I
J. 0. HIRSCHFELDER, R. J. BUEHLER’, H. A. McGEE, Jr,2, and J. R. SUTTON3 Naval Research Laboratory, The University of Wisconsin, Madison, Wis.
Generalized Equation of State for Gases and Liquids The equations, based on a modified principle of corresponding states, include noble gases, hydrocarbons, and the highly polar substances water and ammonia. The framework i s believed adequate for practically all pure substances. The full range of gases and liquids i s covered where experimental data exist: temperatures as low as half and as high as three times the critical; densities up to four times the critical; and pressures up to 190 times the critical. Standard input data required are: the three critical constants, the normal boiling point, and (for the liquid rbgion only) the density of the saturated liquid at two temperatures. When experimental data are not available, known procedures can be used to estimate them: a main advantage of a correspondingstates approach. When extensive data are available, they may be used to modify the equations. Both vapor pressure and liquid density are given by explicit formulas. One can choose these as arbitrary functions of temperature and obtain any required degree of accuracy, when true curves are known. The pressure (or compressibility factor) for a given temperature and density can be calculated in 3 or 4 minutes with a desk calculator for all densities less than the critical. The complexity of calculation increases for gases above the critical density, for liquids, for determination of thermodynamic properties, for inverse calculations in which pressure is known but density i s not. Some more difficult applications might involve hours of hand computation, but all are entirely practical on a high-speed computing machine, such as the IBM Type 650.
ACCURATE
methods of prediction of volumetric and thermodynamic properties of gases and liquids are badly needed, although countless equations and tables exist which meet engineering requirements with varying degrees of success. The present equations are believed to have a unique position, making them more suitable than earlier works for certain applications. Principle of Corresponding States
Pressure, volume, apd temperature in arbitrary units are denoted by P, V, and T; reduced units (small letters) are defined by: Reduced pressure p = P/P, (1) Reduced density p = 1 / v = V,/V ( 2 ) Reduced temperature t = T / T , ( 3 ) Subscript c refers to the value a t the critical point. According to the van der Waals principle of corresponding states, the compressibility factor, z = PV/RT, is a universal function of p and t for all substances: = = Z(P, t ) (4) Equation 4 is not valid within the accuracy of experimental measurements; nevertheless it has provided the basis for useful approximations, such as the equation of van der Waals, and generalized tables and charts developed by Hougen and Watson (70),Dodge (4), and Cope, Lewis, and Weber (3). From these one can make rapid but rather rough estimates of volumetric and thermodynamic properties. The principle of corresponding states can be deduced by statistical mechanical arguments, if it is assumed that the molecules of all substances interact pairwise with a two-parameter intermolecular potential (9,Chap. 4). These two parameters are related to the size of the molecules and the maximum energy of their attraction. Equation 4 implies that the value of the compressibility factor at the critical Present address, Statistical Laboratory, Iowa State College, Ames, Iowa. a Present address, Research Laboratories, Redstone Arsenal, Huntsville, Ala. a Present address, Mechanical Engineering Research Laboratory, East Kilbride, Glasgow, Scotland.
point is a universal constant for all substances: z, = Z ( p = 1, t = 1). The fact that z, is not constant but is observed to vary between 0.23 and 0.30 for known substances is one indication of the limitation of the van der Waals principle of corresponding states. Most probably, the value of z , depends upon the coordination number (or the number of nearest neighbors) in the liquid lattice. For noble gases where the coordination number is 12, z, = 0.298; for water where the coordination number is 4, z, = 0.23. Corresponding-states charts can give greater accuracy if a single chart is replaced by several, each characteristic of a special class of substances having similar properties (the “principle of mechanical equivalence” of Kamerlingh Onnes). T o do this is to generalize the corresponding-states relations by introducing an additional parameter; from the molecular point of view it is equivalent to assuming that the molecules interact according to a three-parameter potential, with the third parameter related perhaps to the shape or to the dipole moment of the molecule. Such generalized corresponding-states charts and tables have recently been developed by Lydersen, Greenkorn, and Hougen (78) using the critical compressibility factor, .tc, as the third parameter [use of 2, for this purpose was suggested earlier by Meissner and Seferian (27)]. Similar generalized tables have been developed by Rowlinson (39) and Pitzer, Lippman, Curl, Huggins, and Petersen (32). Pitzer and others use as a third parameter the quantity w defined by w = -1 - log p7, where p , is the reduced vapor pressure when the reduced temperature is t = 0.7. Still another parameter, CY, the slope of the vapor pressure curve at the critical point, a =d
Inpv/d In
t =
dp,/dt at
= t =1
(5)
has been introduced by Riedel (33, 35-37) in generalizing correspondingstates relationships for vapor pressures, liquid densities, and heats of vaporization. (The quantity defined by Equation 5 is called ahby Riedel; for brevity VOL. 50, NO. 3
MARCH 1958
375
1
\
9.0
to a good approximation, may be regarded as analytic formulations of the tables. For orientation purposes and quick estimates of isolated data points, charts and tables are more practical than equations. Equations are more suitable for high-speed computations, however, because of memory limitations of the machines. The thermodynamic properties determined from equations are smoother than those obtained from the tables, because small errors in the tabular function lead to large errors in the first and second derivatives determined bv numerical differentiation.
' NO
€;OH
:7.0Q
LL
2
6.5-
Regions of Deflnition
-1
-
GO-
Location of the Coexistence Region. P-V- T values have been arbitrarily divided into three regions, with different equations applying in each region. Taken together these expressions define a single consistent equation of state for all values of P, V , and T . The arbitrary division is a compromise which makes possible the use of a relatively simple equation at low densities; it is perhaps a n esthetic defect, but it is nevertheless a practical way of meeting conflicting requirements. The regions of definition are shown in Figure 2 and defined by:
LL
5.51
I
I
0.22
0.23
CRITICAL
Figure 1.
I
0.24
I
0.25
I
I
026
0.27
COMPRESSIBILITY FACTOR
Comparison of parameters,
01
I
I
Q28
0.30
029
LF P,&/R'T,
and z, for different substances
Curve corresponds to Riedel's empirical relation, Equation 7
the k has been omitted.) If Riedel's form is assumed for the vapor pressure,
+
CY In t 0.0838 (CY - 3.75) (36t-1 - 35 - G t f 42 In t ) ( 6 )
lnp,(t) =
one finds that a and w are simply re-
+
lated by 01 = 5.808 4.93 w . If it were possible to express z, as a unique function of a, either quantity would serve equally well as a third corresponding-states parameter. The approximate relation zC-' =
n"
1.4
REGION
_-__. -
I
9a
CRITICAL
W
GAS
[r
3
I
n
// COEXISTENCE LIQu,(D
0 w
'
w
[r 0
I
REGION
'
/
~
\
I REDUCED DENSITY
0
,\
REGION 1
2
?= \/V
REGION
IJ
DENSE GAS
,/
COEXlSTENCE REGION
I 0
I 2 REDUCED DENSITY t=V,/V
Figure 2. Regions of definition of equation of state Equations for Regions I and 111 extend across dashed line into coexistence region
376
3.72
+ 0.26
(CY
- 7)
(7)
has been given by Riedel (37) with the statement that many, but not all, substances conform to it within the limits of experimental error. The existence of exceptional substances suggests the use of z, and 01 as independent parameters for generalizing the principle of corresponding states. I n these equations, 2, and cy in fact enter independently; in this respect the author's approach is similar to that of Martin and Hou (79). Figure 1 shows values of z, and a for typical substances. [Similar plots have been given by Riedel (35) and by Lydersen, Greenkorn, and Hougen (18).] z, has been determined from experimental values of Po, V,, and To. Except for ammonia and water, values of 01 were determined from normal boiling point data by assuming the Riedel form for the vapor pressure. For ammonia and water the Riedel equation is not particularly suitable! and a different method was used. The generalized tables of Lydersen, Greenkorn, and Hougen, Rowlinson, Pitzer, and others have all been used (in addition to considerable experimental data) in deriving the equations given here and in making checks of accuracy; the equations have been found in good agreement with all the tables and thus,
INDUSTRIAL AND ENGINEERING CHEMISTRY
Region I. Gas. All temperatures; density less than the critical, p 1. Region 11. High Density Gas. Temperature above the critical, t >_ 1 ; density greater than the critical, p > 1. Region 111. Liquid. Temperature below the critical, t 5 1; density greater than the critical, p 2 1.
1
(32)
where s is a constant. Equation 32 is more suitable at high densities than the form suggested for densities near the critical by Widom and Rice (43):
p
= 1
+
syp
- l)i fort =
l,p
> 1 (33)
Equation 32 is suggested most strongly by nitrogen data (24)which are represented with a maximum error in pressure of 0.570 in the range 1.6 p 3.0, 1.3 < p 5 68 byputtings = 5.62. Empirical values of s for several substances are given in Table 11. The assumed critical densities are also given, since s is sensitive to these. For an arbitrary substance s may be taken to be
<
Cc./Mole
zc
01
P
33.5 37.47 111.3 218.2 48.0 63.0 52.0 78.0 20.8
90.0 255 72.5 56.6 75.2 167 199 170 602
0.291 0.274 0.243 0.232 0.291 0.248 0.274 0.293 0.247
5.98 6.77 7.95 7.80 5.76 8.98 6.37 6.32 8.18
6.26 6.97 8.50 9.12 6.26 8.23 6.97 6.17 8.28
tl
t2
P11
P12
C
d
0.718 0.737 0.745 0.750
0.862 0.877 0.881 0.889
2.381 2.446 2.540 2.649
2.036 2.061 2.115 2.201
1.839 1.946 2.023 2.334
0.621 0.757 1 * 009 0.715
0.681 0.620 0.579 0.557
0.8196 0.8105
2.670 2.794 2.735 2.930
2.352 2.336
2.283 2.072 1.838 2.053
0.346 0.771 0.850 0.824
...
...
...
0.772
.,..
...
...
2.442
VOL. 50, NO. 3
a
...
...
MARCH 1958
383
Table
Substance Nitrogen n-Butane Ammonia Water Argon Ethyl alcohol Ethyl chloride Carbon disulfide n-Decane
V.
References for Experimental Data
Critical Data
Liquid Density
(42)
(18) (12, 40)
( 1 , 12) (31)
(14)
(14)
(31)
...
(18)
(18)
(11. 18) (28) (18)
(11) (28)
ho, hl, h2, and h3 were based on nitrogen and ammonia data. I n Region I11 no constants were adjusted. I t is fair to say, therefore, that the accuracy in the case of nitrogen is representative of that possible over a very wide range of conditions, when constants are adjusted from P-V-T data (actually, P,, V,, T,, and a were not adjusted). For substances not considered in the choice of constants (butane, decane, argon, ethyl
Table VI.
Temperature
(18)
(34)
chloride, ethyl alcohol, and carbon disulfide) the accuracy is representative of that obtained from the modified corresponding-states method by using only the standard amount of data with no adjustment of constants. Acknowledgment
The authors thank 0 . A. Hougen for encouragement and stimulation through-
out the course of this work, and thank him and his colleagues, A. L. Lydersen and R. A. Greenkorn, for providing copies of their charts, tables, and data prior to publication. The present equation of state represents the culmination of a long process of trial and error, in which many individuals have participated. The authors particularly want to thank Wm. E. Rice, George Town, and John S. Dahler for valuable help. For the efficient numerical work, thanks are due to Elaine Gessert, Marianne Schanzenbach, Helen Ward, Phyllis Reese, and Jean Cattoi. References (1) Beattie, J. A., Simard, G. L., Su, G. J., J . Am. Chem. SOC.61, 24
(1939). (2) Brock, J. R., Bird, R. B., A . I. Ch. E. Journal 1, 174 (1955). (3) Cope, J. Q., Lewis, W. K., Weber, H. C., IND.END. CHEM.23, 887 (1931).
Experimental and Calculated Values for Nitrogen a%‘, PII, or PIII Equation Calcd. yo error pressure, p in pressureo
K.
t
Cc./Mole
P
123.15
0.97661
22389 4477.8 1119.5 279.87 139.95 93.288 55.974 39.980 31.096
0.0040198 0.020099 0,080396 0.32158 0.64317 0.96475 1.6079 2.2511 2.8943
198.15
1.5714
22389 4477.8 1119.5 279.87 139.95 93.288 55.974 39.980 31.096
0.0040198 0.020099 0.080396 0.32158 0.64317 0.96475 1.6079 2.2511 2.8943
Exatl. Pressure Atm. P 0.44800 0.013373 2.1900 0.065373 8.1000 0.24179 23.680 0.70687 30.080 0.89791 29.160 0.87045 31.480 0.93970 262.58 7.8383 1561.5 46.611 0.72400 0.021612 3 * 5949 0.10731 13.980 0.41731 51.520 , 1.5379 2.9134 97.599 146.64 4.3773 331.68 9.9009 949* 93 28.356 2886.6 86.166
22389 4477.8 1119.5 279.87 139.95 93.288 55.974 39.980 31.096
0.0040198 0.020099 0.080396 0.32158 0.64317 0.96475 1.6079 2.2511 2,8943
1.0000 4.9912 19.849 78.648 161.54 262.51 622.16 1569.6 4113.5
0.029851 0.14899 0.59251 2.3477 4.8220 7.8360 18.572 46.854 122.79
0.029926 0.14945 0.59543 2.3658 4.8509 7.7801 18.589 47.009 123.52
0.60 -0.71 0.09 0.33 0.59
273.15
Volume,
2.1661
Density,
0.013434 0.065920 0.24536 0.71788 0.90260 0.86676 0.95369 7.7329 45.975
0.46 0.84 1.48 1.56 .52 - .42 1.49 -1.34 -1.36
0.021682 0.10772 0.42087 1.5467 2.8815 4.3015 9.8417 27.865 84.614
0.32 0.38 0.85
0.57
- 1.09 -1.73 -0.60 -1.73 -1.80
0.25 0.31 0.49
0.77
348.15
2.7609
22389 4477.8 1119.5 279.87 139.95 93.288 55.974 39.980 31.096
0.0040198 0.020099 0.080396 0.32158 0.64317 0.96475 1.6079 2.2511 2,8943
1.2753 6.3811 25.602 104.98 224.35 375.54 898.40 2138.5 5249.4
0.038069 0.19048 0.76424 3.1336 6.6971 11.210 26.818 63.835 156.70
0.0381 71 0.19117 0.76968 3.1818 6.8174 11.275 26.873 64.089 160.17
0.27 0.36 0.71 1.54 I .80 0.58 0.20 0.40 2.21
423.15
3.3557
22389 4477.8 1119.5 279.87 139.95 93.288 55.974 39.980 31.096
0.0040198 0.020099 0.080396 0.32158 0.64317 0.96475 1.6079 2.2511 2.8943
1.5506 7.7697 31.342 131.07 286.37 486.35 1164.9 2684.4 6353.3
0.046287 0.23193 0.93558 3.9126 8.5483 14.518 34.774 80.132 189.65
0.046416 0.23288 0.94378 3.9961 8.7824 14.777 34.602 78.790 193.83
0.28 0.41 0.88 2.13 2.74 1.78 -0.49 -1.67 2.20
384
INDUSTRIAL AND ENGINEERING CHEMISTRY
% E~~~~ in Pressure (19)a 0.27 0.61 1.1 0.43 -1.0 - 1.2 24 1000 15000 0.14 0.19 0.63 0.66 -0.41 -0.35 50 1100 19000
0.07 0.08 0.12 0.26 0.06 0.02 67 1200 21000 0.08 0.11 0.28 0.78 0.71 0.92 77 1300 22000 0.08 0.15 0.42 1.3 1.4
1.7 84 1400 23000
EQUATIONS Table VII.
O
a
Temperature K. t
310.89
0.73132
377.60
0.88812
444.26
1.0449
510.93
1.2017
Experimental and Calculated Values for n-Butane
Volume, Cc./Mole
Density,
36777 8646.4 104.17 103.59 101 - 8 9 98.475 95.100 91.945 45086 10925 4080.5 125.83 119.34 110.45 103.84 98.547 53279 13095 5054.0 1459.2 168.39 127.47 114.22 105 * 88 61410 15199 5954.6 1841.8 390.89 153.45 127.43 114.55
0.0069336 0.029492 2 * 4479 2.4616 2.5028 2.5895 2.6814 2 * 7734 0.0056558 0.023341 0.062492 2.0265 2.1368 2.3088 2.4556 2. :876 0.0047861 0.019473 0.050455 0.17482 1.5143 2.0005 2.2325 2.4085 0.0041524 0.016777 0.042824 0.13845 0.65236 1.6618 2.0011 2.2261
Using p = 7 . Percentage error 100 (Pcalod. Using V , = 0.375 RTJP,.
OF STATE
P
Exatl. Pressure Atm. P 0.68046 2.7218 6.8046 20.414 68.046 204.14 408.27 680.46 0.68046 2.7218 6.8046 20.414 68.046 204,14 408.27 680.46 0.68046 2.7218 6.8046 20.414 68.046 204.14 408.27 680.46 0.68046 2.7218 6.8046 20.414 68.046 204,14 408.27 680.46
0.01816 0.07264 0.18160 0.54480 1.8160 5.4480 10.896 18.160 0.01816 0.07264 0.1816 0.54480 1.8160 5.4480 10.896 18.160 0.01816 0.07264 0.1816 0.5448 1.8160 5.4480 10.896 18.160 0.01816 0.07264 0.1816 0,5448 1.816 5.4480 10.896 18.160
Percentage Error in Pressureb a%’, pix, or PIII Equation van der Calcd. 7% Waals , - error in pressureb b’-Equation“ EquationC pressure, p 0.018233 0.073918 - 2.7373 0.17437 1.6297 5.2596 10.040 15.912 0.018173 0.073054 0.18427 0.47879 1.8049 5.2370 9.8982 15.840 0.018151 0.072692 0.18211 0.54864 1.7540 4.8622 9.3007 15.033 0.018144 0.072578 0.18146 0.54336 1.7602 4.9251 9.1281 14.588
0.40 1.76 - 1607 - 68.0 - 10.3 - 3.46 - 7.86 - 12.38 0.07 0.57 1.47 - 12.1 - 0.61 - 3.87 - 9.16 - 12.8 - 0.05 0.07 0.28 0.70 - 3.41 - 10.8 - 14.6 - 17.2 - 0.09 - 0.09 - 0.08 - 0.26 - 3.07 - 9.60 - 16.23 - 19.7
0.78 2.60
0.90 3.27
0.35 0.94 2.06
0.40 1.23 2.90
0.19 0.28 0.44 0.67
0.19 0.38 0.74 1.60
0.13 0.06 -0.07 -0.67 -1.15
0.12 0.07 -0.02 -0.65 -7.29
- peapti.)/pexpti.
Dodge, B. F., Ibid., 24, 1353 (1932). Friedman, A. S., White, D., J . Am. Chem. Soc. 72,3931 (1950). Guggenheim, E. A., J. Chem. Phys. 13, 253 (1945). Hirschfelder, J. O., Buehler, R. J., McGee, H. A,, Jr., Sutton, J. R., “Generalized Equation of State for Both Gases and Liquids,” University of Wisconsin Naval Research Laboratory, Rept. WIS00R-15 (1956). Hirschfelder, J. O., Buehler, R. J., McGee H. A,, Jr., Sutton, J. R., IND. ENG.CHEM.50,386 (1958). Hirschfelder, J. O., Curtiss, C. F., Bird. R. B.. “Molecular Theorv of Gases and ’Liquids,” Wiley, New York, 1954. (10) Hougen, 0. A., Watson, K. M., “Chemical Process Principles Charts,” Democrat Printing Co., Madison, Wis., 1946. (11) International Critical Tables, vol. 111. McGraw-Hill. New York. 1928. ’ W. B., IND. ‘ENG. CHEM. 32, ( I 2 ) K??8 (1940). Kazarnowsk;, J. S., Acta Physicochim. U.R.S.S. 12, 513 (1940). Keenan, J. H., Keyes, F. G., “Thermodynamic Properties of Steam,” Wiley, New York, 1936. Keyes, F. G., J . Am. Chem. SOC.53, 965 (1931). Kobe, K. A., Lynn, R. E.. Chem. Revs. 52. 117 (1953). (17) Lydersen,’ A. ’L., ‘“Estimation of Critical Properties of Organic Compounds by the Method of Group
Contributions,” University of Wisconsin, Engineerin Experiment Station, Rept. 3 (19f5). (18) Lydersen, A. L., Greenkorn, R. A., Hougen, 0. A., “Generalized Thermodynamic Properties of Pure Fluids.” Universitv of Wisconsin. Engineering Expehment Station; Rept. 4 (1955). (19) Martin, J. J., Hou, Y . C., A.I,Ch.E. Journal 1,142 (1955). (20) Mayer, J. E., Mayer, M. G., “Statistical Mechanics,” Wilev, ,. New York, 1940. Meissner, H. P.. Seferian. R.. Chem. Eng. Piogr. 47,‘579 (1951). ’ Michels, A., Geldermans, M., Physica 9, 967 (1942). Michels, A,, Lunbeck, R. J., Wolkers, G. J., Appl. Sei. Research A2, 345 (1950). Michels, A., Lunbeck, R. J., Wolkers, G. J., Physica 17, 801 (1951). Michels. A.. Michels. C.. Wouters. H., Proc. ’Roy. SOC.’(Loidon) A153; 219 (1936). Michefs, A., Wassenaar, T., de Graaf, W., Prins, C., Physica 19, 26 (1953). Michels, A., Wijker, Hub., Wijker, Hk., Ibid., 15, 627 (1949). O’Brien, L. H., Alford, W. J., IND. ENG.CHEM.43, 506 (1951 ). Olds, R. H., Reamer, H. H., Sage, B. H., Lacey, W. N., Ibid., 36, 286 (1944). Osborne, N. S., Meyers, C. H., Bur. Standards J . Research 13, 1 (1934).
(31) Pickering, S. F., Natl. Bur. Standards, Sci. Paper 541 (1926). (32) Pitzer, K. S., Lippman, D. Z., Curl, R. F., Huggins, C. M., Petersen, D. E.. J. Am. Chem. Soc. 77. 3433 (1955). Plank, R., Riedel, L., Zng. Arch. 16, 255 (1948). Reamer, H. H., Olds, R. H., Sage, B. H., Lacey, W. N., IND.ENG. CHEM.34,1526 (1047) Riedel, L., Che, (1954). (36) Ibz’d., p. ’259. (37) Ibid., p. 679. (38) Rossini, F. D., others, “Selected Values of Properties of Hydrocarbons and Related Compounds,” Carnegie Press, Pittsburgh, 1955. (39) Rowlinson, J. S.,Trans. Faraday SOC. 51, 1.317 (1955). (40) Sage, B. H., Webster, D. C., Lacey, W. N., IND.ENG.CHEM.29, 1188 (1937). (41) Thodos, G., Zbid., 42, 1514 (1950). (42) White, IS., Friedman, A. S., Johnston, H. L., J . Am. Chem. SOC. 73, 5713 (1951). (43) Widom, B., Rice, 0. K., J. Chem. Phys. 23,1250 (1955). RECEIVED for review December 26, 1956 ACCEPTED September 26, 1957 Division of Industrial and Engineering Chemistry, 130th Meeting, ACS, Atlantic City, N. J., September 1956. Work carried out at University of Wisconsin Naval Research Laboratory, supported in part by Contract Da-I 1-022-ORD-994. VOL. 50, NO. 3
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