P - V T Relations for Saturated Liquids H. P. MEISSNER AND 0. H. PADDISON, JR. Massachusetts Institute of Technology, Cambridge, Mass. The engineer frequently faces the problem of predicting liquid densities. This can be done with reasonable accuracy by use of the charts presented in this article, if the critical temperature, critical pressure, and some vapor pressure data are available for the liquid in question.
URING recent years it has been demonstrated that the reduced P-17-2' relations of gases and vapors conform closely to a generalized graphical equation of state (2). I n view of the great engineering utility of this correlation, it is surprising that little work has been done on a similar relation for liquids. Weber (8) presents a few lines for liquids under pressures greater than their vapor pressures on his p chart but councils caution in their use, for the correlation in this region is not good. No work of this kind has been done on liquids under their own vapor pressures-that is, on saturated liquids. Perhaps the reason for this neglect of the saturated liquids lies in a widespread recognition of the fact not only that vapor pressure curvea do not conform to a generalized reduced relation (1, 7), as the theory of corresponding states in its simpler forms (6) would require, but that these curves diverge markedly, as Figure 1 shows. In spite of this divergence, however, it seemed possible that saturated liquids, being in equilibrium with saturated vapors, might obey a relation similar to the one which applies to these vapors-namely,
D
rithms of these values of po were then plotted against the reciprocals of the reduced temperature, as shown in Figure 5, and smooth curves were drawn through the points. At a fixed value of M, the corresponding values of l/Tr, for the various substances were obtained from Figure 5 . The corresponding values of P,, were taken from Figure 1. Each of these points was charted on Figure 4 as shown, and the curve most closely fitting each set of points was then drawn. These points, representing values of POcomputed from experimental data, cluster closely around the average curves for any given value of PO, Some of these points are numbered for identification. Others are not numbered because of space limitations. However, the relative position of these numbered points on any given constant p line does not change, and points bearing the same number can nearly be connected by a straight line which passes through the critical where, of course, both P,, and T,,are unity. This makes it possible to identify the points without numbers on Figure 4.
I .O
0.8 0.6 0.4
0.2 0.1
0.08 0.06 0.04
PV = pRT
1.0
=
roRTo
(1)
where Pois the vapor pressure a t the temperature TO,VOis the corresponding volume of the liquid, R is the gas constant, and PO,like p for gases, is dimensionless and is a unique function of T,,, the reduced temperature, and P,,, the reduced vapor pressure. The relation between PO, P,,, and T,,is presented graphically in the work charts, Figures 2 and 3*. These charts were prepared asfollows: Various values of PO were computed from Equation 1 for the two dozen substances listed in Figure 4, using data reported in the International Critical Tables (8). The loga-
*
Large detailed copies of Figures 2 and 3 may be procured at coat by writing t o the Chemical Engineering Department of Massachusetts Institute of Technology.
1.2
1.3
1.4
1.5
1.6
1 /Tr,
where p is a unique function of the reduced temperature and pressure ( 2 ) . A careful examination of the P-V-T data for pure saturated liquids was therefore made, and it was discovered that the following correlation does exist
PoVo
1.1
VAPORPRESSURE CURVES FIGURE 1. REDUCED FOR SELECTED SUBSTANCES
Figure 2 is merely an elaboration of Figure 4. Figure 3 presents exactly the same information as Figure 2 and differs from it only in that the coordinates have been changed around. Either Figure 2 or 3 may be used as a work chart with Equation 1. The advantage of Figure 2 lies in the fact that vapor pressure curves are practically straight lines when plotted on the coordinates of log P,, us. l/Tro. Hence two known points on the vapor pressure curve of a given substance can be located on Figure 2 and be connected with a straight line, and values of Pro,T,,, and PO can be interpolated direct. However, the value of po can be determined much more easily on Figure 3 than on 2 when the values of P,, and T,,are already known. But since it is not so easy to predict the course of the vapor pressure curve on Figure 3 as on 2, the latter is perhaps to be preferred in certain cases. 1189
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
1190
Vol. 33, No. 9
only when pseudocritical values are used (4). The deviation in none of these cases exceeded 35 per cent. These substances show values of p~g a t the critical which differ widely from the average value of 0.27. On the other hand, paraffin and aromatic hydrocarbons, esters, alcohols other than methanol, ethers othe than methyl ether, amines, hydrogen chloride, and chlorine obeyed this relation to within * 8 per cent. It is clear, therefore, that, the use of Figure6 2 and 3 together with Equation 1 makes it possible to compute the density of most pure saturated liquids within the accuracies indicated above, if the vapor pressure and corresponding temperature are known. Or, given the density of the saturated liquid, it is possible t o compute the corresponding condition of temperature and pressure if the vapor pressure curve is known. While this method is approximate only, nevertheless it makes possible the computation of liquid densities with a precision adequate for much engineering work.
I O
08 0.6 0.4
0.2
pro
0.1 0 08
0.06 0.04
0 02
Liquids under Pressure and Liquid Mixtures The relation developed above applies only to saturated liquids and not to liquids under pressures greater than their vapor pressures a t any given temperature. However, as a
0 01 1.0
1.2
I .4
13
1.6
1.5
1.7
I/Tro
FIGURE 2
Accuracy The accuracy with which po can be determined from Figures 2 and 3 can be judged by noting the divergence of the plotted values of yo from the average constant pa lines, in Figure 4. For the substances plotted, this is of the order of magnitude of * 8 per cent over the entire range of P,, and Tr8. At the critical point itself, the value of p becomes the same for the liquid and gas phase, since at the critical the density of the liquid equals that of the gas. For the substances plotted in Figure 3, the computed values of po at the critical point vary from 0.29 to 0.25. This again is a variation of about * 8 per cent from the average value for po of 0.27 for most substances. As a further test of the accuracy of the relation proposed here, the values of p~gfor about fifty different substances at values of P,, of approximately 0.5 and 0.015 were computed from data in the International Critical Tables and compared with the values read from Figure 4. It was found that highly polar liquids or associated liquids, such as water, ammonia, methanol, acetic acid, and acetone, showed disagreement. Hydrogen and helium also failed to obey the correlation. This was t o be expected, since the values of p for gaseous hydrogen and helium can be accurately computed
0.2
0 .I
0.08 0.06
0.04
0.02
ii0 0.01
0.008 0.006 0.004
0.002
0.00 I 0.01
0.02
0.04
0.06
0.08 0.1 pro
FIGURE 3
0.2
0.4
0.6
0.8 1.0
INDUSTRIAL AND ENGINEERING CHEMISTRY
September, 1941
1191
A CHART OF p o F 0 R SATURATED LIQUIDS
-
18 BENZENE
1.0
1.2
1.1
. .-
1.3
1.4
1.5
1.6
1.7
I / It-,
FIGURE 4
first approximation it may be assumed that the volume of a liquid under moderately high pressure equals the volume 0.2
I
, , ,
0.1
0.06 0.06
this liquid would occupy a t the same temperature, but under its saturation pressure. This, in turn, assumes that liquids are practically incompressible, which is true only at temperatures well below the critical. Similarly, this method cannot be applied to liquid mixtures without modification, except when the volume of the mixture equals the sum of the volumes of the separate constituents. Work is continuing on the P-V-T relations of liquids under pressures greater than their saturation pressures, and on liquid mixtures.
0.04
Literature Cited
NO
0.02
(1) Bingham, J.Am. Chem. SOC.,28,723 (1906). E N G . CHEM.,23,889 (1931). (2) Cope, Lewis, and Weber, IND. (3) International Critical Tables, Vol. 111, pp. 201 ff., New York McGraw-Hill Book Co., 1928. (4) Newton, IND.ENO.CHEM.,27, 302 (1936).
0.0 I 0,006 0.006 1.0
/.I
1.2
1.3
1.4
I 1 Tr,
FIGURE 5
1.5
1.6
1.7
(6) Waal, van der, “Kontinuitit des Gasformigen und Flussigen Zustandes”, Leipzig, 1899. (6) Weber, “Thermodynamics for Chemical Engineers”, p. 108, New York, John Wiley & Son, 1939. (7) Young, Phil. Mug., [ 6 ]33,163 (1892).
