Vapor Phase Imperfections in Vapor-Liquid Equilibria. Semiempirical Equation

CLINE BLACK

Shell Development Co., Emeryville, Calif. I Vapor Phase Imperfections in Vapor-Liquid Equilibria Semiempirical Equation

.

Because separation processes in the chemical and petroleum industries depend to a great ex- tent on knowledge of deviations from laws for perfect gases and ideal solutions, methods for calculating these deviations are of primary importance. In this and the study which follows, equations are developed for application to vapor imperfections and nonideal liquid mixtures

IT HAS long been recognized that real gases and vapors deviate from the simple law formulated for the. perfect gas. A knowledge of these deviations as well as of methods for calculating or pre- dicting them is important for describing the thermodynamic properties of substances. Even the basic problem of de- termining the thermodynamic consist- ency of vapor-liquid equilibrium data, in general, depends upon the availability of such information. Most practical problems require that the methods be capable of treating not only the pure substances but binary and multicom- ponent mixtures as well.

Following earlier work of G. A. Hern in 1867, van der Waals attempted to correct the perfect gas relation for volume and for the attraction forces of molecules. These effects are denoted

by the coefficients b and a, respectively. in his classical equation :

(1)

which gives pressure P as an explicit function of volume V and temperature T.

Although this equation gives satisfactory qualitative behavior it does not furnish the necessary quantitative results for real gases. Numerous attempts to improve upon the van der Waals equation have led to many modified equations. Most of these represent the behavior of substances best in the region above the critical point, but the main region of interest in vapor-liquid equilibria is below the critical pressure. Here the equation of Berthelot (6) has been frequently used, although it fails significantly for some conditions and especially for polar substances.

The most successful in representing the low pressure region is the method of virial coefficients suggested by Onnes in 1901. I t gives the compressibility factor, PVIRT, as a power series in the density. or in the pressure. It can be formulated from the principles of kinetic theory or from statistical mechanics. Substantial progress has been made in recent years in calculating second (73, 24, 26, 35, 37, 47, 52) and third (8,24,40,46) virial coefficients by taking into consideration the forces between the molecules.

A generalized equation of state in reduced temperature and pressure only is necessarily restricted by the limitations of the law of corresponding states. Tests made in the region above the critical point with several of the two-

P = RT/ (v - b ) - a / V 2

constant equations have shown that their failure to represent adequately the experimental data is due more to the inadequacy of the equation than to the limitations of the law of corresponding states. The recent work of Pitzer (40, 47) and his associates in- dicates that many nonpolar liquids closely conform to the principle of corresponding states if a slightly more complex equation involving one additional parameter is used.

A completely generalized relation, in reduced temperature and pressure, only, is not valid for the entire region below the critical temperature. Indeed, the volume of each vapor becomes specific as the saturation pressure is +ap- proached. Because the region below the critical pressure is of particular importance in vapor-liquid equilibria it is represented with the aid of conven- ient algebraic relations. These equations must correctly furnish the molal volumes for pure vapors and their mixtures as a hnction of temperature, pressure, and composition to provide the derived thermotlynamic quantities, such as partial molal volumes and fugacity coefficients. These derived thermodynamic quantities are of primary interest here.

Approximate Equation of State

An approximate equation of state has been developed specifically for the pressure region zero to the saturation pressure Po for temperatures up to the critical. I t is applied to nonpolar substances with the aid of the critical

VOL. 50, NO. 3 MARCH 1958 391

3.5

3.0

2 . 5

2.0 UJ

1 . 5

1.0

0. 5 1

I I I I I I I 0. b 0 . 7 0 . 8 0 . 9 1 . 0 1 . 1 1 . 2 1

REDUCED TEMPERATURE, Tr

Figure 1. Temperature dependence of attraction coefficient f "

1 . Nonpolar, generalized coefficients 2. Chloroform, D' = 0.447 3. 4. 5. 6. 7.

Water, E' = 0.026, m = 4.75 Ammonia, E' = 0.051, m = 4.45 Acetaldehyde, E' = 0.092, m = 4.7 Methanol, E' = 0.120, m = 4.75 Acetonitrile, E' = 0.1 23, m = 4.94

temperature To, the critical pressure Po and the vapor pressure Po for the pure substance. For polar substances one or two individual constants are required.

The equation is explicit in volume and is based on a van der Waals-type equation simplified for low pressure and including an attraction coefficient which expresses the effects of temperature and pressure on the molal cohesive energy af . The attraction coefficient is defined implicitly in terms of the volume V, pressure P, and temperature T according to

V = ( R T / P ) -j- b - a t / R T ( 2 )

in which the van der Waals constants are calculated from the critical temperature T, and critical pressure P, with the aid of

b = RT0/8P , and a = 276 R T , / 8 ( 3 )

If p is defined as the negative value of the residual volume or the difference between the experimental and ideal vapor volumes,

p = V - R T / P ( 4 )

one obtains with the aid of Equation 3 the relation

p = b - a t / R T (5 )

I t is assumed at zero pressure that the molal cohesive energy a t o is a generalized function of the reduced temperature for nonpolar substances. Polar substances require an individual attraction coefficient. Accordingly, the temperature dependence of the attraction coefficient is given by the empirical relation

E" = A' + B ' / T , - C ' / T V 2 + D'/T,s + 64E' /27TrnL ( 6 )

The first four terms are used for representing nonpolar substances with the aid of generalized constants A', B' , C', and D'. I n addition to the generalized constants, an individual D' or the individual constants E' and m are used for polar substances.

The influence of pressure on the attraction coeficient is expressed em- pirically as a function of the reduced pressure, P,, and the ratio of pressure to vapor pressure P/Po according to

$ = E " + F'P, + G'Pr2 + "P,3 + K(P/P0)3 ( 7 )

(For T, > 1 the last term of Equation 7 is to be taken equal to zero.) The coefficients F', G', and H' are temperature dependent, namely,

F' = F / T V 4 , G' = G / T v F H ' = H/T,6 ( 8 )

The last term of Equation 7 contains the vapor pressure and is specific for each substance. This furnishes individual values for the molal volume of any vapor at saturation pressures. The significance of this term is illustrated later in Figure 9 which gives molal volumes of saturated propane vapors.

The equation of state comprising the three relations in Equations 2, 6, and 7 represents the region of greatest interest in vapor-liquid equilibria. This is the pressure region below the critical value. As given, Equation 7 is applicable to pressures up to 90 to 95% of the critical value. An additional term which furnishes values up to the critical point has not been included as little practical use has been found for it. However,

the value of f a t the critical point can be estimated for nonpolar substances with the aid of the approximate empirical relation given in Equation 10.

Equation 2 is of the same form as the relation of Berthelot. I t differs because the constants b and a have different numerical values, and T in the last term appears to the first rather than the second power.

The Berthelot equation can be used in place of Equation 2 if the constants b and a are

b = 9RTc/128P, and a = 6bRTC2

and if a modified attraction coefficient is calculated according to

En = tT7 - 7TV2/54

Similarly, the equation of Redlich and Kwong (45) can be used in place of Equation 2 if

b = 0.0867RT0/P, and a = 0.4278R2T02 6/Po

and if the modified attraction coefficient i s

t(z3 K. = 0.98615ETr0.5 - 0.089.53Tr*J

Equation 6 applies over the whole temperature range. I t is probably more accurate in the reduced temperature range 0.5 to 1.5. Equation 7 gives the influence of pressure on the attraction coefficient. I t furnishes a good approximation of this effect in the pressure range of zero to the saturation pressure Po, for temperatures below the critical isotherm. Along the critical isotherm it is applicable up to about 90 to 95% of the critical pressure.

Generalized and Individual Coefficients

Sonpolar substances can be represented approximately with the aid of a single set of constants in Equations 2, 6, and 7. Accordingly, generalized constants have been set up from available experimental data. Approximate values for these are

A' = 0.396 B' = 1.181 C' = 0.864 D' = 0.384 E' = 0,000 (for nonpolar substances)

and

F = 0.148 G = 0.103 H = 0.091 K = 0.177 E c = 1.9374 + 0.0001892T0 (for

With the aid of these generalized constants the equation of state predicts the Boyle temperature to be

1 r9) i ( l o ) nonpolar compounds)

T B o y l e = 2.5348Tc

This is in reasonable agreement with the approximate experimental value of 2.5 quoted by Glasstone (22).

392 INDUSTRIAL AND ENGINEERING CHEMISTRY

V A P O R - L I Q U I D EQUILIBRIA

Figure 2. Temperature

. D E3 I

dependence of the’ second virial coefficient ,Bo for nonpolar substances

a. 0. 0.

b. 0. A. 0. 0. 0. d .

n-Hexane Cyclohexane lines calculated, generalized coefficients Ethane ( 7 6, 32) Ethylene (76, 32, 381 Carbon dioxide (7) Argon (7, 38) Nitrogen (7, 38) Nitrous oxide ( 7 I ) lines calcu- ‘m. lated, generalized coefficients ’

0 I 1 I I I I I

Based on Equation 10, for the attraction coefficient Ec for nonpolar substances, the critical ratio (PoVc/RT,) is found with the aid of Equations 4 and 5 to be

(P,Y,/RT,) = (27/64)(0.7293 - 0.0001892Tc) (11)

Values predicted by this relation are compared later (Table VII) with data taken from the literature.

For nonpolar substances the generalized constants are to be used in Equa- tions 2, 6, and 7. Polar substances are represented with the aid of individual values of E’ in addition to the generalized values of Equations 9 and 10. However) substances with only weak polar properties can be represented with the aid of an individual D‘, taking E‘ to be zero.

Values for both E‘ and m are given in Table I for a few polar substances. While values for E’ vary significantly for the individual substances, the values for m are about the same. A single value for m-, e.g., 4.75, will furnish a reasonable approximation for these polar substances. Thus, a single experimental vapor density for each of these polar substances is sufficient information to furnish an approximate value for the individual constant E‘.

The limiting value of the attraction coefficient a t zero pressure 5” is plotted as a function of the reduced temperature T, in Figure 1. The lower curve, 1, represents the results according to Equation G with the aid of the generalized constants of Equation 9. The higher curves, 2 through 7 , represent the polar

0 . 6 0.7 0 . 8 0.9

substances, chloroform, water, ammonia, acetaldehyde, methanol, and acetonitrile, with individual D’ or E’ constants as indicated in Table I. Figure 1 shows that polar effects decrease rapidly as temperature increases and become rather insignificant a t reduced temperatures of 1.3 or higher. This is in agreement with the often observed fact that the compressibilities of both polar and nonpolar gases a t high temperatures can be represented by generalized charts which conform to the principle of corresponding states.

Interpolating and Extrapolating Data. Equations 2, 6, and 8 can be used for the purpose of interpolating and extrapolating experimental data. De- pending on the quality and quantity of data available, any or all of the constants can be made individual. How- ever, it is recommended that the generalized value of B‘ given in Equation 9 be used in all cases. A special advan- tage of this technique is that algebraic results are obtained also for the derived thermodynamic quantities which are of primary interest in problems concerning phase equilibria. This technique is recommended for‘use with the “quantum gases” such as hydrogen and helium.

Combining Coefficients for Mixtures

I t is usually assumed on the basis of kinetic theory that the constant for a binary mixture of gases is related to those of the individual gases by the relation

cm = ClYlZ + C Z Y Z + 2ClZYlYZ (12)

1.0 1 . 1 1 . 2 1 . 3

in which c12 is the so-called “interaction constant.’’ Several assumptions have been used to evaluate this constant. The simplest of these is the arithmetic mean

c12 = (c1 + c2)/2 (13)

which upon substitution into Equation 12 leads to

cm = z C,Yi (14)

Assuming the “interaction constant” as the geometric mean

cis = (c1cz)O-S (15)

one obtains a second simplification of Equation 12-, namely,

c,o.5 = zcio.syi (16)

For mixtures the b , constants are calculated by the linear combination, Equation 14, and the ( a t ) , coefficients are derived by the linear-square-root combination expressed by Equation 16. The value of ti is calculated from Equations G and 7 with

(17) TTi = T/Tci and PTi = Yi PIPci

Second Virial Coefficient

The second virial coefficient 6” for a pure substance is calculated at zero pressure with the aid of Equations 5 and 6 according to

(18) 0“ = b - a.$”/RT

It is related to the second virial coefficient B1 given by the usual virial equation in pressure by the simple relation

0” = RTBi. (19)

VOL. 50, NO. 3 MARCH 1958 393

Table 1. Constants for Polar Substances

Compound TO PO D' E' m Data Source

Acetonitrile 547.86 47.7 0.384 0.123 4.94 (31,47) Acetaldehyde 461.16 63.2" 0.384 0.092 4.7 (@) Acetone 508.16 47.0 0.384 0.053 4.75 (47,61) Ammonia 405.6 112.75 0.384 0.051 4.45 (I,$", 47, 62) Water 647 218 0.384 0.026 4.75 ( I & , 47,62) Methanol 513.16 78.6 0.384 0.120 4.75 ( 5 2 , 4 7 ) Ethanol 516.3 6 3 . 1 0.384 0.089 4.75 (18, 25, 27, SO) Dimethyl ether 400 51.99 0.474 0 ... (11) Diethyl ether 467.8 36.7 0.425 0 . . . (34)

Ethyl chloride 460.0 51.99 0.440 0 . .. (52) Methyl fluoride 317.6 58.0 0.520 0 . * . (11) Sulfur dioxide 430.6 77.78 0.470 0 * . * (11)

Methyl chloride 416.0 65.9 0.456 0 . . . (26)

Chloroform 533.17 54.9 0.447 0 . . . (19, 20, 52, 36, 47)

Assumed value for acetaldehyde. Quoted by Rowlinson (47) and given graphically by Lambert and coworkers (32). Quoted by Rowlinson (47).

Tr

Figure 3. P o for polar substances

Temperature dependence of the second virial coefficient

A. Acetonitrile (32, 47) 0. Methanol (29, 32, 47) 0. Ammonia ( 1 4, 47, 52) 0. Water (14, 47, 52) lines calculated, individual E'

BENZENE-

600 300 3 2 0 340 360 3 E

To K

Figure 4. Second virial coefficient Po for benzene

- - - - . Calculated, generalized

-. Calculated, individual co-

0. Scott and coworkers (50) 0. Allen and coworkers ( 3 ) 0. Baxendale and Enustun (5) A . Eucken and Meyer (75) 0. Lambert and coworkers (32 ) 0. Casado and coworkers ( 1 0 ) 0. Francis and McGlashan (27 ) 0. Homann and McManamey (27)

coefficients

efficient D' = 0.396

I O

With the aid of Equations 14 and 16, In harmony with the results of Stock- Equation 18 furnishes the second virial mayer (53) the nonpolar and polar coefficient for a mixture of nonpolar fractions of energy of attraction are gases-namely , combined separately for mixtures of p m O = ZbiYi - [Z(ai.$io)o.6Yi]Z/RT nonpolar and polar gases according to (21))

* p," = Zb<Y{ '- [Z(a&")0.6Yz]2/RT --

[q )0.5yi] Z / R T (21 )

The first four terms on the right-hand side of Equation 6 with the aid of Equa- tion 9 give io, the nonpolar part of the attractive coefficient E " . The polar part 'io is given by the difference E" - 4". If each of the gases is a nonpolar substance, the p's are each equal to zero and Equation 21 reduces to Equa- tion 20.

For treating polar and nonpolar mixtures, Scatchard and Ticknor (49) assume that the interaction between polar and nonpolar molecules is determined only by the nonpolar repulsion and dispersion terms and that these are given by the Berthelot equation using a Lorentz combination to calculate b i , and a geometric mean to furnish ( u k j / b 6 j 2 ) .

For mixtures of polar gases in which special "chemical" effects are present, individual binary coefficients may be required. These coefficients are properties of the binaries and are not readily predicted from the data for the pure components. T o take these into account: binary terms with empirical coefficients are added to Equation 21 to furnish

p," = Equation 21 $.

*

0.526ijYiYi/XT (22) ii

in which the binary coefficients depend on temperature according to 6,,/RT L=

Eli'/ TI , jm'. In the last term of Equation 22 the summation counts every term twice, thus the coefficient 0.5. Coeffi- cients aii or a,, equal zero. The constants E'IZ and m' are determined em- pirically from binary data at a minimum of two temperatures. For the binary terms, T r i j = T/Toi, and Tcdi = ( T,<Tcj)0.5s

There is one type of mixture in which the binary coefficient 64, is always required, although its contribution may completely disappear a t high temperatures. This is a mixture in which neither pure component is associated through hydrogen bonding but in which mod- erately strong interassociation between the unlike molecules takes place because of hydrogen bonding. Here the coefficient 612 is negative and becomes especially significant at low temperatures. Examples of such mixtures are halogen- ated hydrocarbons with an ('active" hydrogen, as in chloroform, mixed with an ether or with a ketone. Binary coefficients are probably required in any mixture in which strong interassociation or %ompound" formation takes place between unlike molecules one or more of which is relatively nonassociated in the pure vapor state.

For mixtures of gases in which each pure component is associated and in which interassociation also takes place


VAPOR-LIQUID EQUILIBRIA

Tc 1

Substance K.

Argon 150.65

Carbon 304.3 dioxidea

Carbon mon- 134.2 oxide

Nitrous oxide 309.6

Nitrogen

Methane

Ethylene

Ethane

126.06

191.1

282.17

305.44

Cyclohexane 554.16

n-Hexane 507.16

Benzeneb 561.66

Carbon tetra- 556.3 chlorideC

Table II. Second Virial Coefficients for Nonpolar Substances

pc , Atm.

48.0

73.0

35.0

71.7

33.5

45.8

50.3

48.45

40.6

29.5

47.7

44.98

Po, Cc./Mole T , O K.

100 150 273.16 323.16 373.16 -219 -87.8 - 19 - 8 -1 - 178.6 -85.5 - 174.5 -85.4 -21 - 11

T, O K. 273.16 300 323.16 373.16 423.16 500

- 145.4 - 119 - 102 - 74 - 55 - 35 - 151 - 103 - 74 - 52

T , O K. 134.2 200 250 300 - 106.6 -42.3 -20.6 -7.3

-44.5 -8.5

T , ' K. 273.16 294.16 309.16 - 156.3 - 133.4 - 119.6 - 157 -136.7 - 126.9

T, O K. 130 150 200 273.16 373.16 423.16

-97.8 -34.8 - 9.2 7.2 12.1 - 97 - 74 -34.8 - 10.3 6.6 12.3

T , OK.

273.16 323.16 423.16 - 51 - 32 - 9 -53.9 -34.6 -11.6

T, OK.

181.16 211.16 263.16 323.16 373.16 423.16 - 426 - 293 - 181 - 117 - 84 - 59 - 430 - 306 - 190 - 119 - 86 -62.7

T,,O K. 214.16 273.16 350.36 - 383 -222 - 130 - 390 - 227 - 145

T , K. 324.16 350.96 380.16 403.16 554.16 - 1318 - 1068 - 880 - 757 - 379 - 1318 - 1099 - 900 - 745

T , K. 304.16 313.16 335.16 395.16 507.16 - 1554 - 1204 - 817 -477 - 1600 - 1260 - 870

T , ' K. 297.68 318.96 335.16 359.36 - 1515 - 1239 - 1079 - 897

T , ' K. 318.16 333.7 343.2 351.16 - 1274 - 1117 - 1036 - 975

Source

Calcd. (7)

Calcd. (7)

Calcd. (7)

Calcd. (16, 3%?, $8)

Calcd. (16, $3)

Calcd.

Calcd.

a Polar. See data plotted in Figure 4 (3, 6, 10, 16, 39, 60). See data plotted in Figure 5 (9, 16, 90, 81, 38).

VOL. 50, NO. 3 MARCH 1958 395

Substance

Acetonitrile

Table 111. Second Virial Coefficients for Polar Substances

Acetaldehyde

Acetone

Ammonia

Water

Methanol

Dimethyl ether

Diethyl ether

Methyl chloride

Ethyl chloride

Methyl fluoride

Sulfur dioxide

Chlorof ormC

bo, Cc./Mole Source T, OK.

324.16 364.96 383.50 438.29 493.07 -3695 -2084 -1664 -950.4 -615.0 Calcd. -4000 -2100 -1700 (8% 47)

T. K. 298.16 313.16 373.16 461.16 533.16 -1201 -978 -503 -257 -172 Calcd. -1230 -960 -540

T , O K. 303.2 323.2 349.16 373.16 403.16 423.16 -1907 -1470 -1096 -866 -672 -578 Calcd. -1860 -1560 - 700 (47)

243.16 273.16 303.16 333.16 373.16 473.16 -570 -371 -262 -197 -143 -78 Calcd. -560 -367 -261 -197 -143 -77 (14?47,62)

T , O K. 343.16 373.16 423.16 473.16 573.16 700 -625 -449 -287 -201 -118 -73 Calcd. -638 -450 -284 -197 -112 -69 (14,*47,62)

T, K. 319.16 351.36 393.16 404.16 461.84 513.16 -1521 - 965 -530 -344 -242 Calcd. -1430 -1010 -620 -535 (52, 47)

T , OK. 273.16 294.16 313.16 -608 -499 -425 Calcd. -600 -499 -421 (11)

T , K. 319.61 350.16 405.16 -906 -721 -512 Calcd. -918 -720 -490 (54)

T , O K. 273,l6 373.16 473.16 -535 -245 -142 Calcd. - 535 (25)

T . O K. .~ 324.16 368.16 406.16 -600 -437 -356 -593 -460 -348

T, a K. 273.16 294.16 317.66 -256 -214 -177 -256 -212 -182

303.16 323.16 -400 -388 -411 -329

T, a K. 315.7 333.7 349.3 363.2 382.1 397.4 -1081 -921 -811 -730 -640 -579 Calcd.

a Quoted by Rowlinson (47) and given graphically by Lambert and coworkers (38). * Quoted by Rowlinson (47) .

See Figure 6 for literature data (19, 80, 82, 86, 47).

between the unlike molecules, it is ex- pected that the binary coefficients, & i s , will be very small and for most practical purposes may be neglected. The few data available to us support this view, but more data are needed for these systems.

Equation 18 has been used with the aid of Equation 6 and the generalized constants of Equation 9 to calculate the second virial coefficients for a number of nonpolar substances. The results are compared with values given by other investigators or with values derived directly from experimental data in Table 11. A better comparison with the experimental results is obtained graphically in Figure 2 as a greater number of experimental data points are given in the plots.

The calculated results for polar compounds are compared with data in Table 111. Calculated curves for water, ammonia, methanol, and acetonitrile are compared with experimental results also in Figure 3.

The experimental results of several investigators are compared for benzene in Figure 4. The dashed curve gives the second virial coefficient calculated with the aid of the generalized constants of Equation 9 (Table 11). The full line is the corresponding result calculated with an individual constant D' = 0.396 as data for mixtures involving benzene, given by some of the same investigarors, are calculated later. The varied results obtained by the several investigators for benzene show that to calculate the mixture one must start with values somewhat consistent with the same investigators' results for the pure substances.

Similarly, the data of several investigators are compared with the calculated results for carbon tetrachloride in Figure 5 and for chloroform in Figure 6.

For benzene and also for carbon tetrachloride the calculated results based upon the generalized constants are within the scatter of the results of the several investigators. For chloroform the generalized constants provide, in general, lower second virial coefficients than obtained experimentally.

The second virial coefficients for mixtures of benzene and chloroform and of carbon tetrachloride and chloroform are reported by Francis and McGlashan (20). These are compared in Table IV with results calculated according to Equation 20. The pure component curves are the full lines given in Figures 4, 5, and 6. The individual D' coefficients for benzene and for carbon tetrachloride were used to make the pure component results consistent with those for the mixtures determined by the same investigators.

The application of Equation 21 to mixtures of nonpolar and polar gases is illustrated with the three binary systems, acetone-cyclohexane, diethyl-



1800

1 4 0 0

a

1000

6 0 0

amine-cyclohexane, and acetonitrile-cyclohexane, reported by Lambert and coworkers (37). The generalized coefficients for nonpolar compounds were used for cyclohexane and the individual coefficients given in Table I were used for acetonitrile and acetone. The experimental point given (37) was used for diethylamine, although it is about 50 ml. higher than data reported earlier (33) by one of the same investigators The value for cyclohexane reported (37) with this same binary system also appears high. Indeed, it is more than 150 ml. higher than the value reported for cyclohexane at the same temperature with the binary system involving acetonitrile and about 50 ml. higher than the corresponding value reported with the binary involving acetone. The value for cyclohexane calculated from the generalized constants is almost halfway between the upper and lower experimental values. A compromise of data from two sources (47, 57) for acetone gives values somewhat lower than those reported (37) with the mixtures.

The calculated results are compared with the experimental values for the mixtures in Figure 7,A, B, C, and D. Calculated results are believed to be well within the accuracy of the experimental data for the systems acetone- cyclohexane and diethylamine-cyclohexane.

In Figure 7,C and D, the calculated and experimental results are compared for mixtures of acetonitrile and cyclohexane. The combination rules do not predict these mixtures quite as well as for the other two systems. However, it is difficult to say whether the cilculated values are slightly high because of real deviations from the combination rules or whether the differences are due to combined inaccuracies in the pure component values for acetonitrile and in

I20[

CARBON TETRACHLORIDE

Figure 6. Second virial coefficient for chloroform 1 0 o c

. Generalized caef- - - - - flcients

cient D' A'.. -. Individual coeffi- ~

cI 8 0 0

A A +. 0. Francis and McGlashan

0. Rowlinsan (47) quoting (20)

1 I '1 A A lambert and coworkers 600

:bb\

Table IV. Second Virial

T, Mixture O K.

Benzene-chloroform 315.7 323.2 333.7 343.2 349.3

Carbon tetrachloride- 337.2 chloroform 323.2

333.7 333.7 323.2 315.7

323.2 343.2

a Data of Francis and McGlashan (20).

Coefficients for

Y2 0.4992 0.4992 0.4992 0.5000 0.5058 0.2363 0.2363 0.2363 0.5034 0.5034 0.5034 0.5034 0.4932 0.4932 0.4932

Mixtures (Bm

Calcd. - 1199 -1120 - 1023 - 947 - 901 - 1121 - 1240 - 1152 - 1068 - 1157 - 1239 - 1313 - 1242 - 1160 - 979

Exptl." - 1145 - 1120 - 1030 - 950 - 945 - 1115 - 1265 - 1180 - 1080 - 1155 - 1225 - 1310 - 1180 - 1155 - 995

the measured values for the mixtures. Figure 3 shows that the uncertainty in the values taken for the pure component can be of the order of f.200 ml. for this temperature region. The accuracy of the data for the mixtures is probably of this same order of magnitude, except a t high concentrations of cyclohexane where they are probably good to about &IO0 ml.

The application of Equation 22 to mixtures having special "chemical" in- teractions is illustrated with the system chloroform-diethyl ether. For this system the third term in Equation 22 is quite small. Indeed, the pure components are calculated with sufficient accuracy neglecting the polar term (go = 0 in Equation 22 and E' = 0 in Equation 6) and making use of an individual coefficient D'. Coefficients for the pure components are given in Table I. The binary coefficients aI2 and m' were derived from the data of Fox and Lambert (79). The calculated and experimental values are compared

in Figure 8 for the five temperatures 53.2', 65O, 78', 90°, and 120' C. The equations can be used to interpolate or extrapolate the data. The data of Fox and Lambert are believed to be low at the high temperatures, as shown in Figure 6 where the results of several investigations are compared for chloroform.

Vapor Volumes of Pure Substances and Their Mixtures

Equations 2, 6, and 7 with the aid of suitable constants, generalized or individual, furnish the vapor volumes of pure substances for pressures from zero to about 90 to 95% of the critical value. In view of Equations 2, 14, 16, and 21 the molal volume of a gas mixture becomes

V, = RT/P + Zbd Yd - [Z(ui&)oJ YslZ/

RT - [Z(ui/)OJ Yil2/RT (23)

For nonpolar gases Zi is zero, and for

CHLOROFORIT I -'os,

I I I I 320 340 3 6 0 3 8 0 4 0 0

To K

VOL. 50, NO. 3 MARCH 1958 397 ,

1 4 0 0

1300

1 2 0 0 . E a

1100

1000

900 I I 1 I I I 0 . 2 . 4 . b . 8 1 0

MOLE FRACTION C Y C L O H E X A N E

0 . 2 . 4 b 8 1 0 MOLE F R A C T l O K CYCLOHEXANE

1 0 0 0 1 I I I I I o , 2 . 4 . b . a 1 . 0

MOLE FRACTION CYCLOHEXANE

3000

5001-1 0 . 2 4 b . 8 1 0

MOLE FRACTION CYCLOHEXANE

Figure 7. Second virial coefficients for mixtures

A. Acetone-cyclohexane 8. Diethylarnlne-cyclohexane C. Acetonitrile-cyclohexane D. Acetonitrile-cyclohexane Data from (32) calculated with Equation 21 Broken lines indicate estimated reliability of cal-

culations; vertical lines through data points indicate accuracy claimed by lambert and coworkers; values reported for cyclohexane shown at right side of flgures

mixtures of these the last term is zero. The special binary terms of Equation 22 can be added when applicable.

Vapor volumes for the nonpolar substances propylene and propane have been calculated according to Equation 2 with the aid of Equations 6-8 and the generalized constants of Equations 9 and 10. The calculated results are in reasonable agreement with the corresponding experimental data (77, 44) in Table V. For propylene the average deviation of the calculated from the experimental results is +0.02%, with a maximum deviation of -0.77%. For propane the average is +0.03% with a maximum deviation of ~ k 0 . 6 7 7 ~ .

Figure 9 compares calculated and experimental molal volumes of saturated propane vapors over the temperature range 37.78' to 90' C. The experimental values reported by Clegg and Rowlinson (72) are lower than those of Reamer, Sage, and Lacey (44) at temperatures above 73' C., but a t temperatures below 60" C. the results are reversed. While the calculated values agree better with the experimental results of Reamer and coworkers, they fall between the two sets of experimental results at both high and low temperatures. The average deviation of the calculated from the 10 experimental values beIow 90" C. is +0.36%. Values should not be calculated with the aid of Equation 7 above about 90" C. as the reduced pressure is already about 0.9. However, values at the critical point are given according to Equation 11 (see Table VII).

The significance of the last term of Equation 7 involving the vapor pressure is apparent from the dashed line in Figure 9. Values represented by

D 90'C /

" 1 50C ' O 0 W 0

YOLE F R A C T I O N CHL,OROFORM

Figure 8. Second virial coefficients for chloroform-diethyl ether mixtures according to Equation 21 with E'rz = -7.34 and m' = 12

Data from (19)

this curve are those obtained if one neglects this term in the calculation of the molal volume at the saturation pressure. They range from 4 to 14% higher than those given according to the complete equation. This last term of Equation 7 has less influence on the fugacity coefficient than on the molal volume.

Vapor volumes for the polar substance water were calculated with the aid of an individual constant E'. These calculated results, shown as a function of temperature and pressure, are compared with experimental data in Table VI. The results are in fair agreement with the tabulated values reported by the

Table V. Calculated and Experimental Vapor Volumes for Propylene and Propane

Molal VoIumes of Vapors, Cc./Gram Mole T, p, Propylenea Propaneb K. Atm. Calcd.c ExptLd Calcd.e ExptLL

277.60 1.0000 2.7218 5.4437

310.94 1.0000 2.7218 5.4437

10.2069 13.6091

22,371.8 7,950.9 3,722.6

25,197.9 9,054.0 4,358.7 2,143.8 1,485.2

344.27 5.4437 4,929.3 13.6091 1,797.9 26.2704 ... 27.2183 694.2

13.6091 1,929.7 27.2183 803.5 34.0229 554.4

360.94 5 , i 4 3 7 ...

a To = 365.0' K.; Po = 45.4 atm. b T, = 370.0' K.; Po = 42.0 atm.

Calculated with aid of generalized constants. Farrington and Sage (17).

e Reamer, Sage, and Lacey (44).

22,371.2 7,946.5 3,724.5

25,190.5 9,044.6 4,349.4 2,143.8 1,485.8

h , 925.0 1,792 I 3

693.0

1,925.9 803.5 558.7

...

...

25,158.6 9,014.7 4,313.8 2,080.7

4,894.4 1,754.2

661.7

5,175.3 1,885.3

750.4

...

e . .

25.171.1 9,020 a 9 4,313.8 2,080.1

4,894.4 1,748 .O

661.7

5,175.3 1,897.8

745.4

I . .

e . .



National Bureau of Standards (38). The calculated values are least accurate a t 200 atmospheres and 800' K. where they are about 4% low. Up to 100 atmospheres the agreement is within about 1.801, or less.

Values of the critical ratio (PcVo/RT,) have been caIculated with Equation 11 for a number of organic and inorganic substances. They are compared in Table VI1 with values taken from the literature. The average deviation of the calculated from the literature values is +0.40j, for the 28 organic substances with a maximum deviation of about +6% for ethylene. The maximum deviation for the inorganic substances given in the table is +3.8% for carbon monoxide.

The critical ratios for the polar substances for which E' values have been derived are given to about &5% by subtracting 0.6E' from the values cal-

Table VI. Calculated and Experimental Vapor Volumes for Water

T, O K.

'% Deviation p, V , Co./Mole of Calcd.

Atm. Cdcd." Expt1.b from Exptl. 400 1 32,472 32,459 0.04

500 1 40,856 40,857 10 3,927 3,919

600 1 10 40 100

700 1 10 40 100 200

800 1 10 40

100 200

49,128 4,816 1,119

359.9

57,366 f

5,670 1,360

494.2 194.5

65,592 6,511 1,587

600 266

49,131 4,818 1,118

353.5

57,371 5,675 1,365

499.6 203.8

65,598 6,516 1,593

607 278

culated according to Equation 11. If the influence of pressure on the attraction coefficient .$ is the same for polar and nonDolar substances and if the values .

a Calculated, Equations 2, 6, 7 with E' = 0.026, m = 4.75. National Bureau of Standards (38).

of E' and rn derived from low temperature data are 100% correct, the value of the critical ratio should be given by subtracting E' from the value calculated according to Equation 11. This is exactly correct for water, the polar substance for which most reliable data are available.

The molar volumes for two different mixtures of propylene and propane are given in Table VIII. The calculated results were obtained with the aid of Equation 23 using generalized constants. The experimental results are those of Reamer and Sage (43). The agreement between calculated and experimental volumes is also satisfactory for these mixtures.

Fugacity Coefficient

f, is the relation An adequate definition for fugacity,

R T In f = RT In P 4- @ d P (24) L I P

DefiGing the fugacity coefficient for a pure gas as

4 = f / P (25)

Equation 24 becomes

or

In 4 = ( l / R T ) ( V - RT/P)dP (27)

its equivalent in terms of the volume V. If Equation 5 is substituted into Equa-

tion 26, or Equation 2 into Equation 27,

LP

0.00 0.20

-0.01 -0.04

0.09 1.81

-0.01 -0.09 -0.37 -1.08 -4.56

-0.01 -0.08 -0.38 -1.15 -4.35

Organic Compounds

Methane Acetylene Ethylene Ethane Propylene

Propane 1,3-Butadiene Isobutylene 1-Butene 2-Butene Isobutane

n-Butane Isoprene Isopentane n-Pentane Cyclopentane

2-Methylpentane Benzene n-Hexane Methylcyclopen-

tane

Toluene n-Heptane 2,2,4-Trimethyl-

pentane

n-Octane n-Decane Dimethyl ether Diethyl ether Ethyl chloride

Table VII. The Critical Ratio (P,V,/RT,)

Calcd.c 0.292 0.283 0.285 0.283 0.278

0.278 0.274 0.274 0.274 0.273 0.275

0.274 0.269 0.271 0.270 0.267

0.268 0.263 0.267 0.265

0.260 0.265 0.264

0.262 0.258 0.275 0.270 0.271

% Devi- ation of Calcd. from API

No. 44a 1.0 3.3 5.9

-0.7 1.5

0.4 1.1

-0.7 -1.1 -1.1 -2.8

0

1.1 0.4

-3.3

-0.4 -4.0

0.7 -2.6

e . .

0 1.9

-4.0

2.3 4.8 2.2b 4.2b 0.7b

American Petroleum Institute (4). 6 (39). 0 Calculations according to Equation 11.

Inorganic Substances

Nitrogen Oxygen Carbon monoxide Argon Carbon dioxide

Hydrogen sulfide Chlorine Sulfur dioxide Helium Hydrogen

% Devi- ation of

(22) Calcd.c 0.298 0.295 0.297 0.296 0.283

0.278 0.274 0.273 0.307 0.305

Calod. from Lit.b

2.1 1.0 3.8 1.7 1.1

... -0.4

0.7 1.7 0

VOL. 50, NO. 3 MARCH 1958 399

Table VIII. Calculated and Experimental Vapor Volumes for Mixture of Propylene and Propane

V , &./Gram Mole 310 94' K. 344.27' K. 380.94' K.

Mole Fraction P ,

Propylene Atm. Calcd." ExptLb Calcd." Exptl.* C a l ~ d . ~ Exptl! 0.2411 1.0000 25,164.9 25,146.1

2.7218 9,020.9 8,995.9 5.4437 4,324.4 4,300.1 4,904.4 4,897.5 10.2069 2,097.6 2,093.2 13.6091 1,766.7 1,758.6 1,904.1 1,897.2 27.2183 763.5 766.6

0.6289 1.0000 25,183.0 25,164.9 2.7218 9,039.0 9,033.4 5.4437 4,341.9 4,335.0 4,917.5 4,906.2 10.2069 2,121.9 2,124.4 13.6091 1,783.0 1,771.7 1,904.1 1,897.2 27.2183 763.5 766.6

a Calculated with the aid of generalized constants. * Reamer and Sage (43) .

and the integration with respect to pressure is carried through at constant temperature remembering Equations 3, 7, and 8, one obtains

I n 9 = (Pr /8T, )[1 - (27/81; ) { E o + F" P, + G" P: + H" P," +

Krf ( P / P 0 ) 3 ) ] ( 2 8 )

with F" = F'/2, GI' = G'/3, H" = H'/4, and K" = K/4.

This equation furnishes the fugacity coefficient for a pure component as a function of Pr, T,, and (P /P" ) for pressures up to about 90 to 95% of the critical value.

The fugacity Ji of component i in a gas mixture is

h f i = ( l / R T ) J p ( v i - R T / P ) d P + In Y , P (29)

If the fugacity coefficient of i is defined as

4% = f % / Y % P (30)

in Lvhich Y,P replaces P in Equation 25, then

In 9% = (l /RT) (vt - R T / P ) d P (31)

The partial molal volume Vi of any component in a gaseous mixture of n components is derived at constant temperature and pressure according to

LP -

vi = v, + BV,/BY, -

The summation extends over all components including i. Expressing V, with the aid of Equation 23, one finds the partial molal volume to be

pi = R T / P + bi - a&,/RT +

in which

G~~ = (aiii)0.5 - ( a j ! j ) o . 6 , * - - - Gij = (a&)oJ - (aj(j)0.6 (34)

and the corresponding values a t zero pressure are denoted by Giio and Gijo.

If one substitutes Equation 33 into Equation 31 and carries through the integration with respect to pressure, In $i is, with the aid of the reasonable assumption, noted

In @a = ( P / R T ) [ b i - aiwi/RT +

if

wi = ( i o + F"P,j + G''P,,' + H"P,,3 + Kr'(YjP/Pi0)3 ( 3 6 )

and Ppi = YiP/Pci. Values of the coefficients are given according to Equations 8, IO, and 28. For integrat- ing the composition term, leading to the last two terms of Equation 35, it is assumed that the Ea's are independent of pressure and equal to the

Because the composition term is usually small relative to the total value, the discrepancies arising from this assumption are small and usually negligible except in the retrograde region for systems in which the composition term assumes a dominant role.

Vapor Phase Imperfections in Vapor-liquid Equilibria

In vapor-liquid equilibria the liquid compositions are denoted by xI)s and the vapor compositions by Y ~ s . The molal and partial volumes in the liquid phase are designated as Vi' and pi', respectively. Further, if the standard reference pressure is denoted by P*, the vapor-liquid equilibrium is given by

The fugacity coefficients +io and (pi cor- respond to the pressures Pio and P, respectively, and are properties of the vapor. The second and third terms of Equation 37 comprise the effect of pressure on the liquid.

Unless Pit can be distinguished from Vi', the third term of Equation 37 cannot be evaluated and there is little reason for assigning a reference pressure P". I n cases for which the liquid phase activity coefficient yi cannot be neglected, the usual convention that y = 1 for the pure component is adopted, in which case the equilibrium pressure is then the vapor pressure Pio. Moreover, the terms involving Vi' and Vi' are usually small relative to the fugacity coefficients $io and $(. For a first approximation of the influence of pressure on the liquid, it is assumed that Vi' = Vi' and that Vi' is independent of pressure. With the aid of these assumptions Equation 37 simplifies to

In y i = In (Y iP/x i Pi") +

-..

In (@i/@,') -Vi' ( P - P i " ) / R T (38)

Together with Equation 35, Equation 38 furnishes the required relations for calculating the effects of the vapor phase imperfections and of pressure on vapor- liquid equilibria. These relations can be used also to calculate the effect of an inert gas on the equilibrium. The last term of Equation 38 is an approximation for the effect of pressure on the liquid. The middle term is the influence of vapor phase imperfections. If a sufficient number of reliable data are available t o distinguish clearly between Vi' and Vd', Equations 35 and 37 are to be used in which case a reference pressure P* must be chosen.

Special Approximations. If the total pressure of the system and the individual vapor pressures for the pure components differ by only a few atmospheres, Equa- tion 38 can be safely simplified in the following way. For wi in Equation 36 at pressure Po the same value it has at pressure P is used. With this approximation and with the aid of Equation 35, Equation 38 simplifies to

In yi = In (Y iP/x iPi ' ) + In 0; (39)

In which the last term is given by

In ei = [(P - P i " ) / R T ] [ b i - Vi' - aiwi/RT + (ZGi j 'Y j ) ' /RT +

( 5 G i j 0 Y j ) ' / R T ] (40)

The effects of vapor phase imperfections and of pressure on the liquid phase are combined in the single coefficient Oi which is called the "imperfection- pressure," or "IP," coefficient.


VAPOR-L IQUID EQUILIBRIA

TEMPERATURE, O C

Figure 9. Molal volumes for saturated propane vapors

-. Calculated with Equation 7 - - - - . 0. A.

Calculated neglecting last term of Equation 7 Data of Reamer, Sage, and Lacey (44) Data of Clegg and Rowlinson (12)

P R E S I U R C , L B /$Q IN.

Figure 10. pane ( I )-isopentane (2)

Vapor/liquid equilibrium ratios for pro-

-. Calculated Data from (54) 0. 167' F. (75' C.) A. 122' F. (50' C.) 0. 77' F. (25' C.) 0. 32' F. (0' C.)

The additional restriction to low pressures gives a further simplification in Equation 40 as wi can be replaced by E". This is usually an adequate approximation for ordinary vapor-liquid equilibria under total pressures of a few atmospheres. If the cohesive en-, ergies of the components are not mark- edly different, the last two terms of Equations 35 or 40 become insignificantly small.

The special case in which binary coefficients are required, as noted in Equation 22, leads to In 6i = Equation 40 +

(P - P')[?GijY, (1 - Y$) - 1

O.586,kYj Yk] /R2 T2 (41 ) Jk #

if the 6's are independent of pressure.

The symbol j k denotes that j and k

each assumes all values except i. #

Applications In the binary mixture propane-iso-

pentane (2-methylbutane), the liquid phase activity coefficients are very close to unity. Vapor-liquid equilibria have been calculated with the aid of Equation 40 taking wi = 5". The calculated values of the equilibrium ratios (Yr/xi) at Oo, 2 5 O , 50°, and 75' C. are compared in Figure 10 with the experimental results reported by Vaughan and Collins (54). For propane only three points of the 26 exceed a deviation of 2y0 from the experimental values. For isopentane only five points exceed a deviation of 3.5%.

. 1

I ' 1 I

100 1.000 1 10,000

PR E I I U R f, d . I Q . IN.

Figure 1 1. Vapor/liquid equilibrium ratios

-. Calculated

0. Methane-n-butane at 160' F.; data from (48) A. Methane-propane at 2' F.; data from (55)

. Ideal - - - -

Of particular importance in petroleum processes is the influence of a volatile gas on the vapor-liquid equilibrium ratio of the less volatile components. Four binary systems have been selected as examples, each containing one volatile component which is present in solution above its critical temperature.

If the equilibrium ratio is defined in the usual way to be K, = Yi/xj, it can be expressed also with the aid of Equa- tion 38 according to

If the temperature of the mixture is

Figure 1 2. Vapor/liquid equilibrium ratios

-. Calculated

0. A.

. Ideal - - - - Methane-n-decane at 460' F.; data from (42) Hydrogen-propane at 0' F.; data from ( I )

vol. 50, NO. 3 MARCH 1958 401

above the critical temperature of any one of the pure components, the Ri value for that component is not readily calculated from Equation 42 because the vapor pressure Pio and the liquid partial molar volume Vi’ become am- biguous above the critical temperature. O n the other hand, Ki values for components which are present a t reduced temperatures less than unity are readily calculated with the aid of Equation 42.

Equilibrium ratios K< calculated according to Equation 42 with the aid of Equation 36, taking wi = to, are compared with experimental values reported in the literature for the less volatile component in each of the systems methane-propane and methane-n-butane in Figure 11 and methane-n-decane and hydrogen- propane in Figure 12. The full lines represent the calculated values wliile the ideal values R$ = P,”/P are indicated by the dashed lines.

The equations correctly predict the K values in each system up to pressures corresponding to those for which the equilibrium ratio goes through its minimum value. Beyond this the calculated values may deviate from the experimental as the value of IT rapidly approaches unity, the limiting value at the conver- gence pressure.

Nomenclature

a = van der Waals attraction constant,

b = van der Waals covolume constant,

c1, cp, c, = constants for individual gases

c12 = an interaction constant B1 = second virial coefficient in pres-

A’, B‘, C’, D’, E‘ = constants in Equa-

F’, G’, H’ = temperature dependent co-

F, G, H, R = generalized constants

see Equation 3

see Equation 3

1 and 2 and for a mixture m

sure series

tion 6

efficients: see Equations 7, 8

= an attraction coefficient-an empirical function which gives the effects of temperature and pressure on the van der Waals cohesive energy

t* = nonpolar part of attraction coefficient

= polar part of attraction coefficient = limiting value of attraction co-

efficient 5 at zero pressure f = fugacity P = total pressure PO = vapor pressure P* = reference state pressure P, = critical pressure P, = P/P, = reduced pressure of a pure

P, = YtP/Pc* = reduced pressure of

R = universal gas constant T = absolute temperature T, = critical temperature Ti = T / T , = reduced temperature V = molal volume, vapor V’ = molal volume, liquid Vt = partial molal volume of com-

substance

component i in a mixture

-

ponent i in a vapor mixture

- Vi‘ = partial molal volume of com-

x = liquid composition, mole fraction Y = vapor composition, mole fraction Z = PV/RT = compressibility factor

Po = second virial coefficient in volume or limiting value of P at zero pressure

ponent i in a liquid solution

p = V - R T / P = (RT/P)(Z - 1)

= binary coefficient in Equation 22 y = liquid phase activity coefficient 3 = f / P = fugacity coefficient for a

pure substance = f i / Y t P = fugacity coefficient for

component i in a vapor mixture = “imperfection-pressure coefficient”

for vapor-liquid equilibria which includes approximations for both the effect of imperfections in the vapor phase and the effect of pressure on the liquid phase

w = see Equation 36

Acknowledgment

The author expresses his appreciation to Mott Souders, Clarence L. D u m ? and Otto Redlich for their cooperation and helpful suggestions.

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RECEIVED for review March 21, 1956 ACCEPTED July 18, 1957

Division of Industrial and Engineering Chemistry, 129th Meeting, ACS, Dallas, Tex., April 1956.


Documents

Vapor Phase Imperfections in Vapor-Liquid Equilibria. Semiempirical Equation