ratio is not a restriction or a criterion for the applicability of the modified law.
I
N 1881 van der Waals proposed the well known law of corresponding states. The original law of van der Waals has been shown to be inexact in many cases, and it§ practical applications are often regarded with reserve. The present paper aims to ago tliis term was called the pseudo-critical volume (IS). V/Vci
1 Present address, Joseph E. Seagram A Sons, Inc., Louisville, Ky.
is called the ideal reduced volume and designated by p. Reduced temperature (T/Tc) is 8, and reduced pressure (p/pc) is ir.
According to the proposed modified law of corresponding states, a generalized relation exists such that
f(rr, 0, Ip) = 0 (1)
In other words, the ideal reduced volume is a universal function of the reduced pressure and the reduced temperature, irrespective of the nature of the gas.
One immediate advantage of the modification is that the use of the term “ critical volume” is avoided. The critical volume is much more difficult to determine than critical pressure and criti
cal temperature, and in many cases is lacking. Any uncertainty involved in the critical volume term is thus avoided. The ideal critical volume is defined in terms of the critical pressure and critical temperature. It is believed that the modification affords a better correlation of compressibility behavior both in accuracy and in scope of application.
One point is an exception— the critical point. The original law as well as the present modified form requires the same critical ratio, rc — RTc/pcVc, for all gases; it varies approximately from 3 to 5 with an average value of about 3.7. On account of the
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August, 1946 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y 805
continuity of physical properties, the immediate neighborhood of the critical point (the critical region) has to be excepted also.
G RA P H IC A L REPRESENTATION representation of Equation 2 is often called the ¿¡-chart. It may be regarded as a graphical representation of the generalized equa
tion of state. The validity of the ¿¡-chart proves the validity of the modified law.
. Several authors have made contributions to the ¿¡-chart (2, 8, 4, 5, 8, 9, 16). It would appear to be better to obtain a ¿¡-chart based on the average compressibility behavior of the gases, rather than based on the compressibility behavior of a single substance.
Deviations from one gas to another in the generalization are ex
pected. To reduce the average deviations of several gases and to have a common ¿¡-chart applicable to all gases, it would be ad
visable to present an average ¿¡-chart based on the compres
sibility data of the gases in line with the generalization. The
¿¡-charts published by other authors seem to be based on data for only a relatively few gases.
A few years ago (18) the author presented a general or aver
age ¿¡-chart based on the compressibilities of seven hydrocarbons.
An extensive comparison was made of compressibility factors de
rived from experimental data with the generalized curves (18).
The over-all average deviation of the seven hydrocarbons from the general ¿¡-chart is 1% . The chart applies to carbon dioxide, nitrogen, and steam with about the same degree of ac
curacy. The figure shows part of the chart with some experi
mental points for ten gases plotted as illustration. The present
¿¡-chart agrees fairly well with the results of earlier workers.
Other ¿¡-charts based on the author’s data have appeared else
where (2, 6, 17).
Two other ¿¡-charts were made (18) with density-temperature and density-pressure as independent variables. A chart of gener
alized isometrics was also prepared (18). The data of generalized isometrics were used to determine the constants of a generalized Beattie-Bridgeman equation of state (15).
AN A LY T IC A L EXPRESSION'S
Several investigators have presented contributions on the re
duced equations of state (7, 10, 12). A generalized form of the Beattie-Bridgeman equation of state (1, 15) is given as follows:
9(1 - «) A
(5) A — /1q(1 — a/tp)
B = Bo(l - b/<p) e = c'/<p$3
The constants were determined from the data of generalized iso
metrics (15): A „ = 0.4758, a = 0.1127, B0 = 0.18764, b = 0.03833, c = 0.05. These are universal constants, irrespective of the nature of gases. The equation has been shown to hold well for densities nearly up; to the critical, and for temperatures as low as the critical.
Table I shows the average deviations of seventeen gases from the present modification of the law of corresponding states. The
first eight gases arc directly compared to the general ¿¡-chart (18); the next nine gases are compared with the generalized Beattie-Bridgeman equation of state (15). The over-all average deviation of the seventeen gases is 1% . For hydrogen and helium the pseudo-critical temperature and pressure are em
ployed as defined by Newton (11).
Ethylene 3.5 8 0.82 Carbon dioxide 3.58 0.86
n-Heptane 3.99 2.47 Ethyl ether 3.82 1.61
Methane 3.46 1.44 Helium 3.28 0.73
Another generalized equation of state of the van der Waals type has also been presented (14):
(6) where a and ¿3 are universal constants. The numerical values of the constants are deduced by imposing on Equation 6 the two
able, it is believed, with the original van der Waals reduced equa
tion of state: ratio, t c — RTc/pcVc, would restrict the applicability of the law of corresponding states. The answer is negative if the law is modi
fied as described above.
The critical ratio enters only at the critical point, at which the modified law as well as the original law fails. Other than the critical point, the critical ratio plays no role in the present treat
ment. The criterion of constancy of the critical ratio is not a to be a universal function of x and 0:
prt = f(x, 0) (9)
For substances which have the same or approximately the same critical ratio, ¿i and ¿¡rc have the same function. For substances which have appreciably different values of the critical ratio, the crucial test would be -whether ¿i or ¿¡rc is a function of x and 8.
Investigation so far points to the conclusion that p, not prc, is a universal function of ir and 8. For instance, steam has a critical ratio of 4.3 as contrasted to the average value of 3.7, a difference of 16%. It conforms to the yu-chart with an average deviation of even less than 2 % . If we had correlated prc instead of p as a function of n and 0, the result would not have been so good as shown by Table I or the chart.
An indirect proof of the validity of the present modification is the work of Newton (11) on the fugacity chart. He showed that for twenty-four substances, deviating within 4 % from the stand
ard curves, with some exceptions the following relation holds:
f/ p = r(7T, o) GO) same question is not involved in the present generalization.
-The present mollification of the law of corresponding states, which shows that p and not pre is a function of ir and 0, may be considered to serve as a rational basis for the ¿i-chart and other related correlations. That this is a valid modification is seen from the rather wide divergencies of the known values of r„ from constancy. A number of other investigators used the law essen
tially in the modified form without explicitly pointing out that any modification was involved. Probably some of these investi
gators assumed that the validity of their correlation was limited by the lack of constancy of the critical ratio rc.
With the removal of the constancy of the critical ratio as a- necessary condition for the validity of the law of corresponding states, one may feel that the validity of the ju-chart and other related correlations is not affected by the variation of critical ratio.
The use of the ideal reduced volume enables one to correlate the thermodynamic properties in terms of <p and 8 or <p and ir with the same success as in terms of ?r and 0. It is hoped that the present discussion may help in paving the way for more extensive and confident use of the law of corresponding states in practical as well as in theoretical treatment of the thermodynamic proper
ties of real gases.
(2) Beattie, J. A., and Stockmeyer, W. H., Phys. Soc. Rcpt. Progress Physics, 7, 195 (1940).
(17) Weber, H. C., Thermodynamics for Chemical Engineers, p. 108.
New York, John Wiley & Sons, Inc., 1939.