Ind. Eng. Chem. Process Des. Dev. 1083, 22, 313-322
313
Correlation and Prediction of the Vapor Pressures of Pure Liquids over Large Pressure Ranges Jack McGerry Department of Chemical Engineering, University of Saiford, Salford A45 4wT, England
Liquid vapor pressures of 72 substances are available over pressure ranges extending from about 1 kPa up to the particular critical pressure. For correlation of these data the Wagner equation yields an average root mean square percentage error of 0.09 % . The results obtained allow the formulation of constraints applying to the behavior of vapor pressure for values of the reduced temperature in the range 0.5 to 1.O. Using these constraints Wagner coefficients were generated and are presented for a further 179 pure substances for which only limited range data are available (the limits are usually about TR = 0.55 and 0.65). It is considered that these coefficients will yield vapor pressures in the range TR = 0.5 to 1.0 which are suitable for process design work.
Introduction The calculation of accurate liquid vapor pressures is vital for the success of many chemical engineering design techniques. Numerous equations have been proposed for thia purpose, and eight of these together with modifications of four of them are considered in this paper. The widely used Frost-Kalkwarf equation (Frost and Kalkwarf, 1953) is not included in these eight because it is not explicit in vapor pressure. It may be worth mentioning, however, that this equation was fitted to the 14 sets of full range data initially involved with an average % RMS error of 0.12. The selected eight equations may be divided into two groups on the basis of the method by which the constants (coefficients) of the equations are calculated. (Group I): Constanta are calculated by the regression of experimental vapor pressure. The Antoine, Wagner, and Thomas equations belong to this group (Antoine, 1888 Wagner, 1973: Thomas, 1976). Representation by Chebyshev polynomials also falls into this group (Ambrw et al., 1970). (Group 11): Constants are calculated from critical temperature and pressure together with one or two further basic data. The Riedel, Miller, Thek-Stiel, and GomezThodos equations belong to this group (Riedel, 1954: Miller, 1965: Thek and Stiel, 1966: Gomez-Nieto and Thodos,1977). It seems obvious that it could be beneficial to use regression of data for evaluation of the constants of the equations of the second group. The Equations Involved Antoine Equation.
B
In P = A - T+C This equation can yield highly accurate values of vapor pressure in a pressure range of about 1to 200 P a . Values of constants valid for this range and yielding vapor pressure in millimeters of mercury are available for many substances (Reid et al., 1977; Boublik et al., 1973). The equation was developed from the Clausius-Clapepon equation and is used extensively. Wagner Equation. 1 In PR = -[A(1 - TR) + B(l - TR)'" C(l - T R ) ~ TR D(1 - T R ) ~(2) ] Statistical considerationswere involved in the development of this equation, which was initially derived to describe the vapor pressures of argon and nitrogen from the triple
+
+
point to the critical temperature (Wagner, 1973). Thomas Equation. p=- 253312 (3) ex - C where In X = A - B In T (4) Equation 3 is based on the observation that "the ratio of the value of RT d(ln P)/dT for any nonassociated compound to the value of the function for any other such compound at the same vapor pressure is constant over a range from a few millimeters" of mercury to the critical pressure (Thomas, 1976). Chebyshev Polynomials. Vapor pressures of liquids from triple points to critical temperatures may be described with high accuracy by (Ambrose et al., 1970)
where
EJx) = cos (s cos-' x )
(6) (7)
The Chebyshev polynomials (E,(x)) are related by the recurrence relation E,+,(x) - 2xE,(x) + E,-,(x) = 0 (8) It is easily shown that E,(x) = x (9) E&) = 2x2 - 1 (10) and then eq 8 allows calculation of the higher polynomials. The use of up to a total of six polynominals gives excellent agreement with data; i.e., up to seven constants (ao,...,a,) are used. Although the number of constants is higher than that used by any other method discussed here, standard computer programs allow easy generation of the constants and evaluation of vapor pressures. Riedel Equation. B In PR= A - - C In T R DTR6 (11) TR where A = -35Q
+
019~-4305ia3ii 122-0313$01.50io @ 1983 American Chemlcal Society
+
314
Ind. Eng. Chem. Process Des. Dev., Voi. 22,
No. 2, 1983
B = -36Q C = 42Q + a, D = -Q Q = 0.0838(3.758 - a,)
Table I. Input Data Normally Required for Eq 1, 2, 3, 5 , 11, 16,20,and 25 equation
(12) where a, is called the Riedel parameter and is defined by a, = d (In PR)/d (In TR) at TR = 1 (13)
The value of a, is best obtained (Reid et al., 1977) from a knowledge of the normal boiling point (TB). Substitution of eq 12 into eq 11 yields 0.3149f(T~~) - In (1O1.325/Pc) a, = (14) 0.0838f(TBR)- In TBR
Antoine Wagner Thomas Chebyshev Riedel Miller Thek-Stiel Gomez-Thodos
eq no.
1 2
3 5
TC,pC,
20 25
Miller Equation. This equation is referred to by Reid et al. (1977) as the Reidel-Plank-Miller equation
where
A = 0.4835 + 0.4605h TBR In (1O1.325/Pc) h= 1 - TBR
(17) (18)
Thek-Stiel Equation.
where
f(TR) = 1 1.14893 - - - 0.11719TR-0.03174T~~ - 0.375 In TR TR
(22) (23) (24) a, is again obtained from a knowledge of normal boiling temperature, and h is given by eq 18. Gomez-Thodos Equation.
B = 1 . 0 4 2 ~- ~0.462844 ~ C = 5.2691 + 2.0753A - 3.1738h
(25) where 7.0109
380900
C = 2.4186 - h he(123.21/h) +
DE-
a,
A = -(B
(26)
+ BC 7
+ D)
(29)
TB
Tc, P c , TB,AHB Tc, P c , TB,LY,
Table 11. Modified Input Data equation Wagner Riedel Miller Thek-Stiel Gomez-Thodos
where
fitted constants A , B, C Tc, Pc, fitted constants A , B, C, D fitted constants A , B, C TL,TH,fitted constants a, to as Tc,P c , TB
T,, T,, T,, T,, T,,
fitted constants Po A , B, P,, fitted constants A , B, fitted constants Po A, B fitted constants Pc, A , B, P,, fitted constants A , B,
C, D C, D
C, D C, D
and h is given by eq 18. Gomez-Nieto and Thodos (1977) give values of a, for use in eq 28 for 138 substances. The Riedel, Miller, and Thek-Stiel equations are all based on the integration of the Clausius-Clapeyron equation using various relationships for enthalpy of vaporization and the difference between saturated vapor and liquid compressibilities. The data input for the above eight equations are summarized in Table I. Adjustment of P, and Modification of Group I1 Met hods Ambrose (1978) has pointed out that measured values of critical pressures are usually of lower accuracy than the related critical temperatures. He therefore proposed that In P, should be treated as an adjustable constant in the Wagner equation and the resultant value used to obtain critical pressure for use with this equation. This procedure is followed here and, while generating critical pressures very close to the measured ones, allows significantly better data correlation. Equations 11, 16, 20, and 25, in addition to being used in the normal manner, were also used as models for regression of vapor pressure data. For eq 16 and 20 the value of Pc was again treated as an adjustable constant. There is no advantage to be gained by this treatment of eq 11 and 25, since in both cases In Pc would merely be merged with the adjustable constant A. The Thek-Stiel eq 20 was further modified by replacement of the constant 0.04 by an adjustable value. It is only to be expected that constants obtained by data regression will yield better results than those obtained from critical properties. The modified input data are summarized in Table 11. Processing the Data Highly accurate vapor pressure data for 12 liquids exist in the range -5 kPa to the critical pressure. Similar data for two light gases are available over a range extending from the triple point to the critical point. These 14 substances and their data sources are listed in Table 111. The equation constants obtained for these substances were those which minimized the sum of the squares of the fractional deviation at each data point, the fractional deviation being given by (experimental vapor pressure calculated value) i(experimental vapor pressure). With the exception of Chebyshev polynomial representation this minimization was carried out by a computer program based on algorithm E04GAF of the Numerical Algorithms Group, Oxford, England. For the exception the algorithms
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 2, 1983 315 Table 111. Sources of High Accuracy Data in Range
-
5 kPa to PC
ID
source
benzene
1
toluene
2
ethylbenzene o-xylene
3 4
m-xylene p-xylene diethyl ether acetone cyclohexane
5 6 7 8 9
Willingham et al. (1945) Bender et al. (1952) Ambrose et al. (1967) Willingham et al. (1945) API 44 Tables (1966) Ambrose et al. (1967) as for toluene Willingham et al. (1945) Ambrose et al. (1967) as for o-xylene as for m-xylene Ambrose et al. (1972) Ambrose et al. (1974) Willingham et al. (1945) Kerns et al. (1974) Hugill and McGlashan (1978) E.R.A. “1967 Steam Tables” Skaates and Kay (1964) Ambrose and Sprake (1970) Ambrose et al. (1975) Ambrose and Sprake (1970) Ambrose et al. (1975) Wagner (1973) Wagner (1973)
substance
water methanol
10 11
ethanol
12
nitrogen argon
13 14
EO2ADF and EO2AEE were used. In all cases iteration continued until the first five significant figures for all adjustable constants remained unchanged. The results for each of the fourteen substances are given in Table IV, which is submitted together with Tables V and VI as Supplementary Material. (Seeparagraph at end of paper regarding this material.) Average root mean square percentage deviations resulting from eq 1 , 2 , 3 , 5 , 11, 16, 20, and 25 were 0.74, 0.046, 0.95, 0.040, 0.15, 0.41, 0.055, and 0.090, respectively. In all cases the constants were obtained by iteration. Values of TB and AHB were obtained from the book by Reid et al. (19771, while values of Pc and Tc were given with the data. It is clear from Table IV that the data are most closely represented by Chebyshev polynomials. However, the Wagner equation gives results which are not significantly different. Since the representation by Chebyshev polynomials requires seven coefficients for the above performance it was decided to use the more tractable Wagner equation for correlation of full range data, and Chebyshev polynomials take no further part in this discussion. Table V (Supplementary Material) summarizes the results obtained by eq 11, 16, 20, and 25 when the constants were calculated from the input data indicated in Table I, as opposed to the use of regression of vapor pressure data. The average root-mean-square percentage deviations resulting from eq 11, 16, 20, and 25 were 1.90, 1.31, 0.85, and 2.93 respectively. The Wagner coefficients for seventy two compounds are presented in Table VII. These compounds include the 14 substances referenced in Table In, the remainder being compounds for which full range data of mixed precision are available. The values of Tc and Pc are those to be used in the Wagner equation. Data for oxygen, carbon monoxide, helium, hydrogen, and neon were taken from the W.A.D.D. Technical Report 60-56 (1961),while the data of Kemp and Giauque (1937) and of Robinson and Senturk (1979) were used for carbonyl sulfide. The work of Mastroianni et al. (1978) is the source of data for trifluorotrichloroethane,and the data of Ambrose (1968) and Ambrose et al. (1975) were used for the fluorobenzenes and pentafluorotoluene.
no. of data points
total data points
19 19 9 20 4 9
47
20 9
17 9 12 12 20 12 25 23
33 33 29 29 29 40 47 38 49 44 48 49 49
Data for the remaining compounds were generated from Chebyshev coefficients published by the Engineering Sciences Data Unit, London (1972,1973,1974,1975,1976, 1977,1978). In all cases the value of Pc was taken to be adjustable. The Pitzer acentric factor (w) is extensively used to quantify the nonideal behavior of gases and is easily calculated from the Wagner coefficients of a substance as given in Table VII. w =
-1.0 - 0.620417[0.34 + (0.3)’“B
+ (0.3)3C+(0.3)60]
(30) Unconstrained Fit to Limited Ranges of Data There are many substances for which data only exist over a small range of pressure, the usual upper limit lying in the range 100 to 200 kPa. The use of such data for successful prediction of vapor pressures outside the particular range is obviously desirable. Consequently, data were extracted from the lower ends of the full ranges of data for the substances of Table 111, and equation constants were obtained by regression on these samples of data. Since no data near the critical points are involved, it w i l l be evident that critical pressures should not be subjects of iteration in eq 2, 16 and 20. By use of these sample constants, vapor pressures were calculated for the full range of data and the root mean square percentage deviations were obtained. The results are presented in Table VI (Supplementary Material). Outstandingly poor results are obtained for argon, and this is due to the fact that the temperature range yielding vapor pressures up to about 100 P a is too small a fraction of the range up to the critical pressure (0.06 compared with about one quarter for the other substances). Excluding argon, the average root-mean-square percentage deviations resulting from eq 1, 2 , 3 , 11, 16, 20, and 25 were 2.3, 0.3, 71.0, 9.1, 0.75, 0.20, and 8.0 respectively. When generating Thek-Stiel eq 20 constants from the samples of data, it was necessary to stop the iteration procedure prematurely to avoid excessive computation. This had the effect of preventing the equation fitting the sample data more accurately but gave a better fit over the full range. In consequence, the percentage deviations given
316
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 2, 1983
Table VII. Wagner Coefficients from Full Range Data
formula Ar
name
A
B
C
argon -5.90501 1.12627 -0.767869 co carbon -6.20798 1.27885 -1.34533 monoxide carbon dioxide -6.95626 1.19695 -3.12614 CO, cos carbonyl sulfide -6.40952 1.21015 -1.54976 fluorine -6.18224 1.18062 -1.16555 F, He helium -3.97466 1.00074 1.50056 hydrogen -5.57929 2.60012 -0.855060 HZ water -7.76451 1.45838 -2.77580 HZ 0 Ne neon -6.07686 1.59402 -1.06092 nitrogen -6.09676 1.13670 -1.04072 NZ oxygen -6.28275 1.73619 -1.81349 0 2 CCl,F, dichlorodi-7.01657 1.73224 -2.97909 fluoromethane CC1, carbon -7.07139 1.71497 -2.89930 tetrachloride CHClF, chlorodi-6.99913 1.23014 -2.49377 fluoromethane CHC1, chloroform -6.95546 1.16625 -2.13970 CHF, trifluoro-7.41994 1.65884 -3.14962 methane difluoromethane -7.44206 1.51914 -2.75319 CHZFZ CH3C1 methyl chloride -6.86672 1.5227 3 -1.92919 methyl fluoride -6.78099 CH-F 0.828379 -1.41137 CH; methane -6.00435 1.18850 -0.834082 CH,O methanol -8.547 96 0.769817 -3.10850 CH,N methylamine -7.527 7 2 1.81615 -4.20677 trichlorotri-7.36666 1.81971 -3.94233 C2C13F3 fluoroethane ethylene -6.32055 1.16819 -1.55935 1,2-dichloro-7.36864 1.767 27 -3.34295 ethane ethane -6.34307 1.01630 -1.19116 'ZH6 ethanol -8.51838 0.341626 -5.73683 'ZH6' ethyiamine -7.20059 C,H,N 1.20679 -3.71972 dimethy lamine -7.9 02 9 5 2.81577 -6.31338 propylene -6.64231 -1.81005 1.21857 acetone -7.45514 1.20200 -2.43926 prop an e -6.67833 1.15437 -1.64984 1-propanol -8.05594 4.25183E-2 -7.51296 2-propanol -8.1 6927 -9.4321 3E-2 -8.10040 propylamine -7.23587 1.22853 -3.75004 trimethylamine -6.88066 1.15962 -2.18332 butene -6.88204 1.27051 -2.26284 isobutene -6.95542 1.35673 -2.45222 methyl ethyl -7.71476 1.71061 -3.68770 ketone butane -6.88709 1.151 57 -1.99873 isobu tane -6.95579 1.50090 -2.52717 diethyl ether -7.29916 1.24828 -2.91931 1-butanol -8.007 56 0.537 826 -9.34240 2-butanol -7.80578 0.324557 -9.41265 diethylamine -7.267 96 1.15810 -3.91125 1-pentene -7.04875 1.17813 -2.45105 pentane -7.28936 1.53679 -3.08367 isopentane -7.12727 1.38996 -2.54302 2,2-dimethyl-6.89153 1.25019 -2.28233 propane 1-pentanol -8.97725 2.99791 -12.9596 chloropenta-8.0217 2 1.54665 -3.78361 fluorobenzene pentafluoro-7.79730 1.35271 -3.50409 benzene 1,2,4,5-tetra-7.79740 1.57406 -3.82060 fluorobenzene benzene -6.94739 1.25253 '6 H6 -2.53686 cyclohexane -6.96009 1.31328 -2.75683 hexane -7.51650 1.54797 -3.38541 isohexane -7.28750 1.29015 C6H14 -2.97853 3-methyl-7.27084 1.26113 6' -2.81741 pentane 2,2-dimethyl-7.25933 1.69602 -3.18124 butane 2,3-dimethyl-7.27870 1.56349 -3.05387 butane
C~HIN
D -1.62721 -2.56842 2.99448 -2.10074 -1.50167 -0.430197 1.70503 -1.23303 4.06656 -1.93306 -2.536453-2 -0.377232
pc
4857.99 3501.15 7374.99 6346.45 5214.72 230.029 1309.60 22122.3 2724.55 3399.61 5089.87 4132.03
Tc
rms % error
150.651 0.023 132.91 0.28
approx lowest data
P
T
69 26
84 71
304.15 378.8 144.31 5.20 33.19 647.35 44.38 126.200 154.7 384.95
0.011 530 0.18 2 0.012 4 0.26 5 7 0.27 0.026 0.7 0.57 43 0.025 1 3 0.28 0.2 0.038 0.2
217 162 64 2 14 275 25 63 54 155
-2.49466
4550.78
556.40
0.027
1
250
-2.21052
4983.31
369.30
0.042
1
170
-3.44421 -0.849379
5365.76 4840.92
536.40 299.06
0.14 0.075
0.1 215 0.2 125
-0.979495 -2.61459 -2.41700 -1.22833 1.54481 -1.22275 0.625601
5826.99 6697.18 5557.36 4596.42 8 0 8 5.0 5 7433.32 3425.71
351.54 416.27 315.0 190.53 512.64 430.0 487.7
0.066 0.11 0.079 0.019 0.12 0.073 0.32
-1.83552 -1.43530
5050.88 5362.00
282.55 566.00
0.034 0.071
0.1 1 0 5 1 260
-2.03539 8.32581 -4.33511 -0.224073 -2.48212 -3.35590 -2.70017 6.89004 7.85000 -4.33990 -2.94707 -2.61632 -1.46110 -0.751692
4869.71 6130.87 5641.37 5308.28 4605.23 4699.93 4255.76 5151.11 4742.44 4806.74 4083.96 4017.60 4007.06 4221.77
305.42 513.92 456.35 437.70 364.85 508.10 369.82 536.78 508.30 497.0 433.30 419.57 417.90 536.78
0.13 0.077 0.089 0.043 0.018 0.086 0.043 0.19 0.22 0.086 0.073 0.094 0.086 0.11
2 6 1 1 0.1 4 0.2 0.2 0.1 1 1 0.2 0.2 1
133 293 215 240 140 259 145 260 250 235 200 170 170 255
-3.13003 -1.49776 -3.36740 6.68692 2.64643 -1.17981 -2.21727 -1.02456 -2.45657 -4.74891
3790.62 3658.01 3646.10 4412.63 4189.75 3705.40 3536.85 3378.62 3385.90 3197.88
425.18 408.14 466.74 563.05 536.01 496.45 464.78 469.74 460.43 433.77
0.11 0.092 0.072 0.14 0.10 0.043 0.095 0.074 0.039 0.036
0.1 0.2 7 2 0.2 1 0.1 0.1 2 41
170 165 250 275 265 240 190 195 220 260
8.84205 -2.99849
3909.45 3235.98
588.15 570.81
0.096 0.024
-3.76856
3535.63
530.97
0.012
27
322
-2.45398
3801.01
543.35
0.017
6
294
-3.49284 -2.45491 -2.36767 -2.17234 -2.17642
4895.60 4075.26 3036.17 3032.52 3121.71
562.10 553.640 507.90 498.10 504.40
0.084 0.051 0.028 0.046 0.037
-0.805183
31 12.72
489.40
0.065
1
225
-1.57752
3145.80
500.30
0.066
1
235
0.6 0.8 0.7 12 10 1 2
155 175 135 91 288 200 238
0.2 290 4 309
8 288 10 293 0.2 220 1 240 1 235
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 2, 1983 317 Table VI1 (Continued) approx rms % lowest data formula C,H,F, C,H, C,H,, C,H,, C,H,, C,H,, C,H,, C,H,, C,H,, C,H,, C,H,, C,H,,O
B
3123.89
566.52
0.019
5
313
-2.79168 -3.20243 -2.42625 -1.10598
4106.45 2732.37 2891.48 2949.13
591.72 540.10 540.64 531.17
0.038 0.063 0.022 0.057
6 0.1 1 1
309 240 265 250
-2.23048 -2.85992 -2.40376 -2.78710 -4.44565 -2.22001
3601.90 3732.98 3536.78 3513.00 2482.82 2561.55
617.12 630.25 616.97 616.15 568.81 543.96
0.041 0.026 0.014 0.021 0.10 0.11
6 6 6 6 0.1 1
330 337 332 331 260 265
-3.59254
2873.76
652.5
0.13
0.1 325
pentafluorotoluene toluene heptane 3-ethylpentane 2,2,3-trimethylbutane ethylbenzene o-xylene m-xylene p-xylene octane 2,2,4-trimethylpentane 1-octanol
-8.04616
1.43971
-3.76736
-3.00179
-7.28607 -7.67468 -7.58305 -7.22017
1.38091 1.37068 1.58587 1.44914
-2.83433 -3.53620 -3.567 3 2 -3.11808
-7.48645 -7.53357 -7.592 22 -7.63495 -7.87867 -7.38890
1.45488 1.40968 1.39441 1.50724 1.32514 1.25294
-3.37538 -3.10985 -3.22746 -3.19678 -3.78494 -3.16606
-9.71763
4.22514
in Table VI (SupplementaryMaterial) for eq 20 are in fact the results of a crudely constrained fit to the sample data. Furthermore, the worst percentage deviation for eq 20 was 2.28 while that for the Wagner eq 2 was only 0.7 (excluding argon). Bearing these facts in mind and reflecting that the Wagner equation has been chosen to represent full range data, it was decided to use the Wagner equation for the constrained fitting of limited data. A further point in favor of the Wagner equation is that the adjusted values of Pc involved in compiling Table IV (Supplementary Material) show equal numbers of small positive and negative deviations from the measured values, while the Thek-Stiel model yields 13 small negative ones and one positive one. It will be seen from Table VI (SupplementaryMaterial) that the unconstrained fitting of the Wagner equation to the limited range data gives good results when the range of the data is not too small. It is also evident that the unconstrained fitting of the Wagner equation yields more accurate vapor pressures than any of the generalized equations whose results are given in Table V (Supplementary Material). Constrained Fit to Limited Range Data First Constraint. The Clausius-Clapeyron equation rigorously relates liquid vapor pressure to latent heat of vaporization, temperature, and volume change accompanying vaporization. It may be written d (In P) AH -(31) dT RP(Zv-ZL) Rearranging
Waring (1954) observed that for a wide variety of substance a plot of the left-hand side of eq 32 against reduced temperature exhibits a minimum value at a value of TR in the range 0.80 to 0.85. Ambrose et al. (1978) observed that the minimum occurs at a somewhat lower value of TR for very low-boiling substances and at about TR = 0.95 for alcohols. Differentiation of the left-hand side of eq 32 and equating the result to zero will yield the minimizing value of TR. If the Wagner equation is used, the following equation is obtained for the minimizing value of TR 0.75B(1 - T R ) ~ 'i . '6C(1 - TR) 300(1 - T R )=~0 (33)
+
Using eq 33, minimizing values of T R for each of the 72 substances of Table VI1 were obtained, and it was found
T
error
A
-12.9222
P
Tc
D
C
name
PC
Table VIII. Full Range Deviations from Constrained Fit of Wagner Equation t o Limited Sample of Data
frattional temp, temp value K - range max % dev
rms % dev substance ID
sample
all
1 2 3 4 5 6 7 8 9 10 11 12 13 14* 14 av value (excluding 14*)
0.002 0.019 0.016 0.014 0.010 0.006 0.028 0.017 0.015 0.014 0.018 0.015 0.043 0.023 0.028 0.018
0.29 0.13 0.05 0.11 0.12 0.07 0.23 0.11 0.12 0.19 0.50 0.20 0.10 0.99 0.21 0.17
-0.55 -0.34 -0.22 -0.24 0.30 -0.17 0.33 0.43 0.33 -0.33 -1.1 -0.54 -0.12 -1.6 -0.42
500 480 600 500 560 500 400 480 470 490 480 490 120 130 140
0.24 0.27 0.28 0.28 0.28 0.29 0.27 0.28 0.24 0.26 0.22 0.27 0.23 0.06 0.24
that the values fell into quite narrow ranges. With the exception of alcohols, it was possible to classify the ranges of TR according to the normal boiling temperatures (TB) of the substances. In fact, for 50 K C TB C 100 K the range is 0.70 to 0.77; 100 K C TB C 273 K, 0.78 to 0.85; 273 K C TB, 0.82 to 0.88; and for alcohols, 0.90 to 0.98. Substances with values of TB below 50 K do not exhibit such minima. When fitting the Wagner equation to limited range data, the values of the coefficients were restricted to those sets yielding a solution for eq 33 that lies within the appropriate range for the substance. Second Constraint. At low pressures the term LvI/(Zv - 2,) in eq 31 and 32 varies only weakly with temperature. Assuming this function is constant and integrating eq 31 yields In P34= A - (B/T) (34) The constants A and B may be obtained from vapor pressure data at T = TCand T = 0.7Tc. Ambrose et al. (1978) show that In evaluated at TR = 0.95 falls into a narrow range for many substances. Making use of the Pitzer acentric factor (w), eq 34 becomes In ( P 3 4 / P ~ = )- 0.28278 (1 w ) for TR = 0.95 (35)
+
Equations 2 and 35 were used to calculate the value of In (PIPa)at TR = 0.95 for the seventy two substances of Table VII.
318
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 2, 1983
m
8
w
m
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 2, 1983 319
320
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 2, 1983
a)
3
E 0
aJa
a ) a ) a ) a ) a ) a ) a ) w a ) a ) w
""C cow
C C C E C E C C C C C E 0 c 0 E 0 E 0 c 0 E 0
E C
L ~ S G ~ S NrrWE0 c0 c0 c0 c0
N
m
z
a)aa E C C W
0 0 E ~ r i l " c j r l0 crr E 0 E 0 0 cei
W
d
a ) a ) a ) a )
N M E C E C rr-l"cjc 0 E 0 c 0 0 C
2m N C 0~
