The Rayleigh Equation Revisited

What to Do When Alpha (α) Isn’t Constant

Robert G. Kunz

RGK Environmental Consulting, L.L.C.

Hillsborough, North Carolina

Paper 118 b

2012 AIChE Spring Meeting

April 1-5, 2012
Houston, Texas

Abstract

Integration of the Rayleigh Equation for batch distillation in closed analytical form has
heretofore required that relative volatility (α) be assumed constant. A new technique
presented in this paper produces a continuous analytical function for the Rayleigh Equation
integral whether α is constant or not. The newly developed equation reduces algebraically to
the traditional function when α is absolutely constant.
Both the traditional expression at constant α and the new equation are evaluated
against numerical integration for a number of ideal and non-ideal binary vapor-liquid
equilibrium (VLE) systems. The new method has proven especially useful for the non-ideal
systems investigated, in which α varies widely with composition.
The method is then utilized successfully in a computer-simulated batch distillation of an
ethanol-water solution at 1 atm pressure, a popular student laboratory exercise and one of the
original systems studied by Lord Rayleigh. Results compare well with expected values.
A summary of VLE relationships for ideal and non-ideal systems plus temperature-
dependent methods for analysis of constituent composition specific to the ethanol-water
system are discussed in separate appendices. These latter include, among others, refractive
index measurements assembled from various sources and estimates of liquid density for
ethanol-water mixtures extended beyond the range of published data.
The paper consists of new material supported by background information tutorial in
nature.

Keywords. Alcohol “Proof” Levels; Boiling Point Curve; Density of Ethanol-Water Solutions;
Ethanol-Water Flash Points; Ethanol-Water Refractive Index; Gas Chromatography (GC);
Numerical Integration; Rayleigh Equation; Simpson’s Rule; Systems: Acetone-Water, Benzene-
Toluene, Ethanol-Water, Ethylene Dichloride (EDC)-Toluene; Vapor-Liquid Equilibrium.
 Disclaimer
(“Some restrictions apply; batteries not included; your mileage may vary”)

The following caveat applies: The information contained herein is offered in good faith
but without guarantee, warranty, or representation of any kind (expressed or implied) as to its
usefulness, correctness, completeness, or fitness for any particular purpose. The user
assumes all risk for its implementation and should seek independent professional verification
of its accuracy. The author assumes no responsibility and shall not be liable for any loss of
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use of any of the information contained in this presentation, be it oral or written. Any
statements concerning design, construction, operation, what constitutes regulatory
compliance, and/or how to achieve such compliance should not be construed as
recommendations on the part of the author and/or his organization.

Introduction

Distillation is a separation technique utilizing the difference in composition of a liquid

and the vapor in equilibrium with that liquid [1,2]. Simple, or differential, distillation employs
a single equilibrium stage to effect that separation. A batch distillation operates on an
unreplenished charge of material being boiled off from a still pot and then condensed as
product in another vessel. It is an unsteady-state process, where the amount and composition
of feed and product vary with time. The charge of feed material boils away and gradually
becomes richer in the less volatile/higher boiling component(s). The condensed vapor product
(distillate) increases in volume and is enhanced in the more volatile/lower boiling
component(s). These are also known as heavy and light components, respectively.
This paper discusses the simple, differential batch distillation of a two-component
(binary) system, in which none of the condensed vapor is returned, or refluxed, to the still pot,
a configuration referred to by one author as a Rayleigh Distillation [3]. A typical laboratory
setup is shown in Figure 1. It is a favorite student experiment in chemistry and engineering
laboratories throughout the world.
Simple batch distillation is not practiced extensively in industry. It is used commercially
for separating small quantities of high-value chemicals, when high purities are not required, as
a preliminary step to be followed by further processing, for separations where the equipment
requires frequent cleanout, or where the feed mixture is very easy to separate [1,2]. A rule of
thumb for ease of separation is that the feed components differ in atmospheric boiling point by
at least 25°C [4, pp.167-168], 30-40°C [5, p.61], or 75°C [6, p.89; 7, p.89]. Whatever the
precise numerical value, the intent is that the lighter component be much more easily
vaporized in preference to the heavier component in order to distill a decent amount of high-
purity product before too much of the feed material boils away.

2
 Figure 1. Typical Laboratory Setup for Simple, Differential Batch Distillation

The Rayleigh Equation

Simple batch distillation without reflux is described by the Rayleigh Equation, first
advanced by Lord Rayleigh in 1902 [8]. In the derivation below [2; 9, p.580-581; 10; 11],
one begins with an overall mass balance:

F=W+D (1)

where F = total moles of feed in the still pot/distilling flask at time (t) = 0
W = total moles left in still pot (bottoms)
and D = total moles boiled off and condensed into the distillate at the same
point in time

and a component mass balance:

FxF = WxW + DxD (2)

where x = the mol fraction of the more volatile component (MVC) in the feed (F),
bottoms (W), and distillate (D).

Unsteady-state mass balance over a differential amount of time yields:

initial amt amt remaining amt being

in still in still vaporized
Wx = (W – dW) (x – dx) + ydW (3)

3
 in which y denotes the mol fraction of the more volatile component in the vapor
phase in equilibrium with its mol fraction in the liquid phase (x).

Expanding the equation above, one obtains:

Wx = Wx – xdW – Wdx + (dW) (dx) + ydW (4)

Simplifying, rearranging, and neglecting the second order differential (dW) (dx) results in:

Wdx = ydW – xdW (5)

From which:
W xW
∫dW/W = ∫dx/(y – x) (6)
F xF

and upon integration of the left-hand side between the stated limits, one obtains the Rayleigh
Equation:
xW
ln (W/F) = ∫dx/(y – x) (7)
xF

The integrand is not a simple function of x. At constant pressure, every value of x

corresponds to a different temperature and hence vapor pressure. Except for ideal systems
which follow Raoult’s Law in the liquid and the Ideal Gas Law and Dalton’s Law in the vapor, y
is also not related to x in a simple manner.
The standard procedure is to evaluate the integral numerically or graphically,* or to
come up with a suitable substitution in an attempt to produce an analytical solution in closed
mathematical form. Numerical integration involves repeatedly fitting the numerical values of
the integrand with a mathematical function over short ranges, integrating those functions
mathematically, and then summing. This numerical fitting and summing process results in a
series of discrete values, rather than a continuous function.

Relative Volatility

An analytical approach makes use of a parameter known as the relative volatility and
denoted by the symbol alpha (α ). It is defined as the ratio of the vapor-phase mol fraction of
the more volatile component, denoted as component 1, to its liquid-phase mol fraction, this

*
In previous times, in addition to counting boxes under the curve of the integrand or using a planimeter device
[12], the curve was drawn carefully on high quality graph paper of uniform density and then was weighed and
calibrated against a rectangular section of that same graph paper. This method is noted in Ref. [13, p.75].

4
ratio being divided by a similar ratio for the less volatile component 2. For a successful
separation by distillation, α must be greater than 1.0. Mathematically:

α = (y1/x1) / (y2/x2) = (y1/x1) / [(1 – y1) / (1 – x1)] (8)

The alternate expression above for α comes about since y2= (1 – y1) and x2 = (1 – x1).
Rearranging Equation (8), one obtains for y1:

y1 = α x1 / [1 + x1 (α – 1)] (9)

and for 1 / (y1 – x1):

1 = 1 + (α – 1) x1 = 1 + 1 (10)
(y1 – x1) (α – 1) x1 (1 – x1) (α – 1) x1 (1 – x1) (1 – x1)

The relative volatility is evaluated at every point along the vapor-liquid equilibrium x-y
curve. In general, relative volatility is not absolutely constant across the entire range of x,
even for the most ideal of systems. However, in those cases where α is considered to be
reasonably constant, it is taken outside the integral sign to allow the Rayleigh Equation
integral to be evaluated analytically by a closed-form mathematical function.

Now substituting the extreme right-hand side of Equation (10) into Equation (7) with
the assumption of a constant α yields a new set of integrals for ln (W/F):

xW xW
ln (W/F) = 1 (12)
(α – 1)
∫ dx1
x1 (1 – x1)
+
∫ dx1
(1 – x1)
xF xF

in which each term can be integrated analytically [14, #40 and #29] to yield:

ln (W/F) = 1 { α ln [(1 –xF)/(1 – xW)] + ln (xW/xF) } (13)

(α – 1)

A mathematically equivalent form [2] is:

(14)
ln (W/F) = 1
(α – 1) {
ln
[ x(1 (1–x–) xx ) ] +
F
F W

W
ln
[ (1 –xF)
(1 – xW)
] }

5
 The percent of original charge remaining in the still pot is then given by:

100 (W/F) = 100 exp [ln (W/F)] (15)

Relative Volatility from Vapor-Liquid Equilibrium (VLE) Relationships

Equations for low-pressure binary vapor-liquid equilibrium (VLE) are summarized in

Appendix A. Important variables are temperature (t), total pressure (PT), partial pressure (p),
vapor pressure (pvap), liquid-phase (x) and vapor-phase (y) mol fractions of each component,
plus so-called activity coefficients (γ’s) when a given system cannot be considered to be ideal.
Pure-component vapor pressures are conveniently calculated using the Antoine
Equation:

log10 pvap (mmHg) = A – B / [t (°C) + C] (16)

Antoine constants are listed in Table 1 for chemicals of interest in this paper.
By manipulation of these equations, the relative volatility (α) in Equation (8) can be
calculated at constant pressure for any temperature or the corresponding value of composition
along the VLE curves from a ratio of vapor pressures and, if necessary, activity coefficients:

α = (y1/x1) / (y2/x2) = (p1vap γ1) / (p2vap γ2) (17)

in which the subscript 1 denotes the more volatile component (MVC).

A system can be considered to be ideal when γ1 = γ2 = 1.0 for all values of x. Relative
volatilities for a number of systems – some ideal, some non-ideal – are listed in the second
column of Table 2. Alpha is not perfectly constant for any of these systems across the range
of interest. In general, α shows a wider variation in a system exhibiting non-ideal behavior
than for an ideal system.

Other Quantities Are Determined by Material Balance

The Rayleigh Equation in whichever form – for example, Equation (7), Equation (13), or
Equation (14) – describes the relationship between W and xW for a given starting value of F.
The other quantities of interest are determined by material balance from Equations (1) and (2)
and Equation (18) below, derived from them:

xD = ( FxF – WxW ) / ( F – W ) (18)

The composition of the first drop of distillate, when W equals F and xW = xF and the
expression in Equation (18) becomes mathematically indeterminate, is provided by the x-y
equilibrium curve, as is the instantaneous composition of the distillate at every moment during
the distillation. The situation will become clearer by way of the following example.

6
 (a)
Table 1. Antoine Equation Constants Used in the Analysis

Chemical Name Formula MW (b) BP Applicable A B C

and Synonyms (˚C) (c) Range(˚C)

Acetone H3CCOCH3 58.08 56.2 –– 7.02447 1161.0 224

2-Propanone
Propanone
Dimethyl Ketone

Benzene C6H6 78.12 80.1 8 to 103 6.90565 1211.033 220.790

Benzol

1-Butanol C4H9OH 74.12 117.7 15 to 131 7.47680 1362.39 178.77

n-Butyl Alcohol
Butanol-1
n-Propyl Carbinol

Ethanol C2H5OH 46.07 78.3 –– 8.04494 1554.3 222.65

Ethyl Alcohol
Grain Alcohol
Methyl Carbinol

Ethyl Benzene C6H5C2H5 106.17 136.2 –– 6.95719 1424.255 213.206

Phenyl Ethane

Ethylene ClCH2CH2Cl 98.96 83.5 –– 7.18431 1358.46 232.2

Dichloride
(EDC)
Ethylene Chloride
1,2-
Dichloroethane

Isobutyl (CH3)2CHCH2OH 74.12 107.9 20 to 115 7.32705 1248.48 172.92

Alcohol
Isobutanol
2-methyl-1-
propanol
2-methyl-
propanol-1
Isopropyl Carbinol

Methanol CH3OH 32.04 64.7 –20 to +140 7.87863 1473.11 230.0

Methyl Alcohol
Wood Alcohol
Carbinol

Methyl Ethyl H3CCOC2H5 72.12 79.6 –– 6.97421 1209.6 216

Ketone (MEK)
2-Butanone
Butanone

7
 (a)
Table 1 (continued). Antoine Equation Constants Used in the Analysis

Chemical Name Formula MW (b) BP Applicable A B C

and Synonyms (˚C) (c) Range(˚C)

1-Propanol C3H7OH 60.11 97.2 –– 7.99733 1569.70 209.5

Propanol
Propanol-1
n-Propyl Alcohol
Ethyl Carbinol

2-Propanol H3CCH(OH)CH3 60.11 82.4 0 to 113 6.66040 813.055 132.93

Propanol-2
Isopropanol
Isopropyl Alcohol
Dimethyl Carbinol

n-Propyl C6H5C3H7 120.20 159.2 –– 6.95142 1491.297 207.140

Benzene
Propyl Benzene

Toluene C6H5CH3 92.14 110.6 6 to 137 6.95464 1344.800 219.482

Methyl Benzene
Phenyl Methane

Water H2O 18.02 100.00 0 to 60 8.10765 1750.286 235.0

Hydrogen Oxide 60 to 150 7.96681 1668.21 228.0

Notes:
(a) Antoine Constants from “Lange’s Handbook of Chemistry”, 11th Edition [15, Table 10-10], except
for toluene and the butanols, which are from the 12th Edition [16, Table 10-8].
(b) Molecular Weight (MW) from “Handbook of Chemistry and Physics,” 62nd Edition [17, pp. C-65 to
C-576], rounded for water [17, p.B-105] to 2 decimal places.
(c) Boiling point (BP) at 760 mmHg from “Lange’s Handbook,” 11th-13th Editions [15, p.4-59 and
Table 7-4; 16, p.4-59 and Table 7-4; 18] or “Handbook of Chemistry and Physics,” 62nd Edition [17,
pp. C-65 to C-576, p.B105], chosen to be the most consistent with boiling point calculated using the
Antoine Constants listed.

8
 Illustrative Example. An illustrative example for batch/differential distillation is
presented in McCabe and Smith [11]. In this example, an ideal solution of 50 mol percent
benzene (MVC) and 50 mol percent toluene is subjected to batch distillation at 1 atm pressure.
(Refer again to Figure 1.) Instructions are to take 1/α as constant at 0.41 for this system over
the entire range of the distillation and to plot the calculated results for temperature and
benzene composition at various locations against the mol fraction of charge distilled (D/F).
Instantaneous mol fraction benzene is desired in the still (xW) and in the vapor leaving the still
(y1), along with the cumulative average benzene composition in the distillate (xD).
Temperatures and compositions newly calculated here via electronic spreadsheet and
plotted in Figures 2 and 3 reproduce the curves in the cited reference [11]. Calculations
performed here assume the familiar benzene-toluene system to be ideal, that is, to follow
Raoult’s Law. Pure component vapor pressures were computed using the Antoine Equation
with constants listed for benzene and toluene in Table 1. Any differences between the present
computations and the tabulated values accompanying the plotted curves in Ref. [11] are
Inconsequential and can be attributed to the graphical procedure employed there.

110
BOILING TEMPERATURE
IN STILL ( C)

100
o

90

80
0.0 0.2 0.4 0.6 0.8 1.0
MOLES DISTILLED / MOLES CHARGED

Figure 2. Calculated Temperature during Batch Distillation of 50-50 Mol Fraction

Benzene-Toluene Solution at 1 Atm Pressure

Atmospheric boiling temperature for an ideal mixture depends on the vapor pressures
and mol fractions of the pure components. The temperature in the still pot (Figure 2), initially
just above 92 °C, approaches 110.6 °C, the boiling point of toluene present by itself in the still
pot at the end of the distillation. Benzene concentration in the still pot (bottoms) approaches
zero as the distillation proceeds (Figure 3). The last drop in the bottoms is pure toluene,
although for safety reasons one would not want actually to distill the mixture to dryness.
Benzene in the distillate, highest at its initial instantaneous value, decreases somewhat in the

9
cumulative distillate as more toluene comes over. Instantaneous benzene concentration in the
vapor condensing into the distillate declines more rapidly as the benzene is boiled off, leaving
the still-pot composition richer in toluene.

1.0

TOP CURVE: AVERAGE DISTILLATE

MOL FRACTION BENZENE (MVC)

0.8 MIDDLE CURVE: INSTANTANEOUS VAPOR

0.6

0.4

0.2 BOTTOM CURVE: LIQUID IN STILL

0.0
0.0 0.2 0.4 0.6 0.8 1.0

MOLES DISTILLED / MOLES CHARGED

Figure 3. Calculated Compositions from Rayleigh Equation during Batch Distillation of

50-50 Mol Fraction Benzene-Toluene Solution at 1 Atm Pressure

Testing of Numerical Evaluation of the Integral

Using as the basis a starting value of 0.5 mol fraction in the distilling flask inspired by
the illustrative example discussed above, results from numerical integration of Equation (12)
by Simpson’s Rule were compared with the exact values obtained from Equation (13). Since
Equation (13) with an absolutely constant value of α is an exact analytical integration of
Equation (12), it becomes a primary standard against which to test the accuracy of a
numerical integration approximation.
Simpson’s Rule for numerical integration [12]:

Area under Curve ≈ (Δ abscissa / 3) (ordinate 1 + 4 × ordinate 2 + ordinate 3) (19)

approximates a small section of a curve with a parabola and computes the area under that
parabola (an integral). Summing the areas from its repeated application to one adjacent
section after another produces a series of values for cumulative area.† The process is
terminated when the final desired abscissa is attained.

†
In the present case, the area is negative since the integration is conducted from right (higher values of the
abscissa) to left (lower values). That makes sense because logarithms are negative when the argument, as here,
is a fraction less than 1.0.

10
 It has been found here that repeated application of Simpson’s Rule in steps of 0.02 mol
fraction and an interval size of 0.01 mol fraction (the Δ abscissa above) reproduces the exact
integral for ln (W/F), even when exponentiated, to a degree better than can be measured
experimentally. This situation applies down to a mol fraction bordering on 0.02. Narrowing
the interval size at the lower end of the mol fraction range, where the integrand begins to
increase rapidly, maintains the accuracy of the numerical integration even further. Eventually,
however, the integrand grows without bounds as mol fraction remaining in the still pot
approaches zero.
Simpson’s Rule numerical integration (the points) is compared with Equation (13) (the
curves) in Figures 4 and 5 for typical values of constant α. The curves pass directly through
the points, and one cannot detect any difference between points and curve at the scale of the
figures. The same is true for even greater values of α, although those curves are not shown
here. Note that the amount of bottoms remaining in the distilling flask drops steeply for the
smallest value of α and begins to decline more slowly with each increase in α. The conclusion
is that numerical integration with a properly chosen interval size is coincident with the results
of Equation (13), the exact solution of Equation (12) with a constant α.

100

PARAMETER: RELATIVE VOLATILITY (ALPHA)

80
PERCENT OF ORIGINAL
CHARGE REMAINING

60

40 2.2

1.7
20 1.2

0
0.00 0.10 0.20 0.30 0.40 0.50

MOL FRACTION OF MVC IN BOTTOMS

Figure 4. Comparison of Numerical Integration with Exact Solution of Rayleigh Equation at

Constant α (Lines Analytical, Points Numerical) – Low Values of α

11
 100

PARAMETER: RELATIVE VOLATILITY

(ALPHA)
80 TOP CURVE 8.0
MIDDLE CURVE 5.0
PERCENT OF ORIGINAL

BOTTOM CURVE 3.5

CHARGE REMAINING

60

40

20

0
0.00 0.10 0.20 0.30 0.40 0.50

MOL FRACTION OF MVC IN BOTTOMS

Figure 5. Comparison of Numerical Integration with Exact Solution of Rayleigh Equation at

Constant α (Lines Analytical, Points Numerical) – Higher Values of α

A New Twist

Alpha need not necessarily be constant to come up with a solution to Equation (7) as a
closed-form analytical expression. A less restrictive condition is that the factor 1/(α – 1) can
be fitted by a quadratic function of mol fraction (x):

1 / (α – 1) = a x2 + b x + c (20)

over the range of interest, in much the same way as the expression of gaseous heat capacities
by an empirical power series in temperature. With 1/(α – 1) so represented, the integrand of
Equation (10) becomes a new series of terms in the second equation below, each having an
analytical solution upon integration term by term [14, #32, #29, #40]:

xW xW
ln (W/F) = 2 (21)
∫ (a x1 + b x1 + c) dx1
X1 (1 – x1)
+
∫ dx1
(1 – x1)
xF xF

12
 xW xW xW
ln (W/F) = (22)
∫ a x1 dx1
(1 – x1)
+
∫ (1 + b) dx1
(1 – x1)
+
∫ c dx1
x1(1 – x1)
xF xF xF

When the smoke clears and the dust settles, one is left with the following:

ln (W/F) = a (xF – xW) + (1 + a + b + c) ln [(1 – xF) / (1 – xW)] + c ln [ xW / xF ] (23)

This form too is an exact solution to Equation (7). The only source of error is the goodness of
fit of the quadratic function in Equation (20). When α is constant for all values of composition,
a = b = 0, c = 1/ (α – 1) identically, and Equation (23) reduces to Equation (13).

Testing of the New Function against Numerical Integration

A number of real vapor-liquid equilibrium systems were chosen to test how well
Equation (23) works in practice. These are broken up in Table 2 between Systems Taken to
Be Ideal and Non-Ideal Systems, where coefficients are tabulated for Equation (20) over the
range of α from x1 = 0.02 to x1 = 0.5. Chemical synonyms for the individual constituents of
these systems can be found in Table 1.
Here, the previously validated Simpson’s Rule integration is the standard, using the
same interval size found suitable for the numerical integration of Equation (12) with constant
α. The overall conclusion is that agreement of the newly derived Equation (23) with numerical
integration is as good as in the previous graphs where α is constant (Figures 4 and 5). In
addition, one has a continuous function in Equation (23) as opposed to numerical integration,
which is valid only at discrete points. Results are discussed separately below for the Ideal and
the Non-Ideal systems.
Ideal Systems. The system ethylene dichloride (EDC)-toluene, midway down the
listing in Table 2, is representative of the ideal systems investigated. Percent of initial charge
remaining in the still pot on a molar basis is plotted in Figure 6 against mol fraction EDC. The
discrete points trace the numerical integration of Equation (7) or Equation (12) with a variable
1/(α – 1) inside the integral sign. The continuous curve through the points is the integrated
Equation (23) with values of a, b, and c for this system from Table 2.
The upper and lower curves represent the integrated Equation (13) with two different
constant alphas bracketing the range of values in Table 2. The range has been expanded a bit
to widen the difference between the curves. In this case, inserting the arithmetic average of
the upper and lower alphas into the integrated Equation (13) for constant α would result in a
curve virtually coincident with the middle curve calculated from Equation (23), passing through
the points from numerical integration of Equation (21). Results from averaging of α are better
for the Ideal Systems above EDC-toluene in Table 2 and not quite so good for those following.
However, Equation (23) using proper values of a, b, and c was found to work regardless.

13
 Table 2. Coefficients of 1/ (α – 1) Function

System Range of α from 1/(α-1) = ax12 + bx1 + c

x1 = 0.02 to 0.5 a b c

Systems Taken as Ideal:

Benzene(1) – EDC(2) 1.108 – 1.110 0.0249847 –0.414753 9.26702

Ethanol(1) – 2-Propanol(2) 1.163 – 1.167 –0.00895551 –0.344138 6.15889

Methanol(1) – Ethanol(2) 1.66 – 1.71 0.077259 –0.251001 1.51560

Ethanol(1) – 1-Propanol(2) 2.03 – 2.11 0.0549304 –0.172904 0.977133

Acetone(1) – MEK(2) 2.09 – 2.17 0.0564870 –0.168884 0.924787

EDC(1) – Toluene(2) 2.20 – 2.26 0.0381021 –0.112977 0.838362

Benzene(1) – Toluene(2) 2.36 – 2.49 0.072534 –0.178451 0.741501

Methanol(1) – 1-Propanol(2) 3.16 – 3.65 0.125203 –0.241046 0.466673

Ethanol(1) – 1-Butanol(2) (a) 4.0 – 4.6 0.090100 –0.162840 0.338006

Benzene(1) – Ethyl Benzene(2) 4.3 – 5.3 0.138854 –0.210900 0.302701

Methanol(1) – Isobutyl Alcohol(2) 4.4 – 5.5 0.129731 –0.215647 0.297965

Benzene(1) – n-Propyl Benzene(2) 7.1 – 10.7 0.183439 –0.217725 0.167047

Non-ideal Systems: (b)

Methanol(1) – Water(2) 7.6 – 3.6 0.228138 0.363189 0.144093

Ethanol(1) – Water(2) 9.8 – 2.0 3.5716 –0.07977 0.136641

.
Acetone(1) – Water(2) 28.6 – 5.0 0.91519 –0.04840 0.039525

Notes:

(a) May not be strictly ideal, but the ideal x-y curve is bracketed by the data of two different
investigators [19,20].
(b) Coefficients A1-2 and A2-1 of van Laar Equations for Methanol-Water and Acetone-Water given
in Perry’s 3rd Edition [9, p.528] are 0.36, 0.22, and 0.89, 0.65, respectively (log10 based) or 0.83,
0.51 and 2.05, 1.50 (ln based). For the Ethanol-Water system, coefficients of 0.68381, 0.41724
(log10 based) or 1.5745, 0.9607 (ln based) were determined [21] by fitting the azeotrope of 89.43
mol % ethanol and 10.57 mol % water at 78.15°C [9, pp.631,633] using vapor pressures calculated
from the ethanol and water Antoine Constants of Table 1.

14
 Non-Ideal Systems. Of the three non-ideal systems evaluated, acetone-water
displays the widest variation in α. Accordingly, this system has been picked to illustrate how
Equation (23) can deal with a widely varying α. Amount of solution remaining in the distilling
flask (W) is shown in Figure 7 as a function of acetone concentration (xW). As in Figure 6, the
points represent the results of numerical integration.
The middle curve drawn through those points is calculated from Equation (23) using the
a, b, and c constants from Table 2 for this system. The curve provides an excellent fit to the
numerically integrated points even though the 1/(α – 1) fit itself is not quite so good as any of
those obtained for the ideal systems. The fit of 1/(α – 1) tends to improve for shorter ranges
of xW. If a better fit of 1/(α – 1) is needed, the range of xW can be subdivided into smaller
sections to piece together the integration all the way from the starting point to the desired
final composition. Alternatively, a cubic term can be added in the fit of 1/(α – 1) against
composition. In that case, Equation (22) would then contain an additional term, whose
integrated form can be found in Ref. [14, #36]. That exercise is left to the reader.
The upper and lower curves in the figure depict the analytically integrated function of
Equation (13) with α constant at each of its extreme values. Clearly neither one of these
curves falls anywhere near the middle curve and the points from numerical integration, and it
is not clear a priori how to average the α values to achieve such a fit. In addition, with values
of α so far apart, it turns out that even the best “average” α utilized in Equation (13) does not
reproduce the correct track of the middle curve and its associated points. Once again, use of
Equation (23) with proper values of a, b, and c is the method of choice to obtain an analytical
expression relating total moles remaining in the distilling flask with molar composition.

100
LEGEND:

TOP CURVE ALPHA = 2.4

80
MIDDLE CURVE VARIABLE ALPHA
PERCENT OF ORIGINAL
CHARGE REMAINING

BOTTOM CURVE ALPHA = 2.1

60
POINTS FROM NUMERICAL INTEGRATION

40

20

0
0.00 0.10 0.20 0.30 0.40 0.50
MOL FRACTION EDC (MVC) IN BOTTOMS

Figure 6. Newly Derived Function vs. Numerical Integration – EDC-Toluene at 1 Atm

15
Testing of the New Function in Simulated Laboratory Experiment

To demonstrate the ease of use of the newly developed technique, batch distillation of
a pure ethanol-water solution was simulated at a standard atmospheric pressure of 760 mmHg
(1.013 bar). This is a favorite system employed in batch distillation as a student laboratory
exercise and one of the systems studied by Lord Rayleigh [8].

100

80
PERCENT OF ORIGINAL
CHARGE REMAINING

60

40 LEGEND:

TOP CURVE ALPHA = 28.6

MIDDLE CURVE VARIABLE ALPHA
20
BOTTOM CURVE ALPHA = 5.0

POINTS FROM NUMERICAL INTEGRATION

0
0.00 0.10 0.20 0.30 0.40 0.50
MOL FRACTION ACETONE (MVC) IN BOTTOMS

Figure 7. Newly Derived Function vs. Numerical Integration – Acetone-Water at 1 Atm

The Laboratory Experiment. An initial 200-mL charge of a pure ethanol-water solution,

20% ethanol by volume (at 60 °F, 15.56 °C) in an apparatus similar to Figure 1 at standard
atmospheric pressure (760 mmHg, 1.013 bar) is typical [6,7,22]. Volume of the distilling flask
is some 500 millimeters (mL). Boiling stones are added to prevent bumping [4, pp.146,170].
Samples of distillate and perhaps still bottoms are taken as every 5 mL of distillate is
collected, and temperature is read at such times. The experiment is over when 50 mL of
distillate has been collected. Sometimes the total collected distillate is subjected to further
purification in a second step using a separate refluxed fractionating column. This experiment
has also been conducted using denatured alcohol and water, fermented sugar solutions, and
wine being distilled into brandy.
The student is cautioned to wait to read the temperature until droplets condense on the
thermometer bulb [4, p.172]. The experiment then proceeds drop by drop. The
recommended distilling rate varies, but numbers in the range of 2 [6,7] to 10 [4, p.171] drops
per minute are stated. At an average 6 drops per minute (1 drop every 10 seconds), and 20
drops to the millimeter, collection of 50 mL would take just under 3 hours, exclusive of

16
analytical time. This would allow completion of the first step of the experiment within a typical
standard laboratory period.
Quantitative analysis is done by refractive index, density, or possibly gas
chromatography. A qualitative analysis for ethanol in the distillate is often conducted by
attempting to ignite several drops of solution on a watch glass, since ethanol and its aqueous
solutions greater than about 50 % by volume are combustible at room temperature [23].
These methods are reviewed in some detail in Appendix B. Whatever the method of analysis,
it is good practice to prepare one’s own calibration curve(s) using the materials and analytical
equipment employed rather than relying exclusively on literature data.
The Simulation. To demonstrate the ease of use of the technique, a computer
simulation of the batch distillation laboratory experiment described above was conducted. The
simulated experiment is valid only for a binary mixture of pure ethanol and water, with an x-y
VLE curve as in Figure A3. It does not apply quantitatively to denatured alcohol containing
methanol and other ingredients, fermented sugar solutions, or wine being distilled into brandy.
The simulation begins when equilibrium becomes established as the temperature in the
laboratory experiment stabilizes and the first drop of liquid falls into the receiving vessel. The
simulation also assumes that the distillation takes place slowly enough so as to maintain true
vapor-liquid equilibrium with a drop of liquid condensed on the thermometer at all times.
Results are summarized in Table 3. Distilling flask temperature, amount of bottoms
remaining and distillate formed, plus composition of bottoms and distillate are recorded at
every 5 mL of liquid distillate collected at 20 °C. There are two ways to conduct such an
experiment: save each 5-mL increment separately for individual analysis, or let the distillate
accumulate as a composite sample. In the latter case, the distillate composition would follow
the trend of the relationship depicted in Figure 3 for benzene-toluene in the illustrative
example above, but with different chemicals. Results from the simulation are listed both ways
in Table 3.
Gram moles of bottoms (W) and mol fraction of ethanol in the bottoms (xW) are related
by Equation (23), using a function for 1/(α – 1) from Equation (20) fitted from xW of 0 and 0.2
(a = 1.73317, b = 0.511067, c = 0.103302). As one might expect, these different constants
provide a better fit for1/(α – 1) over the limited range of xW in this “experiment” than the
coefficients listed in Table 2 for a larger span of xW.
Moles of distillate (D) and mol fraction of ethanol in the distillate (xD) are computed
from material balances [Equations (1) and (2), rearranged]. Moles are converted to grams
using a molecular weight of 46.07 for ethanol and 18.02 for water from Table 1:

Grams ethanol = 46.07 × moles of ethanol (24)

Grams water = 18.02 × moles of water (25)

Mol fraction (m.f.) ethanol is converted to wt % ethanol by

wt % = 100 × (m.f.) (46.07) / [(m.f.) (46.07) + (1 – m.f.) (18.02)] (26)

17
to make use of weight-percent dependent density relationships for ethanol-water mixtures,
either at an assumed laboratory temperature of 20 °C or as estimated for the solution
remaining in the distilling flask (Appendix B).
Mass is converted to volume by means of density at known temperature and
composition. Tables in Ref. [24, Tables 6.22 and 6.36], interpolated if necessary, relate wt %
and % by volume at the standard temperature of 60 °F (15.56 °C).

Table 3. Summary of Simulated Batch Distillation Experiment

Temp Bottoms Distillate Ethanol in Ethanol in Distillate (xD)

(°C) (W) (D) Bottoms (xW) Cumulative Individual Instantaneous
g moles g moles mol fract mol fract mol fract mol fract
mass (g) mass (g) wt % wt % wt % wt %
mL @ 20 °C mL @ 20 °C % vol % vol % vol % vol
mL @ temp

89.1 9.7367 0.0000 0.0706 0.3704 0.3704 0.3704

194.6994 0.0000 16.269 60.07 60.07 60.07
200.0 209.5 0 20.0 67.8 67.8 67.8
89.6 9.5787 0.1581 0.0658 0.3645 0.3645 0.3577
190.2358 4.4636 15.26 59.46 59.46 58.74
195.2 204.2 5 18.8 67.2 67.2 66.5
90.0 9.4180 0.3188 0.0609 0.3578 0.3513 0.3439
185.7561 8.9433 14.22 58.76 58.07 57.27
190.3 199.0 10 17.5 66.5 65.8 65.0
90.5 9.2542 0.4825 0.0560 0.3508 0.3370 0.3289
181.2583 13.4410 13.17 58.04 56.51 55.63
185.4 193.7 15 16.3 65.8 64.3 63.5
91.0 9.0871 0.6496 0.0511 0.3432 0.3214 0.3128
176.7406 17.9588 12.11 57.20 54.78 53.79
180.5 188.5 20 15.0 65.0 62.6 61.6
91.6 8.9161 0.8206 0.0463 0.3352 0.3045 0.2952
172.2004 22.4989 11.04 56.32 52.83 51.72
175.6 183.3 25 13.7 64.1 60.7 59.6
92.2 8.7409 0.9959 0.0415 0.3265 0.2863 0.2763
167.6352 27.0641 9.96 55.36 50.63 49.40
170.7 178.0 30 12.3 63.2 58.5 57.2
92.8 8.5607 1.1760 0.0367 0.3174 0.2665 0.2559
163.0420 31.6573 8.88 54.31 48.17 46.79
165.8 172.8 35 11.0 62.2 56.0 54.5
93.5 8.3750 1.3618 0.0321 0.3075 0.2454 0.2341
158.4176 36.2818 7.82 53.10 45.40 43.87
160.8 167.5 40 9.7 61.0 53.1 51.5
94.2 8.1830 1.5538 0.0276 0.2971 0.2228 0.2109
153.7584 40.9410 6.77 51.94 42.29 40.61
155.8 162.3 45 8.3 59.8 49.8 48.0
94.9 7.9839 1.7529 0.0233 0.2859 0.1909 0.1867
149.0609 45.6385 5.76 50.59 38.84 36.99
150.8 157.0 50 7.2 58.4 46.1 44.0

18
 The boiling temperature curve relates still-pot composition and the temperature at
which it boils. The boiling temperature curve is predicted from the VLE relationship described
in Appendix A. The complete curve, including the entries in Table 3, Columns 1 and 4, is
plotted in Figure 8 against weight fraction (wt %/100) ethanol as in Ref. [24, p.251].
Experimental data from three different sources judged to be reliable [24, p.251; 25,26] are
shown for comparison. Although the data are better correlated by the broken-line curve in the
figure, agreement with the predicted (solid) curve is within several tenths of a degree Celsius
for the vast majority of the 78 data points plotted. Difference equals or exceeds 1.0 °C for
only 3 of those points, with a maximum temperature difference of 1.3 °C. Maximum deviations
occur in the range of 0.05 to 0.3 weight fraction ethanol. These differences in the boiling-
point curve highlight the minor imperfections in the activity-coefficient model fitted solely on
the azeotropic point (Appendix A).

100
EQUILIBRIUM BOILING TEMPERATURE ( C)

SOLID CURVE CALCULATED USING VAN LAAR CONSTANTS

o

FITTED TO THE AZEOTROPE

95

90

MINIMUM-BOILING AZEOTROPE AT 78.15 oC

85 0.956 WT FRACT ETHANOL
0.044 WT FRACT WATER

EMPIRICAL BROKEN-LINE CURVE

80
FROM A FIT TO THE EXPERIMENTAL DATA SHOWN

DATA FROM SOURCES CITED IN THE TEXT

75
0.0 0.2 0.4 0.6 0.8 1.0

WEIGHT FRACTION ETHANOL (MVC) IN LIQUID

Figure 8. Boiling Temperature Curve for Ethanol-Water at 1 Atm Pressure

Plotting the temperature values in Table 3 against cumulative distillate collected results
in the expected s-shaped curve of Fig. 5.8 of Refs. [6 & 7]. The simulated experiment has
been extended to collect an extra 50 mL of distillate and provide additional temperature data
points in Figure 9. Those temperatures are not listed in Table 3.
The density function developed to prepare Figure B1 in Appendix B can be used to
estimate the hot liquid remaining in the still pot during distillation. Estimated bottoms volume
is listed in Table 3, Column 2. Liquid volumes shown in the table at each boiling temperature
are 4-5 % greater than the volume of the same liquid at 20 °C. Change in volume remaining
in the distilling flask is therefore not accounted for exactly by subtracting the volume of
distillate removed, measured at 20 °C. Even when bottoms and distillate volumes are both

19
expressed at 20 °C, those volumes are not additive because of the difference in density with
ethanol concentration.
Density of the residual liquid at each temperature and composition for the simulated
experiment summarized in Table 3 is plotted against temperature as the points in Figure 10.
As before, additional points were obtained by extending the simulation to collect another
50 mL of distillate (again not shown in the table). The point at the extreme upper right is the
liquid density of pure water boiling at 100 °C and 1 atm. “Experimental” points extrapolate
smoothly to this datum along the curve drawn in the figure.

100

98
DISTILLING FLASK ( C)

96
o
TEMPERATURE IN

94

92

90

88
0 10 20 30 40 50 60 70 80 90 100
CUMULATIVE ML OF DISTILLATE COLLECTED

Figure 9. Still-Pot Temperature vs. Cumulative Volume of Distillate for Simulated

Ethanol-Water Batch Distillation at 1 Atm Pressure

The empirical curve-fit function, forced to pass through the 100 °C point:

ρ BOTTOMS = 0.958380 + 1.35444 × 10 —3 (t – 100 °C) – 8.1550 × 10 —5 [(t – 100 °C)] 2

– 3.23156 × 10 —7 [(t – 100 °C)] 4 (27)

reproduces the points of the simulation with a correlation coefficient (r2) of 100 % and a
standard deviation (s2) of 0.00002. This function allows one to estimate the bottoms density
(and therefore the volume of solution left in the distilling flask), from temperature alone when
batch-distilling without reflux a 20 % by volume (60 °F, 15.56 °C) solution of pure ethanol and
water. Division of mass by density gives volume.
Composition of the distillate is enumerated in the last three columns of Table 3. Three
versions of distillate collection are shown. In Column 5, data are recorded at every 5 mL as
the distillate is allowed to accumulate in the receiving vessel. In Column 6, the receiving
vessel is changed as soon as each 5 mL of distillate is collected, and the individual distillate

20
samples are kept separate for “analysis.” Column 7 represents the instantaneous composition
at each temperature and distillate volume noted. For the first drop of distillate (0 mL
accumulated), all three columns are exactly the same. Composition is also the same, but at a
different value, in Columns 5 and 6 for the first 5 mL of accumulated sample. For other
entries, compositions show a regular progression from top to bottom and from left to right in
Columns 5, 6, and 7.

0.96
ESTIMATED DENSITY OF
STILL BOTTOMS (g/mL)

0.95

0.94

0.93

0.92
88 90 92 94 96 98 100
o
TEMPERATURE ( C)

Figure 10. Estimated Density of Solution Remaining in the Still Pot at Boiling Temperature
for Simulated Ethanol-Water Batch Distillation at 1 Atm Pressure

In the laboratory, distillate solutions cooled to a standard temperature of perhaps 20 °C

would be analyzed quantitatively by any of the methods discussed in Appendix B. For
example, densities at laboratory temperatures are well known. Any samples extracted from
the distilling flask should also be cooled to laboratory temperature before analysis by density.
On a qualitative basis, unheated distillate samples of about 50 % by volume or more
would light off at laboratory temperature with a small flame and leave no residue on the watch
glass (Appendix B). This qualitative ignition test should produce positive results at every 5 mL
increment for the cumulative samples and for most of the individual samples.
In some versions of the experiment, the composite ethanol-rich distillate is saved and
further purified using a separate refluxed fractionating column. In a properly designed
column, composition will approach pure water in the bottoms and the azeotrope in the
overhead. A sample of that distillate would ignite on the watch glass as well, burning more
vigorously with a brighter flame and again leaving no visible residue.
Wrap-up. The newly developed equation has successfully facilitated the simulation of
an unrefluxed batch distillation experiment; however, modeling of any further purification of
distillate samples in a fractionating column is beyond the scope of the present investigation.

21
 Summary and Conclusions
• Batch, or differential, distillation without reflux is described by the Rayleigh Equation:
+ Relates composition and amount of material remaining in distilling flask
+ Other quantities determined by material balance
+ Illustrative example from the literature presented.
• Numerical integration is required for the Rayleigh Equation in its basic form.
• Substitution of relative volatility (α) allows analytical integration for constant α.
• New equation derived here allows analytical integration whether α is constant or not.
• Analytical integration utilizing α has been evaluated here against numerical integration:
+ For constant or nearly constant α
+ For a number of real vapor-liquid systems, where α varies with composition.
• For ideal systems, with α not varying widely, use of an “average” α is adequate.
• For non-ideal systems, only the new equation follows the track of numerical integration.
• The new function was used in simulation of batch distillation of pure ethanol and water.
• Ethanol-water results compare favorably with expectations, for example:
+ Boiling point vs. composition curve
+ Still pot temperature vs. cumulative volume of distillate collected
+ Density of solution remaining in still pot.
• Methods to analyze liquid composition are discussed for the ethanol-water system:
+ Gas chromatography (GC)
+ Refractive index data compiled from various sources
+ Densities of ethanol-water solutions extended beyond range of published data
+ Qualitative ignition test / flash points of ethanol-water solutions.

About The Author

Robert G. Kunz was an environmental engineering manager at Air Products and
Chemicals, Inc., Allentown, PA before retiring after 26+ years of service. He then joined
Cormetech, Inc., Durham, NC as Technical Project Manager in support of sales and marketing
efforts in the petroleum refining and petrochemical industries. He held engineering positions
previously at Esso Research and Engineering Company, Florham Park, NJ and The M.W.
Kellogg Company, New York, NY and is currently an independent environmental consultant.
“Dr. Bob” has earned a BChE degree in Chemical Engineering from Manhattan College, a PhD
in Chemical Engineering from Rensselaer Polytechnic Institute, an MS in Environmental
Engineering from Newark College of Engineering, and an MBA from Temple University. He has
contributed numerous publications to the technical literature, including topics in phase
equilibria, and is a recipient of the Water Pollution Control Federation’s Harrison Prescott Eddy
Medal for noteworthy research in wastewater treatment. He is author or co-author of two
books and holder of one U.S. patent. He is a member of the American Institute of Chemical
Engineers (AIChE), the American Chemical Society (ACS), and the Air & Waste Management
Association (A&WMA) and is a licensed professional engineer in several states.

Acknowledgement
The author is grateful to Roberta Kunz Fox, AIA, of fox2 design for help with Figure 1.

22
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3/3/2011).

53. U.S. Census Bureau, Foreign Trade Statistics, “2011 Schedule B – Chapter 22,”
http://www.census.gov/foreign-trade/schedules/b/2011/c22.html (accessed 3/8/2011).

54. Anon., “At What ‘Proof’ Will Spirits Burn? – Alcohol Fire Chemistry,”
http://ask.metafilter.com/95484/At-what-proof-will-spirits-burn (accessed on 3/8 2011).

55. Spencer, A.B. and G.R. Colonna, editors, “Fire Protection Guide to Hazardous Materials,”
13 ed., p.325-57 (ethanol-water), p.497-8 (pure ethanol), NFPA, Quincy, MA (2002).

26
 APPENDIX A

Summary of Vapor-Liquid Equilibrium (VLE) Relationships

Ideal Systems

Equations. Low-pressure binary vapor-liquid equilibrium for an ideal system is

described by the following relationships involving partial pressure (p), vapor pressure (pvap),
and liquid-phase mol fraction of each component. The total pressure is denoted as PT.

p1 = p1vap x1 (A1)

p2 = p2vap x2 (A2)

and PT = p1 + p 2 (A3)

with x1 + x2 = 1 (A4)

Equations (A1) and (A2) are known as Raoult’s Law.

If the Ideal Gas Law and Dalton’s Law are assumed to apply to the vapor phase, vapor-
phase mol fractions (y1 and y2) can be calculated from:

y1 = p1 / PT = p1vap x1 / PT (A5)

y2 = p2 / PT = p2vap x2 / PT (A6)

with y1 + y2 = 1 (A7)

Vapor pressures for any system, ideal or non-ideal, are conveniently calculated using
the Antoine Equation:

log10 pvap (mmHg) = A – B / [t (°C) + C] (16)

Antoine constants are listed in Table 1 of the main text for chemicals of interest in this paper.
By combining and rearranging the equations above, one obtains:

x1 = (PT – p2vap) / (p1vap – p2vap) (A8)

Chiefly by means of this equation, the entire x-y equilibrium curve and temperature-
composition curves at constant pressure for ideal systems can be mapped out in a
straightforward manner. One first chooses a temperature, computes vapor pressures from
Equation (16), then calculates x1 from Equation (A8) and y1 from Equation (A5). Only if a
specific value of x1 is desired, does the calculation become trial and error, in which a new

27
value of t is repeatedly chosen until the desired value of x1 is obtained to within whatever
tolerance is satisfactory.
Relative Volatility. The relative volatility (α) in Equation (8) can be calculated for
any temperature or the corresponding value of x1 along the VLE curves from a ratio of vapor
pressures:

α = (y1/x1) / (y2/x2) = p1vap / p2vap (A9)

Relative volatilities for a number of systems – some ideal, some non-ideal – are listed in the
second column of Table 2. Alpha is not perfectly constant for any of these systems across the
range of interest. In general, α shows a wider variation in a system exhibiting non-ideal
behavior than for an ideal system following Raoult’s Law.
The systems considered as ideal [9, p.526;25;27-29] are listed in the top section of
the table in order of increasing α. The higher the α, the more the bulge between the x-y
equilibrium curve and the y=x, 45° line. EDC-toluene, midway down the listing, is
representative of the ideal systems investigated, and its x-y curve is drawn in Figure A1 along
with experimental data from several investigators [25,30-32]. Similar plots for methanol-
ethanol and benzene-toluene have been published previously [21]. The other ideal systems
noted here have also been verified with real data. However, those additional data-enhanced
x-y diagrams are not shown since the curves are easily calculated and look largely the same.

1.0

0.8
Y1 MOL FRACTION EDC IN VAPOR

0.6

0.4

0.2

0.0
0.0 0.2 0.4 0.6 0.8 1.0

X1 MOL FRACTION EDC IN LIQUID

Figure A1. Vapor-Liquid Equilibrium Diagram for EDC-Toluene at 1 Atm Pressure

Because of the close similarity in α between EDC-toluene and the adjacent benzene-toluene,
the resulting material balance relationships for EDC-toluene would closely resemble the curves

28
of the illustrative example above for benzene-toluene (Figure 3). Since the pure component
atmospheric boiling point is not quite the same for EDC and for benzene, the temperature
graph analogous to Figure 2 would differ.

Non-Ideal Systems

Equations. For these systems, a so-called activity coefficient (γ) is introduced into
Raoult’s Law to yield:

p1 = p1vap γ1 x1 (A10)

p2 = p2vap γ2 x2 (A11)

and PT = p1 + p 2 (A3)

with x1 + x2 = 1 (A4)

Under the Ideal-Gas and Dalton’s Law assumptions, vapor phase mol fractions are given by:

y1 = p1 / PT = p1vap γ1 x1 / PT (A12)

y2 = p2 / PT = p2vap γ2x2 / PT (A13)

again with y1 + y2 = 1 (A7)

As in ideal systems, the Antoine Equation [Equation (16)] can be used to calculate vapor
pressures. Selected Antoine constants are listed in Table 1.
Activity Coefficients. Activity coefficients (γ’s) account for deviations from ideality.
They are back calculated as point values from experimental data and then fitted to a special
type of continuous function for use in VLE calculations. There are several empirical activity-
coefficient formulations in widespread use; for any given application, some fit the deviations
better than others across the range of data. The van Laar equations [1; 9, p.527, 33], for
example:

log10 γ1 = A1-2 (A14)

2
[1 + ( A1-2 x1) / ( A2-1 x2)]

and log10 γ2 = A2-1 (A15)

2
[1 + ( A2-1 x2) / ( A1-2 x1)]

fit the data well for the non-ideal systems considered here. These particular equations are also
written in terms of natural logarithms (base e), using constants 2.302585… × A1-2 and A2-1.

29
 Relative Volatility. Examples of non-ideal systems are contained in the last three
entries of Table 2. These systems were selected to demonstrate the effect on α as the
behavior becomes more and more non-ideal. The relative volatility (α) in Equation (8) for
these and other non-ideal systems can be calculated by adding activity coefficients to the
vapor-pressure ratio in Equation (A9) as shown below:

α = (y1/x1) / (y2/x2) = (p1vap γ1) / (p2vap γ2) (17)

Additional comments on the three example systems follow.

Methanol-Water and Acetone-Water. Activity coefficients for methanol-water and
acetone-water at 1 atm pressure derived from experimental data are given in Ref. [9, p.528]
and reproduced here in a footnote to Table 2 of the main text. The methanol-water x-y curve
with accompanying experimental data from a number of investigators is shown elsewhere
[21]. It resembles the curves for the ideal systems. It uses the van Laar constants listed in
the Table 2 footnote, Antoine constants from Table 1, and a trial- and-error‡ procedure to
compute each point on the continuous curve at the chosen total pressure.

1.0

0.8
ACETONE IN VAPOR
Y1 MOL FRACTION

0.6

0.4

0.2

0.0
0.0 0.2 0.4 0.6 0.8 1.0
X1 MOL FRACTION ACETONE IN LIQUID

Figure A2. Vapor-Liquid Equilibrium Diagram for Acetone-Water at 1 Atm Pressure

‡
Constant-pressure calculations for non-ideal systems are always trial and error since the activity coefficients are
functions of liquid composition. An efficient method to map out the x-y equilibrium curve is to select a value of x1
and total pressure and then vary the temperature until the calculated total pressure from Equation (18) agrees as
closely as desired with the total pressure selected.

30
 The acetone-water system (Figure A2) is distorted somewhat but does not exhibit
azeotropic behavior. The curve in Figure A2, prepared in a similar manner to the x-y curve for
methanol-water, correlates the data of several investigators [19,28,34-39] reasonably well.
Ethanol-Water. The classic example frequently cited to illustrate an azeotrope is the
ethanol-water system. This system is well investigated in the literature and is a favorite
student exercise in college/university laboratories throughout the world. Its predicted x-y
diagram at atmospheric pressure, shown without data points (Figure A3), was drawn by using
the same trial-and-error procedure described above for methanol-water and acetone-water.
It exhibits an azeotrope composed of 95.6 wt % ethanol and 4.4 wt % water at 1 atm
pressure (760 mmHg, 1.013 bar) and 78.15 °C [9, pp.631,633]. This corresponds to 89.43
mol % ethanol and 10.57 mol % water [9, p.633]. The azeotropic point is different at other
total pressure levels [9, p.631]; water content decreases with decreasing pressure, and at
least in theory becomes zero when distillation pressure is reduced low enough [9, p.631;40].
In the special case of an azeotrope, one can obtain the van Laar constants for use in
Equations (A14) and (A15) from the composition of the azeotropic point alone [21]. Since
the fitting technique to obtain them involves only the azeotropic point, agreement with other
experimental values along the curve is not guaranteed. Although the fit is not perfect, the
myriad of accompanying data points from multiple investigators (shown elsewhere [21]) are,
however, so numerous as to obscure the calculated x-y equilibrium curve almost completely.
Furthermore, the ethanol-water system at constant pressure has been investigated
from below atmospheric pressure [40] up to at least 300 lbf/in2 absolute (psia) [26,41]. It is
well described in Ref. [26] by van Laar constants only slightly different from those obtained
here and differing among themselves but little for huge changes in pressure. Therefore, small
day-to-day variations in barometric pressure encountered in the laboratory would alter only PT
in simulation calculations, and not require modification of the 760-mmHg van Laar constants.

1.0

NOTE:
THIS CURVE, COVERED WITH DATA POINTS,
APPEARS IN THE REFERENCE CITED IN THE TEXT.
0.8
ETHANOL IN VAPOR
Y1 MOL FRACTION

0.6

0.4
AZEOTROPE AT 78.15 oC
89.43 mol % ETHANOL,
10.57 mol % WATER;
95.6 wt % ETHANOL,
0.2 4.4 wt % WATER

0.0
0.0 0.2 0.4 0.6 0.8 1.0
X1 MOL FRACTION ETHANOL IN LIQUID

Figure A3. Predicted Vapor-Liquid Equilibrium Diagram for Ethanol-Water at 1 Atm Pressure

31
 APPENDIX B

ANALYTICAL METHODS FOR ETHANOL-WATER SOLUTIONS

This appendix reviews several ways to analyze for chemical composition in the ethanol-
water system. These include at least three quantitative methods and one qualitative
technique. Quantitative analysis is done by refractive index, density, or possibly gas
chromatography (GC). A qualitative analysis for ethanol in the distillate is often conducted by
attempting to ignite several drops of solution on a watch glass since ethanol and its aqueous
solutions greater than about 50 % by volume are combustible at room temperature [33].
These are discussed in turn below.

Analysis Using Refractive Index

Refractive index data for mixtures of ethanol and water at various temperatures (nDt)
are summarized in Table B1, assembled from a number of sources. Refractive index can
theoretically be determined to 1 part in 10,000 [4, p.319; 6, p.299] but is more likely to agree
with literature values to 1 part in 1,000 because of the presence of impurities [6, p.279].
Refractive index is commonly measured at 20 °C (nD20) [17, pp.D-200,D207]. Minor
excursions from 20 °C are corrected for by adding a mean value of 0.00045 refractive index
units for every °C above 20 °C [6, p.299;42].

nD20 = nDt + 0.00045 (t – 20 °C) (B1)

As seen in Table B1, differences in refractive index for more appreciable differences in
temperature are not exactly accounted for by the general factor.
Refractive index for ethanol-water solutions passes through a maximum point when
plotted against composition. This occurs at 79.3 wt % ethanol and 25 °C, as reported by
Andrews [43], and in the vicinity of 79-80 wt % ethanol at 60 °F (15.56 °C), as determined by
curve-fitting the data in Ref. [24, p.250] combined with differential calculus. A less rigorous
inspection of the other entries in Table B1 detects maxima occurring somewhere in a wider
range between 70-80 wt % ethanol. In the affected range, therefore, more than one value of
ethanol composition corresponds to the same refractive index when used as the independent
variable in a calibration curve.
Determination of refractive index requires only two or three drops of sample [6, p.300].
Once the procedure is set up and the proper technique is developed [4, pp.247-251], it is
possible to analyze a new sample every minute [13, p.237]. Refractive index should be
measured as soon as possible after sampling lest the samples decompose on standing [6,
p.300]. Analysis by refractive index does, however, require specialized equipment [6, p.300].

32
 Table B1. Refractive Index (nD) Values of Ethanol-Water Solutions
at Various Temperatures (°C)

Ethanol
(wt %) nD15 nD15.56 nD20 nD25 nD30 nD40 nD50 nD55

0 1.33345 1.33336 1.33300 1.33250 1.3318 1.3306 1.3290 1.3281

10 1.34020 1.3402 1.3395 1.3389 1.3384 1.3368 1.3349 1.3339
20 1.34778 1.3479 1.3469 1.3462 1.3450 1.3429 1.3406 1.3393
30 1.35470 1.3542 1.3535 1.3520 1.3510 1.3481 1.3452 1.3435
40 1.35948 1.3590 1.3583 1.3565 1.3550 1.3518 1.3484 1.3468
50 1.36290 1.3626 1.3616 1.3598 1.3578 1.3543 1.3506 1.3488
60 1.36505 1.3650 1.3638 1.3620 1.3597 1.3560 1.3522 1.3501
70 1.36645 1.3662 1.3652 1.363038 1.3608 1.3570 1.3528 1.3505
80 1.36690 1.3665 1.3658 1.36331 1.3611 1.3569 1.3525 1.3502
90 1.36626 1.3659 1.3650 1.36239 1.3603 1.3561 1.3515 1.3491
100 1.36332 1.36316 1.3614 1.35941 1.3573 1.3531 1.3487 1.3465

Notes:

(a) Values at 15°C, 30°C, 40°C, 50°C, and 55°C from Ref. [44].

(b) Values at 15.56°C (60°F) for pure water and ethanol from Ref. [24, p.250]. Other
entries at 15.56°C from regression of Ref. [24, p.250] values at other compositions.

(c) Values at 20°C from Ref. [17, p.D-200 (water) and p.D207].

(d) Values at 25°C either reported by Andrews [43], assembled from entries on p.1.95,
p.2.294, and Tables 10-71 and 10-72 of Lange’s Handbook [45], or curve-fitted to
interpolate between those values.

33
Analysis by Density

Published densities of ethanol-water solutions [9,15-18, 45-47] facilitate ethanol

analysis by careful measurement of density at constant temperature. Unlike the refractive
index of ethanol-water solutions, for the most part, solution density at a given temperature
corresponds to a single value for composition. The simplest technique employs a pycnometer,
or specific gravity bottle, a piece of calibrated glassware of precisely known volume [13,
pp.191-193]. It is weighed first empty and then full of solution, and density is calculated by
dividing solution weight by solution volume. Pycnometer volume is typically about 10 mL
[48]. Related studies used a 10-mL [25,40] or 5-mL [49] pycnometer to analyze by density.
Analysis by density has the advantage of using simple laboratory equipment in
widespread use. It takes a minimum of time and provides a unique value for ethanol
composition at a constant known temperature. However, in some cases, required sample size
may be greater than the amount of distillate collected.
Density-Composition Plot. Density data for ethanol-water mixtures across the entire
gamut of compositions and sample temperatures are difficult to find. This is not so much of a
problem for analysis by density, but rather for converting from mass to volume at any given
temperature and vice versa. Density data (g/mL) for liquid water (0-102 °C) and ethanol-
water mixtures are tabulated in NBS Circular 19 [46] and handbooks reproducing said tables.
For pure ethanol, density or its reciprocal, specific volume, is listed in various tables up to
80 °C [47, p.41], and tabular entries and correlating equations are provided for density from
0-78 °C [50]. Density data in lbm/gal is plotted either against °C [24, p.286] or °F [51, p.62]
from 5 to 60 °C, parametric in vol % ethanol. Density data for liquid solutions outside this
temperature range could not be found in the published literature.
However, the parametric plots [24,p.286; 51,p.62] appear to be well behaved,
consistently spaced, and fairly linear at the upper temperatures, suggesting that missing
values might be interpolated or extrapolated with some degree of success. To that end, the
density plot has been reconstructed here in units of g/mL between 0 and 100 °C, with wt %
ethanol as parameter (Figure B1).
The newly developed curves of Figure B1 are based on the data from NBS Circular 19
[46], the equations of Ref. [50], and the aforementioned graph [24, p.286]. The fit is exact
for all values for pure water, for pure ethanol up to 78 °C, and for tabulated mixture data from
10-40 °C [46], and in reasonable agreement with values read with difficulty from the curves of
Ref. [24, p.286] from 5 to 60 °C. Two different methods produce the same extrapolated pure
ethanol curve from 78 to 100 °C.
The top curve (0 % ethanol, 100 % water) contains the actual maximum point in the
density for pure water at about 4 °C. Curve fitting predicts a maximum in density for 20 wt %
ethanol at about 2 °C, which may or may not be the case. As drawn in Figure B1, the other
curves exhibit a continuous decrease in density with increasing temperature.
Estimated densities for pure ethanol beyond 80 °C and for mixtures below 5 °C and
above 60 °C are not based on data and should be used with caution, since extrapolated
densities are most likely not exact. Furthermore, although values are shown in Figure B1 for
pure ethanol and for some mixture compositions above their atmospheric boiling points, these
solutions do not exist as stable liquids at 1 atm pressure. In that case, the predicted densities

34
correspond to whatever slightly elevated pressure is necessary to maintain the solution in the
liquid state. This slight difference in pressure should have a negligible effect on density since
enormous pressure increases are necessary to effect a significant change in liquid density
[47, pp.40-42]. Finally, while estimated values of density may be useful to calculate volume
from mass or vice versa, density values outside the range of data should not be used for
analytical purposes to determine solution composition.

1.05

1.00 0
DENSITY OF SOLUTION (g/mL)

20
0.95
40
0.90
60
0.85
80

0.80
100

0.75

0.70 PARAMETER: ETHANOL COMPOSITION IN WT %

0.65
0 10 20 30 40 50 60 70 80 90 100
TEMPERATURE (oC)

Figure B1. Estimated Liquid Density of Ethanol-Water Solutions

Analysis by Gas Chromatography (GC)

Gas chromatography is a well-documented separation and analysis technique used in a

chemical laboratory. When a liquid sample is injected into the gas chromatograph instrument,
it is picked up by a carrier gas, usually helium but possibly nitrogen, and passed through a
heated column packed with adsorbent material, where different chemicals are adsorbed and
desorbed at different rates. The chemicals being analyzed for arrive at a detector downstream
as a series of peaks traced on a strip chart recorder. The time for the arrival of a peak is
indicative of each particular chemical in the mixture. The width of each peak, or more
properly the area under the peak, represents the amount of that chemical present. More
information to supplement this highly simplified explanation is contained in the cited
references [4, pp.255-274; 6, pp.215-223; 7, pp.165-173; 13, pp.149-151].
Advantages: A gas chromatograph can detect extremely low concentrations in an
unknown sample. It requires a sample size on the order of 1-25 micro liters (μL), smaller or
much smaller than the several drops (20 drops = 1 mL) needed for a refractive index
determination.

35
 Disadvantages: The GC instrument is exceptionally expensive to purchase and may
have to be shared with other investigators, if it is available at all. It needs an hour or so
warm-up time, and requires experience it setting its temperature controls for the particular
system being analyzed and some technique/practice in handling the micro-liter syringe to
pierce the septum at the instrument’s injection port. Residence time for each sample to pass
through the column and show up as peaks at the detector may be several minutes or longer.

Analytical-Qualitative

A qualitative analysis for ethanol in the distillate is often conducted by attempting to

ignite several drops of solution on a watch glass. Like hydrogen, ethanol burns with a bluish
flame not easily visible in ordinary daylight [51, p.101]. Ethanol and its aqueous solutions
greater than about 50 % by volume are combustible at room temperature [23]. These will
ignite and burn to CO2 and water without leaving any residue.
Alcohol Proof. In former times, this phenomenon formed the basis of a test of the
alcoholic spirits given to members of the British Royal Navy as part of their rations [52]. In
that test, a sample lighting off with gunpowder constituted “proof” that the alcohol had not
been watered down. In the U.S. today, proof is defined as exactly twice the alcoholic content
by volume, measured at 60 °F (15.56 °C) [24, Table 6.36; 51, p. 108; 53].
Anecdotal evidence suggests that a certain brand of vodka at 100 proof (50 % ethanol
by volume) lights off, whereas this same vodka at 80 proof (40 % ethanol by volume) does
not [54]. That this should be so is explained by the flash point of ethanol-water solutions
(Table B2, prepared from a single source [55]); flash point is plotted vs. temperature in
Ref. [24, p.254]. The flash point is the lowest temperature at which just enough liquid is
evaporated at the surface to create a combustible concentration in air of gas that will catch
fire if an ignition source is present.§ The table shows that an ethanol-water solution is
combustible at “room temperature” at about the stated 50 % by volume. Heating the solution
first, which generates additional combustible ethanol vapor (and thereby lowers the flash
point), allows solutions of lesser ethanol content to ignite at the same room temperature [54].

Table B2. Flash Points of Ethanol-Water Solutions [55]

Ethanol Temperature Ethanol Temperature

Volume % °F °C Volume % °F °C
5 144 62 60 72 22
10 120 49 70 70 21
20 97 36 80 68 20
30 85 29 95 63 17
40 79 26 96 62 17
50 75 24 100 55 13

§
The autoignition temperature, for which no external ignition source is necessary, is much higher (for example,
363 °C for 100 % ethanol [55]). Degrees F for pure ethanol in table and footnote calculated from entries in °C.

36