ must diffuse to the electrode, along with 2Cl-
▪ Similarly at the anode, to supply 10Cl- per unit time,
8Cl- must arrive by diffusion, along with 8H+.

Diffusion
4.3.1 Balance Sheets for Mass Transfer During Electrolysis

▪ Thus,
the current for H+: id = 2, im = 8 at the cathode
the current for Cl-: id = 8, im = 2 at the anode
The total current, i = 10.
: the migration current is in the same direction as the diffusion current

im

id
4.3.1 Balance Sheets for Mass Transfer During Electrolysis

Example 4.2
▪ Consider the electrolysis of a solution of 10-3 M Cu(NH3)42+, 10-3 M Cu(NH3)2+, and 3
X 10-3 M Cl- in 0.1 M NH3 at two Hg electrodes
4.3.1 Balance Sheets for Mass Transfer During Electrolysis

▪ Assuming the limiting

equivalent conductances of
all ions are equal, that is,

zCu2+ cCu2+ 2 2 1
tCu2+ = = = =
zCu2+ cCu2+ + zCu+ cCu+ + zCl− cCl− 2 +1+ 3 6 3
그림을 표시할 수 없습니다 .

1 1
tCu+ = =
그림을 표시할 수 없습니다 .
2 +1+ 3 6

1 1 1
tCl− = 1 − tCu2+ − tCu+ = 1 − − =
3 6 2
4.3.1 Balance Sheets for Mass Transfer During Electrolysis

▪ With an arbitrary current of 6e per unit time being passed,

the migration current in bulk solution is carried by movement of one Cu(II) and one
Cu(I) toward the cathode, and three Cl- toward the anode.

▪ Thus,
Cu(II): id = 5, im = 1 at the cathode
Cu(I): id = 7, im = -1 at the anode
4.3.1 Balance Sheets for Mass Transfer During Electrolysis

Example 4.3
▪ Let us consider the same cell as in Example 4.2, except with the solution containing
0.10 M NaClO4 as an excess electrolyte
4.3.1 Balance Sheets for Mass Transfer During Electrolysis

▪ Assuming the limiting equivalent

conductances of all ions are equal,
that is,

λCu = λCu = λCl = λNa = λClO

2+ + − + −
4

zCu2+ cCu2+ tCu + = 0.00485

tCu2+ =
zCu2+ cCu2+ + zCu+ cCu+ + zCl− cCl− + zNa+ cNa+ + zClO − cClO − tCl − = 0.0146
4 4

t Na + = 0.485
−3
2 ×10
= −3 −3 −3 −1 −1
= 0.0097 tClO − = 0.485
(2 ×10 ) + (1×10 ) + (3 ×1×10 ) + (1×10 ) + (1×10 ) 4
4.3.1 Balance Sheets for Mass Transfer During Electrolysis

(-) (+)
6Cu2+ + 6e → 6Cu+ 6Cu+ - 6e → 6Cu2+
6e 6e
6Cu2+ 6Cu2+
6Cu+ 6Cu+
[Migration]
0.0291 Cu2+
0.0291 Cu+
0.0873 Cl-
2.91 Na+
2.91 ClO4-
[Diffusion] [Diffusion]
5.97 Cu2+ 5.97 Cu2+

6.03Cu+ 6.03 Cu+

0.0873 Cl- 0.0873 Cl-

2.91 Na+ 2.91 Na+

2.91 ClO4- 2.91 ClO4-

4.3.1 Balance Sheets for Mass Transfer During Electrolysis

▪ Thus, when the solution contains additional 0.10 M NaClO4 as an excess

electrolyte (supporting electrolyte)
i=6
Cu(II): id = 5.97, im = 0.03 at the cathode
Cu(I): id = 6.03, im = -0.03 at the anode

▪ Note that, when the solution contains no supporting electrolyte,

i=6
Cu(II): id = 5, im = 1 at the cathode
Cu(I): id = 7, im = -1 at the anode

Thus, the addition of an excess of nonelectroactive ions (a supporting electrolyte)

nearly eliminates the contribution of migration to the mass transfer of the
electroactive species.