January 1954
20 1
INDUSTRIAL AND ENGINEERING CHEMISTRY
Liquid and vapor densities are compared with those of Young (8) in Tables I11 and IV. A comparison of vapor pressures is given in Table V. It is evident that considerable differences exist between the data of Young and those obtained in this laboratory. The disagreement in vapor pressure data is in the same direction and of the same order of magnitude as that observed in Young’s data on n-pentane when compared with the work of Beattie, Douslin, and Levine ( 9 ) and the work done in this laboratory (6). It can be concluded then, that the information presented here is more reliable. Benedict, Webb, Rubin, and Friend ( 4 ) have fitted an equation of state to the data of Young (8). This equation therefore also deviates from the compressibility data presented here. I n general, the pressures calculated by the equation of state are higher than the observed pressures, Table VI gives a few of the residuals from the Benedict-Webb-Rubin equation in the superheated region. The residual is defined as the calculated pressure minus the observed pressure. No attempt was made to adjust the original equation as presented by Benedict, Webb, Rubin, and Friend.
BENEDICT-WEBB-RUBIN EQUATION FOR ISOPENTANE
P
= RTd
+ (BoRT-Ao-Co/T*)d2+ (bRT-a)d3 + aad6 + $ [(l+ydZ)e-Yd2]
Bo
b
= 2.56386 A0 = 4825.36 Co X 10-6 = 21336.7 P = lb./sq. inch
a
c X 10-6
= 17.1441 = 226902
136025.0 d = 1b.-mole/ou. foot =
a X 102 = 6987.77 y X 102 = 1188.07
R = 10.7335 T = degrees Rankin
LITERATURE CITED
(1) Beattie, J. A., Douslin, D. R., and Levine, S. W., J. Chem. Phya., 20,1619 (1952). (2) Beattie, J. A., Levine, S. W.. and Douslin, D. R., J . Am. Chem. Soc.. 73. 4431 (1951). (3) Ibid.i74,4778(1962).’ (4) Benedict, M.,Webb, G . B.,FRubin, L. C , , and Friend, L., Chem. Eng. Progr., 47,419 (1951). 15) Chernes, B. J., Marchman. H.. and York, R., IND.ENG.CHEM., 41. 2653 (1949). (6) Li, Kun, and Canjar, L. N., Chem. Eng. Progr. Symposium Series No. - -. 7. 49 ~- (1963). . /
(7) Sage, B. H., Lacey, W. N., and Schaafsma, J. G., IND.ENQ. CHEM.,27,48 (1935). (8) Young, S., Proc. Phys. Soe. (London),13, 602 (1894-95). (9)Young, S.,J . Chem. SOC.(London),Trans.,71,446 (1897). RECEIVED for review June 5, 1953.
ACCEPTEDSeptember 17, 1953.
Single-Constant Linear Equation for Vapor-Liquid Equilibria in Binary Systems R. S. NORRISH AND G. H. TWIGG Research and Development Department, The Distillers Co., Ltd., Great Burgh, Epsom, Surrey, England
T
HE importance of vapor-liquid equilibria for the design of distillation columns has directed much attention to the derivation of algebraic expressions for smoothing experimental data, checking the internal consistency of the results, and predicting complete equilibrium curves from a minimum of experimental points. Unfortunately, most of the equations, including the well-known Margules and Van Laar relations, are valid only for conditions of constant temperature, whereas the great majority of distillations are carried out under conditions of constant total pressure. These semitheoretical equations based on the GibbsDuhem equation suffer from the further practical disadvantage that their use requires conversion of the experimental equilibrium results to activity coefficients to give a nonlinear plot and subsequent reconversion to x,y data, involving an accurate knowledge of the boiling point as a function of composition. More recently, entirely empirical linear equations have been proposed by Clark (8)which accurately relate vapor to liquid compositions for many systems a t constant pressure and do not require a knowledge of the boiling points. These equations, however, involve three and sometimes four arbitrary constants. The equation now proposed is linear, is immediately applicable to experimental equilibrium measurements at constant pressure without knowledge of the boiling point curve, contains only one arbitrary constant, and has been verified for 25 different constantpressure systems involving a wide range of chemical types. The equation does not apply to systems containing water as one component. EMPIRICAL EQUATIONS
Z is a function of x and y only, defined by the equation:
where K is the ratio of the molar latent heat of vaporization of the lorer to that of the higher boiling component. When experimental values of 2 a t constant pressure are plotted against xl,it is found that the points lie on a straight line: 2 = Mzt
+c
(2)
where .W and C are constants for a given system a t a given total pressure. Equation 2 has been verified for the 25 systems listed in Table I, and Figure 1 shows some typical plots. The apparently large deviations a t the ends of some of the lines correspond to comparatively small errors in the equilibrium data, as the sensitivity of Z to changes in x and y is a t a minimum near the middle of the plot and increases rapidly to infinity a t the ends. For example, an error in x of 0.001 mole fraction results in an error in 2 of 0.004 a t x = 0.5, but of 0.1 at z = 0.01. In particular, apparent deviations in the systems heptane-methylcyclohexane and 2,2,4-trimethylpentane-methylcyclohexane can be shown to be within the probable analytical error and apparent deviations in the plot for chloroform-cyclohexane disappeared completely when the materials were dried through silica gel and precautions taken to exclude atmospheric moisture from the still (see Figure 2 and Table 11). The only systems tested which did not give linear plots were those containing water as one component. It is not known why aqueous systems show anomalous behavior, but it is probably significant that because of the low
202
INDUSTRIAL AND ENGINEERING CHEMISTRY TABLE I.
COYPARISON OF
CALCULATED AND OBSERVED RESULTS 0.4342 a t 51
System" Acetone-scetio acid Phenol-aceto henone Hept ane-met{ ylcyclohexane 2,2,4-Trirneth Ipentane-octane 2,2,4 Trimetxylpentane - rnetliylcyclohexsne Propane-2-rnethylpropene (200 lb./ sq. inch) Carbon tetrachloride-cyclohexane 1-Butanol-2-chloroethanol Acetone-1 -butanol Carbon tetrachloride-cyclohexane (400 mm.) Carbon disulfide-carbon tetrachloride Toluene-octane Chloroform-cyolohexane Heptane-toluene C clohexane-2 2 3-trimethylbutane Cgloroform-cy~lbhexane(300 rnni.) Benzene-c yclohexane Hexane-benzene Benzene-2,2,3-trimethylbutane 1-Butanol-1-butyl acetate Bromine-carbon tetrachloride Cumene- henol Isopropy ether-2-propanol Toluene-2-chloroethanol 2-Propanol-nitromethane
K
0.434M f O 825 +0.495 0.000 0.000
-
VI a t
-8
XI
=
--B
Obsd. Calcd. Obsd. 0.83 0.91 0.89 0 . 1 5 0 . 6 3 0.57 0.04 0.52 0.52 0.31 0.69 0.70
0.000
1.00
0.500
0.02
0.03
0.51
0.52
-0 03.5
0.524 0.495 0.489 0.564
0.38 0.05 0.16 0.77
0.32 0.05 0.19 0.89
0.73 0.52
-0.066
0.87 1.03 1.07 0.69
0.70 0.52 0.59 0.95
-0.072
1.03
0.495
0.06
0.06
0.53
0.53
-0.205 -0.205 -0.230 -0.239
0.85 0.96 0.98 0.96 1.03 0.98
0.527 0,507 0.504 0.507 0,495 0.504 0.600 0.507 0.492 0,468 0.504 0,537 0.544 0,535 0.471
0.38 0.18
0.37 0.17 0.24 0.14 0.01
0.75 0.61 0.65 0.59 0.50 0.66 0.51 0.59 0.51 0.54 0.64 0.74 0.68 0.68 0.63
0.74 0.61 0.64 0.59 0.50 0.66 0.50 0 60 0.53 0.54 0.65 0.76 0.68 0.68 0.63
-0.063 -0.065
-0.240
-0,295 -0.325
1 .oo
0.96 1.05 1.22 0.98 0.80 0.77 0.81 1.19
-0.360
-0.570 -0.690 -0.622 -0.656
f
Calcd. 0.640 0.79 0.494 0.25 0.495 0.04 0.520 0.30 -0
0.79 1.04 1.03 0.89
=
-0.828
- 1.180 -1.270
0.26
0.15 0.01 0.28 0.01 0.14 0.00 0.14 0.24 0.32 0.20
0.22
0.32
At atmospheric pressure unless otherwise stated. b Equilibrium determinations by the authors (see Table 11).
Q
polarizability and high dipole moment of the water molecule as compared with organic molecules the dipole orientation contribution to the potential energy is greater and the dispersion effect less, so that when an organic liquid is diluted with water, the nature as well as the magnitude of the intermolecular forces is a1tered. I t seems improbable that such a complex mechanism can be described by equations as simple as Equation 2. -4s it stands, Equation 2 contains two arbitrary constants, J!f
0.68
0.93
0.28
0.00 0.16 0.04 0.14 0.25 0.37 0.21 0.22 0.32
Vol. 46, No. 1
and C, but it is found that C can be expressed in terms of the latent heat and boiling points of the pure components to give the single-constant equation:
where
Hence the plot of Z against, x1 is linear with slope M and passes through the point
2 (k1- k)] Lill- E7
[-e,
Table I gives the values of -i; calculated from the vapor pressure data [mostly taken from the collection of Stull ? I d ) ] compared with the values of Z at x1 = -e read off from the experimental plots of 2 against $1. The calculated and observed values of y1 a t 21 = -6 are also given. The agreement is in general within the accuracy of the data except for the systems propane2-methylpropene at 200 pounds per square inch and acetoneacetic acid. The discrepancies for these systems may be due in part to nonideality of the vapors. Although Equation 3 can be regarded simply as an empirical equation which adequately represents the observed facts, it is shown below that it can also be obtained theoretically from Equation 2 by applying the isopiestic equivalent of the Gibbs-Duhem restriction. DERIVATION OF EQUATION 3
C YCLOHEX A N E
Assuming ideal behavior in the vapor phase and neglecting the temperature variation of the latent heat, for any binary system:
-
PI = PIX1 yl = PYl
. 0z0-0-0,
(5)
where
and similarly for component 2, whence
where 2'1 and 2'2 are the boiling points of the pure components a t pressure P. From Equations 2 and 7 :
whence, at constant pressure:
(%)p
- K (a
k
=
M
(9)
The Gibbs-Duhem equation at constant pressure may be written: dln
-04 00
I
I
I
I
I
I
I
I
I
01 01 0 3 04 0 5 06 07 08 0 9 10 MOLE F R A C T I O N MORE VOLATILE C O M P O N E N T I N L ~ Q U I D ~ , )
Figure 1. Typical Plots of Z us. Liquid Composition
YI
dln
YS
-
INDUSTRIAL AND ENGINEERING CHEMISTRY
January 1954
203
with Li. This will generally be the case, few systems having values of W max. greater than about 200 calories per mole, whereas Ll will usually be 7000 to 10,000calories per mole. Equation 14 therefore becomes:
and combination with Equation 9 gives:
z
= M(Z~
+ e) + -'L
(k - A),
which is the proposed Equa-
tion 3.
&
=
-
(3dp,, dln
y
EQUATIONS FOR ACTIVITY COEFFICIENTS
The terms containing Win Equations 13 may also be neglected. These terms are zero a t the concentration limits where In y is also zero and rise to a maximum value where In y is also a maximum. The most unfavorable case will be where the system is nearly ideal over the whole concentration range, while W has a high - ( K - l)x2]value (this case will rarely be met, as in general W max. is small 1)rz when In y max. is small). For such a system we may have Ti = 300' K., T P= 320' K., W Max. = 200 calories permole, In y Max. = 0. Then, since the numerical value of each of the terms in(13) volving W in Equation 13 is less than,
Integrating and setting In y1 = 0 when z2 = 0; In yz = 0 when 21 = 0; and writing wlzl 20~x2= W , the heat of formation of the solution from the pure components, the exact expressions for the activity coefficients become:
+
In
y1 =
.M
- ___ [I(ln (K - 1)2 K
K
- (K -
(16) 0.5
I
I
I
L
I
I
I
I
I
I
the error in -1 is not greater than about 2%. Equations 13 therefore become, to a good approximation:
CHLOROFORM-CYCLOHEXANE MATERIALS NOT SPECIALLY D R I E D
0.4
9
TABLE11. EQCILIBRIUM DETERMINATIONS BY AUTHORS All determinations were made in the Gillespie ( 4 ) still. All analyses were b y density except cumene-phenol (refractive index). Chloroform-Cyclohexane (760Mm.), Phenol-Acetophenone (760 bIin.), Mole Fraction Chloroform in Mole Fraction Phenol in Liquida Vapor Liquid Vapor
0.077 0,082
0.130 0.308 0.498 0.705 0.786 0.874 0.925
Solving for C between Equations 8 and 13 and substitution in 2 gives:
z
=
+ +
M ( X ~ e)
L 2 R I;(
- &)
Xow as K does not differ from unity by more than &SO% for most systems and TV = 0 at infinite dilution of either component, the integral is of the same order of magnitude as:
Vapor
0.270 0.462 0.641 0.859 0.929 0.949
0,880
where W max. is the maximum value of W , 60 that the integral is
(k - &)
if W max. is small compared
0.124 0.194 0.251 0.312 0.315 0.414 0.489 0.627 0,638 0.755 0,858 0.941 0.974
0.263 0.414 0.604 0.764 0.878
Carbon Tetrachloride-Cyclohexane (760 Mm.), Male Fraction Carbon Tetraohloride in Liquid Vapor
0
0.056 0.233 0.3195 0.396 0.454 0.6155 0.7595 0.843 0.886
Materials not specially dried.
0.172 0.269 0.413 0.495 0.544 0.648 0.749 0.781 0.945
Cumene-Phenol (760 Mm.), Mole Fraction Isopropylbenzene in Liquid Vapor
Chloroform-Cyclohexane (300 Mm.) Mole Fraction Chloroform in Liquid b Vapor
0.207 0.286 0.366 0.4195 0.5845 0.7425 0.833
small compared with
0.895 0.935
0.150 0.310 0,513 0.804 0,901 0.932
0.048
L
0.174 0,254 0.365 0.426 0.468 0.558 0.638 0.669 0.896
0.838
Liquid b
0.130 0,235 0.433 0,654 0.826
It can be shown that, in general,
0.154 0.157 0.249 0.458 0.640 0.793
0.388 0.494 0.579 0.610 0.631 0.716 0.733 0.797 0.797 0.849 0.815 0.956 0.979
Carbon Tetrachloride-Cyclohexane (400 Mm.), Mole Fraction Carbon Tetrachloride in Liquid Vapor
0.248 0.306 0.364 0.448 0.566 0.596 0.706 0.9205 8
0.281 0.344 0.404 0.486 0.597 0.6245 0.728 0.926
Material dried through silica gel.
INDUSTRIAL AND ENGINEERING CHEMISTRY
204
Equations 17 refer, of course, to the activity coefficients at the boiling point, but it is interesting to note that when K = 1-i.e.) the latent heats of the components are the same-they reduce to:
Vol. 46, No. 1
It is only in the special case where K = 1 that In a is a linear function of x.
Thus the value of M a t any pressure can be approximated from the value at any other if the heats of mixing a t infinite dilution are known. Equation 25 has been used to calculate the change in M for the system chloroform-cyclohexanewhen the pressure was reduced from 760 to 300 mm. of mercury. The value of M at 760 mm. was found to be -0.525. The value at 300 mm. was calculated by Equation 25 to be -0.667 and the value actually found was -0.685. For this calculation, heats of mixing a t room temperature were taken from the International Critical Tables (6). Strictly, of course, the heats of mixing a t the boiling points are required, but unfortunately there are few data in the literature above room temperature. Available data show that the assumption made in the foregoing argument that w is independent of total pressure (and therefore temperature) will not always be justified.
SIGNIFICANCE OF M IN RELATION TO RAOULT'S LAW
ACKNOWLEDGMENT
1 In vi = - - Mxi 2
which are identical with the equations for regular solutions at constant temperature. The corresponding regular solution equation connecting 2: and y is:
From Equations 17, when M = 0, In yi = In yn = 0 for all values of XI, which is the condition that the system obeys Raoult's law. When M < 0, In 71 and In "/z > 0 for all values of 5 1 and the system exhibits positive deviations from Raoult's law; similarly when M > 0 the system exhibits negative deviations, The value of M , which is a pure number, is thus a logical measure of the departure of the system from ideal behavior and also affords a convenient method of classification. VARIATION OF M WITH TOTAL PRESSURE
The variation of M with total pressure is obtained by combining the approximate Equations 17, to give In y1
- K In
yZ =
M(xl
+ 0)
(20)
where
e= K -L (1 -1
- -K - 1
l n ~ )
(4)
Differentiation at constant composition, assuming K and therefore e to be independent of pressure, gives:
(21)
However, it is known that the variation of the activity coefficients with pressure alone is very small, so that
Thanks are due to the directors of The Distillers Co., Ltd., for permission to publish this paper. NOMENCLATURE
constant in pressure-temperature relation: In P = A L/RT C = arbitrary constant K = ratio of molar latent heats of vaporization of pure components at pressure P = LI/& L = molar latent heat of evaporation M = arbitrary constant, slope of plot of 2 against 21 P = total vapor pressure PA, P B = pressures a t which values of M are M A , and M E , respectively, for a given binary P = partial vapor pressure R = gas constant T = Zemperature, O K. = boiling point of component 1 a t pressure P, ," K. Ti = boiling point of component 2 a t pressure P, K. T2 = integral heat of mixin W = relative partial molal feat of mixing W = molal heat of mixing at infinite dilution W" = mole fraction in liquid X = mole fraction in vapor
A
=
e
= -K t l ( l - K - 1
-
n
In K ) Subscripts 1 and 2 refer to the more and less volatile components, respectively. LITERATURE CITED
where w is the partial molal heat of mixing and writing
RTa Equation 21 becomes
Integration between PA and PB,taking as a first approximation w1 Kwoindependent of pressure, gives
-
Since the variation of M with pressure is the same a t all concentrations, the values of xi = 0 and xi = 1 can be substituted in Equation 24 and combined to give:
(1)
Bromiley, E. C., and Quiggle, D., IND. ENQ.CHEM.,25, 1136
(1933). (2) Brunjes, A. S., and Furnas, C. C., Ibid., 27,396 (1935). (3) Clark, A. M., Trans. Faraday Soc., 41,718 (1945). (4) Gillespie, D.T.C., IND. ENQ.CHEM.,ANAL.ED., 18, 575 (1946). (5) Harrison, J. M., and Berg, L., IND.ENQ.CHEM.,38, 117 (1946). (6)International Critical Tables, Vol. V, p. 155, New York, McGraw-Hill Book Co., 1926. (7) Miller, H. C., and Bliss, H., IND. ENQ.CHEM.,32, 123 (1940). (8)Richards, A.R., and Hargreavea,E.: Ibid. 36,805(1944). (9) Rosanoff, M.A,, and Easley, C. W., J . Am. Chem. SOC.,31, 979 (1909). (IO) ENQ.CBEM.,31, . . Scheeline, H. W., and Gilliland, E. R., IND. 1050 (1939). (11) Sohumacher, J. E., and Hunt, H., Ibid., 34,701 (1942). (12) Snyder, H. B., and Gilbert, E. C., Ibid., 34, 1519 (1942). (13) W. M..and Kruger, J., J. Am. C h m . SOC.,72, 1855 . . SDioer. (1950). (14) Stull, D. R., IND. ENG.CHEM.,39,517 (1947). (15) Tongberg, C.O.,and Johnston, F., Ibid., 25, 733 (1933). (16) York, R.,Jr., and Holmes, R. C., Zbid., 34, 345 (1942). RECEIVED for review April 3, 1953.
ACCEPTED August 18, 1953.
