I
CLINE BLACK Shell Development Co., Emeryville, Calif.
Vapor Phase Imperfections in Vapor-Liquid Equilibria Semiempirical Equation
Because separation processes in the chemical and petroleum industries depend to a great extent 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 t h e . perfect gas. A knowledge of these deviations as well as of methods for calculating or predicting them is important for describing the thermodynamic properties of substances. Even the basic problem of determining the thermodynamic consistency 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 multicomponent 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 : P = RT/(v - b ) a/V2 (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-
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 indicates 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 +approached. Because the region below the critical pressure is of particular importance in vapor-liquid equilibria it is represented with the aid of convenient 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
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
3.5
3.0
2.5
2.0 UJ
1.5
b = 9RTc/128P, and a = 6bRTC2 1.0
and if a modified attraction coefficient is calculated according to I
0. 5
0. b
1
I
I
I
I
I
I
0.7
0.8
0.9
1.0
1.1
1.2
En = tT7 - 7 T V 2 / 5 4
1
REDUCED T E M P E R A T U R E , Tr
Figure 1. ficient f "
Similarly, the equation of Redlich and Kwong (45) can be used in place of Equation 2 if
Temperature dependence of attraction coef1. 2.
3. 4. 5. 6. 7.
Nonpolar, generalized coefficients Chloroform, D' = 0.447 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 a f . The attraction coefficient is defined implicitly in terms of the volume V, pressure P, and temperature T according to
E"
=
(2)
in which the van der Waals constants are calculated from the critical temperature T , and critical pressure P, with the aid of
+ B'/T,
0.4278R2T026/Po
and if the modified attraction coefficient i s t(z3 K.
+
- C'/TV2 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 empirically as a function of the reduced pressure, P,, and the ratio of pressure to vapor pressure P/Po according to $ =
V = ( R T / P ) -j- b - a t / R T
A'
b = 0.0867RT0/P, and a =
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,
(3)
F' = F / T V 4 ,G' = G / T v F H ' = H/T,6 ( 8 )
If p is defined as the negative value of the residual volume or the difference between the experimental and ideal vapor volumes,
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 u p 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,
b = R T 0 / 8 P , and a
p = V
=
276 R T , / 8
- RT/P
(4)
one obtains with the aid of Equation 3 the relation
p = b - at/RT
(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 a n individual attraction coefficient. Accordingly, the temperature dependence of the attraction coefficient is given by the empirical relation
392
INDUSTRIAL AND ENGINEERING CHEMISTRY
= 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 u p 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)
r9)
and F G H K
Ec
= = = = =
1
0.148 0.103 0.091 0.177 1.9374
+
0.0001892T0 (for nonpolar compounds)
i
(lo)
With the aid of these generalized constants the equation of state predicts the Boyle temperature to be TBoyle
= 2.5348Tc
This is in reasonable agreement with the approximate experimental value of 2.5 quoted by Glasstone (22).
VAPOR-LIQUID EQUILIBRIA
.
D
E3 I
Figure 2. Temperature dependence of the’ second virial coefficient ,Bo for nonpolar substances a.
0.
n-Hexane
0. Cyclohexane lines calculated, generalized coefficients
b.
0. Ethane ( 7 6, 32) A. Ethylene (76,32, 381
0. Carbon dioxide (7)
0. Argon (7, 38) 0. Nitrogen (7, 38)
d . Nitrous oxide ( 7 I ) lines calcu- ‘m. lated, generalized coefficients
’
0
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 Equations 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
I
1
I
I
I
I
I
0.6
0.7
0.8
0.9
1.0
1.1
1.2
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. Depending on the quality and quantity of data available, any or all of the constants can be made individual. However, it is recommended that the generalized value of B‘ given in Equation 9 be used in all cases. A special advantage 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‘usewith the “quantum gases” such as hydrogen and helium.
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 ClYlZ
+
CZYZ
+ 2ClZYlYZ
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 = (c1
c12
+ c2)/2
(13)
which upon substitution into Equation 12 leads to cm = z C,Yi
Assuming the “interaction as the geometric mean
(14) constant”
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
TTi= T/Tci and PTi = Yi PIPci
(17)
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
Combining Coefficients for Mixtures
cm =
1.3
(12)
0“ = b - a.$”/RT
(18)
It is related to the second virial coefficient
B1 given by the usual virial equation in pressure by the simple relation
0”= RTBi. VOL. 50, NO. 3
MARCH 1958
(19)
393
Table 1.
p,"
Constants for Polar Substances
Compound
TO
PO
D'
E'
m
Acetonitrile Acetaldehyde Acetone Ammonia Water Methanol Ethanol Dimethyl ether Diethyl ether Methyl chloride Ethyl chloride Methyl fluoride Sulfur dioxide Chloroform
547.86 461.16 508.16 405.6 647 513.16 516.3 400 467.8 416.0 460.0 317.6 430.6 533.17
47.7 63.2" 47.0 112.75 218 78.6 63.1 51.99 36.7 65.9 51.99 58.0 77.78 54.9
0.384 0.384 0.384 0.384 0.384 0.384 0.384 0.474 0.425 0.456 0.440 0.520 0.470 0.447
0.123 0.092 0.053 0.051 0.026 0.120 0.089 0 0 0 0 0 0 0
4.94 4.7 4.75 4.45 4.75 4.75 4.75
*
*
[Z(a&")0.6Yz]2/RT--
[q Data Source
... .. .... .. ..* .*. . ..
= Zb
