Nomograph for P-V-T Relations of Gases below the Critical Point GEORGE W. THOMSON Ethyl Corporation, Detroit, Mich.
nential integral (d), as shown in Equation 4 : FOR VAPOR PHASEONiY --
- .7
following equation: p
log p = -P&o
+
(1)
.95
-t
C9O
Inf/P = p
In (-In
35
.90
3
a5 ‘c
4
- Inp -
p)
+1.577216 Ei (In
p)
(4)
The term Ei (In p) is conveniently obtained by entering the tables (9) of -Ei( - t ) with t = -In p. The simple parabolic equation below gives a satisfactory fit to the values and is much easier to use than Equation 4: f/P
=
0.7
-
0.4p
+ 0 . 7 . ~ (~5 )
W
temperature were presented in their paper. A direct solution for p is not
f,
0 :
w -
a
r: .3 -
3
.65 0 Q
A useful relation between f / P and P, was obtained by plotting f / P against p log p. The resulting equation,
lb
.60
.7 0
f/P = 1 - 2.181fi0P7 (6)
.55
fits the f / P values from Equation 4 to within .40 ~t0.003. .65 The f / P values shown on t h e same axis on the nomograph as the p values were obtained from Equation 4. They are a little higher than the values in the chart presented by Newton (4) and are in good agreement with the charts of Lewis and Kay (3) and Weber (6). Extrapolation of the nomograph beyond the ranges shown is inadvisable. For example, the p scale shown extends down to 0.40. At T, = 1.00, P, = 0.98 at this value of p. At the critical (P, = 1.00),p is about 0.26, indicating a rapid drop in p as P, goes from 0.98 to 1.00, Similarly the data should not be extended into the liquid region where misleading results may be obtained. .50
-
-
an accurate solutionof Equation 1, it is not recommended for a p value closer than *0.01 because of the uncertainties in the basic correlation. The activity coefficient, f / P , equal to the ratio of fugacity to pressure, may be obtained by integration (6),as shown in Equation 2:
The relation between f / P and p can be expressed by Equation 3, obtained by substituting the value of P, from Equation l into Equation 2:
*
LITERATURE CITED
(1) Cope, Lewis, and Weber, IND. ENQ.CHEM.,23, 887 (1931). (2) Federal Works Agency, Work Projects Administration, “Table of the Sine, Cosine and Exponential Integrals”, New York, 1940. (3) Lewis and Kay, OiE Gas J.,March 29, 1934; plot of their correlation in Sherwood’s “Absorption and Extraction”, 1st ed., p. 100, New York, McGraw-Hill Book Co., 1937. (4) Newton, IND. ENG.. CHEM.,27, 302 (1935). (5) Weber, “Thermodynamics for Chemieal Engineers”, 1st ed , p. 198, New York, McGraw-Hill Book Co., 1939. (6) Ibid., p. 197, Equation 12.
(3)
It is to be noted that the function of the reduced temperature, fro, has been eliminated, so that Equation 3 represents a unique relation between f / P and p below the critical point. The integral cannot be evaluated without the use of series, although a direct solution can be obtained by using tables of the expo895
