I
KENNETH A. KOBE,’ Department of Chemical Engineering, University of Texas, Austin 12, Tex.
P. S. MURTI,
Deportment of Technology, Andhra University, Woltoir, India
Ideal Critical Volume for Generalized Correlations Application to the Macleod Equation o f State Too few scientists are aware of the usefulness of the ideal critical volume as a correlating parameter
THE
Macleod equation of state (4) is of the van der ‘12’aals type, but it assumes that the covolume is a function of internal) pressure. the total (external This equation was discussed by Rush and Gamson (5), whose nomenclature is used here. For 1 mole of gas, the Macleod equation is:
+
a =p
+ (alt2)
b’ = A - Ba
(2)
+ Cr2
(3)
Equation 1 is of the third degree in pressure and seventh degree in volume, hence the calculated isotherm follows the actual isotherm u p into the liquid region (whereas the isotherm calculated by the Beattie-Bridgeman equation does not do so, but turns downirard). Macleod showed that Equation 1 was applicable to a varietv of chemical compounds with good agreement between calculated and observed values. H e also shoued that for normal substances b’ = V, 2, so that a could be calculated from the critical values. a
2RTCT’, - j c V c 2
Su (6) has shown that the ideal critical volume, Equation 6, can be used in equations of state in place of the actual critical volume. Vci = R T c / j c
(6)
Thus it is necessary to know only: p , and T , in order to calculate u,i, rvhich may be used in place of Vc. This would make the hlacleod constants functions of pc and T , only (Equations 7). Values of a, A , B , C were calculated by Macleod’s method for 1 3 compounds that had been used by Rush and Gamson for their correlations. The value of the constant was plotted as a function of the respective functional relationships of Equation ?. The results are similar to those of Rush and Gamson, but better correlation was obtained. Analytical expressions rvere fitted to the lines to give the followinp. equations: a =
0.472 ( j c r c i 2 )
.4 = 0.146
B
x 106 = 1.124 (T7cL/p,)(103)f 39 ( 1 ~ ~ ~ ~ p>=~ GO 103)
if B
(4)
X
106 = 1.124 (T’cjlje)(103’1
+
Once a is evaluated, three points can be selected on the critical isotherm and b’ calculated for each point. The three values substituted into Equation 3 gives three equations that can be solved simultaneously for A , B, and C. Rush and Gamson shoMed that the constants of the Macleod equation \+ere functions of the critical constants of the compound. n =
.4
f ~ ( p ~ V (~j a~) )
=
f?(T’,)
a =
(jb) A
J5(pcT7c,z)
(‘a)
J6(T’cij
(7b)
=
B = f 3 ( V c / p , ) ( j c ) B = j 7 ( V c L / j c ) (’c) C = f4(vc/$cz)
(jd) C
f ~ ( V c t / $ c ~ ) (7d)
For the constants a and A , the function was linear, but for B and C the function was a smooth curve. Tabulation of critical data by Kobe and Lynn (3) shows that values of Vc are not knoum for many compounds for Ichich values ofpe and T , are known. I
Deceascd.
332
region it is recommended that values of B and C be calculated horn t\vo experimental points on the isotherm. Rush and Gamson applied their correlations to mixtures by usinp- the pseudocritical concept of Kay (2). They found it necessary to use an em. pirical factor, 1.08, to bring the data into reasonable agreement, giving b,/ ( a / P ) ](I’ - 1.08b’) = R T . With the present correlation it is not necessary to use the empirical factor of 1.08, sho\ring the superiority of the ideal critical volume correlation. -ill calculations are made as given by Rush and Gamson (5). T h e superiority of the correlation based on the ideal critical volume and the larger number of values of critical pressure and temperature available show the desirability of using, or a t least trying, the ideal critical and ideal reduced volume concept for correlations involving V, and V,. Fair and Lerner (7), in their generalized correlation of diffusion coefficients, use the factor
