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 men- tioning, 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 tem- perature and pressure together with one or two further basic data. The Riedel, Miller, Thek-Stiel, and Gomez- Thodos 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 1 to 200 P a . Values of constants valid for this range and yielding vapor pres- sure 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 considerations were involved in the development of this equation, which was initially derived to describe the vapor pressures of argon and nitrogen from the triple

019~-4305ia3ii 122-0313$01.50io

point to the critical temperature (Wagner, 1973). Thomas Equation.

253312 p=- ex - C (3)

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 com- pound 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 de- scribed 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)

E,(x) = x (9)

(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 com- puter programs allow easy generation of the constants and evaluation of vapor pressures.

It is easily shown that

E&) = 2x2 - 1

Riedel Equation.

(11) B In PR = A - - + C In TR + DTR6 TR

where A = -35Q

@ 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,)

a, = d (In PR)/d (In TR) at TR = 1

(12) where a, is called the Riedel parameter and is defined by

(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

where

Miller Equation. This equation is referred to by Reid et al. (1977) as the Reidel-Plank-Miller equation

where A = 0.4835 + 0.4605h (17)

h = (18) TBR In (1O1.325/Pc)

1 - TBR

Thek-Stiel Equation.

where

f(TR) = 1 1.14893 - - - 0.11719TR -0.03174T~~ - 0.375 In TR

(22) B = 1 . 0 4 2 ~ ~ ~ - 0.462844 (23)

C = 5.2691 + 2.0753A - 3.1738h (24) a, is again obtained from a knowledge of normal boiling temperature, and h is given by eq 18.

T R

Gomez-Thodos Equation.

(25)

where

(26) 7.0109 380900 C = 2.4186 - - h + he(123.21/h)

a, + BC 7

D E -

A = -(B + D) (29)

Table I. Input Data Normally Required for Eq 1, 2, 3, 5, 11, 16,20, and 25

equation eq no. 1 fitted constants A, B, C Antoine

Wagner 2 Tc, Pc, fitted constants A, B, C, D Thomas 3 fitted constants A, B, C Chebyshev 5 TL, TH, fitted constants a, to as Riedel Tc, P c , TB

Thek-Stiel 20 Tc, P c , TB, AHB Gomez-Thodos 25 Tc, P c , TB, LY,

Miller T C , p C , TB

Table 11. Modified Input Data

equation Wagner Riedel Miller Thek-Stiel Gomez-Thodos

T,, fitted constants Po A, B, C, D T,, P,, fitted constants A, B, C, D T,, fitted constants Po A, B T,, fitted constants Pc, A, B, C, D T,, P,, fitted constants A, B, 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 relation- ships for enthalpy of vaporization and the difference be- tween 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 re- gression 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 re- placement of the constant 0.04 by an adjustable value. It is only to be expected that constants obtained by data regression wil l yield better results than those obtained from critical properties. The modified input data are summa- rized 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 sub- stances 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 de- viation 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 no. of total

substance ID source data points data points

benzene

toluene

ethylbenzene o-xylene

m-xylene p-xylene diethyl ether acetone cyclohexane

water methanol

ethanol

nitrogen argon

1

2

3 4

5 6 7 8 9

10 11

1 2

13 14

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. (1 945) 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)

1 9 1 9 9

20 4 9

20 9

17 9

1 2

1 2 20 1 2 25 23

47

33 33

29 29 29 40 47

38 49

44

48 49 49

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. (See paragraph 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 poly- nomials requires seven coefficients for the above perform- ance 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 con- stants 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 car- bonyl 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 fluo- robenzenes and pentafluorotoluene.

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 cal- culated 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 par- ticular 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 wi 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 cal- culated 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 co CO, cos F, He HZ HZ 0

NZ 0 2 CCl,F,

CC1,

Ne

CHClF,

CHC1, CHF,

CHZFZ CH3C1 CH-F CH; CH,O CH,N C2C13F3

'ZH6

'ZH6'

C,H,N C ~ H I N

'6 H6

C6H14

'6

name A

argon -5.90501 carbon -6.20798

monoxide carbon dioxide -6.95626 carbonyl sulfide -6.40952 fluorine -6.18224 helium -3.97466 hydrogen -5.57929 water -7.76451 neon -6.07686 nitrogen -6.09676 oxygen -6.28275 dichlorodi- -7.01657

fluoromethane

tetrachloride

fluoromethane

carbon -7.07139

chlorodi- -6.99913

chloroform -6.95546 trifluoro- -7.41994

methane difluoromethane -7.44206 methyl chloride -6.86672 methyl fluoride -6.78099 methane -6.00435 methanol -8.547 96

trichlorotri- -7.36666

ethylene -6.32055

methylamine -7.527 7 2

fluoroethane

1,2-dichloro- -7.36864 ethane

ethane -6.34307 ethanol -8.51838 ethyiamine -7.20059

propylene -6.64231 acetone -7.45514 prop an e -6.67833 1-propanol -8.05594 2-propanol -8.1 6927

dimethy lamine -7.9 02 9 5

propylamine -7.23587 trimethylamine -6.88066

methyl ethyl -7.71476

butene -6.88204 isobutene -6.95542

ketone butane -6.88709 isobu tane -6.95579

1-butanol -8.007 56 2-butanol -7.80578

1-pentene -7.04875 pentane -7.28936 isopentane -7.12727

diethyl ether -7.29916

diethylamine -7.267 96

2,2-dimethyl- -6.89153 propane

1-pentanol -8.97725 chloropenta- -8.0217 2

pentafluoro- -7.79730

1,2,4,5-tetra- -7.79740

fluorobenzene

benzene

fluorobenzene benzene -6.94739

hexane -7.51650 isohexane -7.28750 3-methyl- -7.27084

cyclohexane -6.96009

pentane

butane

butane

2,2-dimethyl- -7.25933

2,3-dimethyl- -7.27870

B

1.12627 1.27885

1.19695 1.21015 1.18062 1.00074 2.60012 1.45838 1.59402 1.13670 1.73619 1.73224

1.71497

1.23014

1.16625 1.65884

1.51914 1.5227 3 0.828379 1.18850 0.769817 1.81615 1.81971

1.16819 1.767 27

1.01630 0.341626 1.20679 2.81577 1.21857 1.20200 1.15437 4.25183E-2

-9.4321 3E-2 1.22853 1.1 5962 1.27051 1.35673 1.71061

1.1 51 57 1.50090 1.24828 0.537 826 0.324557 1.1 5810 1.17813 1.53679 1.38996 1.25019

2.99791 1.54665

1.35271

1.57406

1.25253 1.31328 1.54797 1.29015 1.26113

1.69602

1.56349

C D -0.767869 -1.62721 -1.34533 -2.56842

-3.12614 2.99448 -1.54976 -2.10074 -1.16555 -1.50167

-0.855060 1.70503 -2.77580 -1.23303 -1.06092 4.06656 -1.04072 -1.93306 -1.81349 -2.536453-2 -2.97909 -0.377232

-2.89930 -2.49466

1.50056 -0.430197

-2.49377

-2.13970 -3.14962

-2.75319 -1.92919 -1.41137 -0.834082 -3.10850 -4.20677 -3.94233

-1.55935 -3.34295

-1.19116 -5.73683 -3.71972 -6.31338 -1.81005 -2.43926 -1.64984 -7.51296 -8.10040 -3.75004 -2.18332 -2.26284 -2.45222 -3.68770

-1.99873 -2.52717 -2.91931 -9.34240 -9.41265 -3.91125 -2.45105 -3.08367 -2.54302 -2.28233

-2.21052

-3.44421 -0.849379

-0.979495 -2.61459 -2.41700 -1.22833

-1.22275 1.54481

0.625601

-1.83552 -1.43530

-2.03539

-4.33511 -0.224073 -2.48212 -3.35590 -2.70017

8.32581

6.89004 7.85000

-4.33990 -2.94707 -2.61632 -1.46110 -0.751692

-3.13003 -1.49776 -3.36740

6.68692 2.64643

-1.17981 -2.21727 -1.02456 -2.45657 -4.74891

-12.9596 8.84205 -3.78361 -2.99849

-3.50409 -3.76856

-3.82060 -2.45398

-2.53686 -3.49284 -2.75683 -2.45491 -3.38541 -2.36767 -2.97853 -2.17234 -2.81741 -2.17642

-3.18124 -0.805183

-3.05387 -1.57752

rms % pc Tc error

approx lowest data

P T 4857.99 3501.15

7374.99 6346.45 5214.72

1309.60

2724.55 3399.61 5089.87 4132.03

4550.78

230.029

22122.3

4983.31

5365.76 4840.92

5826.99 6697.18 5557.36 4596.42 8 0 8 5.0 5 7433.32 3425.71

5050.88 5362.00

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

3790.62 3658.01 3646.10 4412.63 4189.75 3705.40 3536.85 3378.62 3385.90 3197.88

3909.45 3235.98

3535.63

3801.01

4895.60 4075.26 3036.17 3032.52 3121.71

31 12.72

3145.80

150.651 0.023 132.91 0.28

304.15 0.011 378.8 0.18 144.31 0.012

5.20 0.26 33.19 0.27

647.35 0.026 44.38 0.57

126.200 0.025 154.7 0.28 384.95 0.038

556.40 0.027

369.30 0.042

536.40 0.14 299.06 0.075

351.54 0.066 416.27 0.11 315.0 0.079 190.53 0.019 512.64 0.12 430.0 0.073 487.7 0.32

282.55 0.034 566.00 0.071

305.42 0.13 513.92 0.077 456.35 0.089 437.70 0.043 364.85 0.018 508.10 0.086 369.82 0.043 536.78 0.19 508.30 0.22 497.0 0.086 433.30 0.073 419.57 0.094 417.90 0.086 536.78 0.11

425.18 0.11 408.14 0.092 466.74 0.072 563.05 0.14 536.01 0.10 496.45 0.043 464.78 0.095 469.74 0.074 460.43 0.039 433.77 0.036

588.15 0.096 570.81 0.024

530.97 0.012

543.35 0.017

562.10 0.084 553.640 0.051 507.90 0.028 498.10 0.046 504.40 0.037

489.40 0.065

500.30 0.066

69 84 26 7 1

530 217 2 162 4 64 5 2 7 14 0.7 275

43 25 1 3 63

0.2 54 0.2 155

1 250

1 170

0.1 215 0.2 125

0.6 155 0.8 175 0.7 135

12 91 1 0 288 1 200 2 238

0.1 105 1 260

2 133 6 293 1 215 1 240 0.1 140 4 259 0.2 145 0.2 260 0.1 250 1 235 1 200 0.2 170 0.2 170 1 255

0.1 170 0.2 165 7 250 2 275 0.2 265 1 240 0.1 190 0.1 195 2 220

41 260

0.2 290 4 309

27 322

6 294

8 288 10 293

0.2 220 1 240 1 235

1 225

1 235

Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 2, 1983 317

Table VI1 (Continued) approx

formula name A B C D rms % lowest data

P C Tc error P T C,H,F, pentafluoro-

C,H, toluene C,H,, heptane C,H,, 3-ethylpentane C,H,, 2,2,3-tri-

C,H,, ethylbenzene C,H,, o-xylene C,H,, m-xylene C,H,, p-xylene C,H,, octane C,H,, 2,2,4-tri-

C,H,,O 1-octanol

toluene

methylbutane

methylpentane

-8.04616

-7.28607 -7.67468 -7.58305 -7.22017

-7.48645 -7.53357 -7.592 22 -7.63495 -7.87867 -7.38890

-9.71763

1.43971

1.38091 1.37068 1.58587 1.44914

1.45488 1.40968 1.39441 1.50724 1.32514 1.25294

4.22514

-3.76736

-2.83433 -3.53620 -3.567 3 2 -3.11808

-3.37538 -3.10985 -3.22746 -3.19678 -3.78494 -3.16606

-12.9222

in Table VI (Supplementary Material) 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 wil l be seen from Table VI (Supplementary Material) 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 (Supple- mentary 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 accompa- nying vaporization. It may be written

(31) AH -- - d (In P)

dT RP(Zv-ZL) Rearranging

Waring (1954) observed that for a wide variety of sub- stance 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 - TR)~' . ' 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

-3.00179

-2.79168 -3.20243 -2.42625 -1.10598

-2.23048 -2.85992 -2.40376 -2.78710 -4.44565 -2.22001

-3.59254

3123.89

4106.45 2732.37 2891.48 2949.13

3601.90 3732.98 3536.78 3513.00 2482.82 2561.55

2873.76

566.52

591.72 540.10 540.64 531.17

617.12 630.25 616.97 616.15 568.81 543.96

652.5

0.019

0.038 0.063 0.022 0.057

0.041 0.026 0.014 0.021 0.10 0.11

0.13

5 313

6 309

1 265 1 250

6 330 6 337 6 332 6 331 0.1 260 1 265

0.1 325

0.1 240

Table VIII. Full Range Deviations from Constrained Fit of Wagner Equation t o Limited Sample of Data

max % dev frat- tional temp, temp rms % dev

substance ID sample all value K- range 1 2 3 4 5 6 7 8 9

10 11 12 1 3 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 0.24 480 0.27 600 0.28 500 0.28 560 0.28 500 0.29 400 0.27 480 0.28 470 0.24 490 0.26 480 0.22 490 0.27 120 0.23 130 0.06 140 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. Sub- stances 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

(34) The constants A and B may be obtained from vapor

pressure data at T = TC and 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.

In P34 = A - (B/T)

318 Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 2, 1983

8 w

m 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) a ) a ) a ) a ) a ) a ) a ) w a ) a ) w a J a a ) a a a ) a ) a ) a ) C"" C C C E C E C C C C C E C m z E C C W N M E C E C 2 m 0 0 E ~ r i l " c j r l 0 c r r 0 0 0 E E c e i rr-l"cjc 0 0 0 0 E c C N ~ C < 0 cow 3 0 E L ~ S G ~ S N W rr 0 0 0 0 0 0 0 0 0 0 0 E c c c c E c E E c E N

d W m d w O N 0 w 0 0 m w m 0 w w m w w w m w w w ~ w 0 r i m w w w w w w w w w w o i w w w w w w w w w w rirrNC\l r i m 0.1 N

* * *

Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 2, 1983 321

Values of w were largely taken directly from the book by Reid et al. (1977). In a small number of cases the Antoine constants presented in the same book were valid for the value of TR relating to the acentric factor; these constants were therefore used to calculate w.

at TR = 0.95 fell into the fol- lowing ranges: for TB < 100 K, the range is -0.024 to

-0.020 to 4.001; and for alcohols, +0.007 to +0.040. The only exception is hydrogen, which exhibited a value of -0.043. The above ranges constitute the second restraint when fitting the Wagner equation to limited range data.

Third Constraint. The Watson correlation (1943) may be written

The values of In

4.018; 100 K C TB C 273 K, -0.017 to -0.006; 273 K C TB,

1 w w w w w a w w ~ w w w w w w w w w w m w w o F

@I

FJ

d 0

5 -

Y

Y

8 c,

I + .- a

E 0 .- Y

w w a w 9 9 9 9 4 2

E .I

2 2 8 Y

2 'i Y

.- e

.y,

I .- w 4 2

U

Since A H B and TBR are constants AH = g(1 - TR)"~ '~ (37)

At low values of TR vapor pressures are low and eq 32 where g is a constant.

approximates to d(ln P)

RP-=AH dT

Combining eq 37 and 38

(39) R P d(ln PR)

1 - TR)0.375 dT g =

In terms of Wagner coefficients this becomes -R Tc

(1 - T R ) ~ ' ~ ~ ~ [A + BV"(1.5 - 0.5Y) + CV(3 - g =

2Y) + DP(6 - 5Y)] (40) where Y = 1 - TR.

Ambrose et al. (1978) suggest that the coefficients of a vapor pressure equation should be restricted to those sets which yield little differences in g at TR = 0.5 and 0.6. The constants of the fitted Wagner equation were used to calculate the value of g at these values of TR for the 72 substances of Table VII.

The ratios g(TR = 0.5) to g(TR = 0.6) fell into the fol- lowing narrow ranges which constitute the third constraint employed for fitting the Wagner equation to limited range data. For TB C 100 K, the range is 1.06 to 1.20; 100 K C TB C 273 K, 1.01 to 1.06; 273 K C TB, 0.99 to 1.03; for alcohols, 0.98 to 1.06. There were no exceptions to these ranges.

By use of the same samples of data that were involved in Table VI and the three constraints detailed above, values of the Wagner equation constants were obtained by a computer program utilizing NAG routine E04VBF for constrained minimization of an objective function. These constants were then used to calculate vapor pres- sures up to the critical pressure, and the root mean square percentage deviations from the full range data were ob- tained. The results are given in Table VIII.

The last entry in the table relates to an argon data sample which represenb a fraction of the full-range data having a similar size to the other entries. While the con- straints caused a dramatic reduction of the overall devia- tion obtained from the smaller sample (0.99 cf. 81) the larger sample yielded an overall deviation (0.21) more in keeping with those of the other substances.

It will be seen from Tables VI (Supplementary Material) and VI11 that the use of constraints improves full range prediction using only limited range data. The uncon-

322

strained fit of the Wagner equation yielded an average RMS % deviation of 0.30 for the whole range while the constrained fit yields a value of 0.17.

The publication of Boublik et al. (1973) was used as the main source of vapor pressure data for constrained re- gression of Wagner coefficients. The data of Sheft et al. (1973), Oguchi et al. (1975), and Messerly et al. (1975) were used to obtain coefficients for hydrogen fluoride, chloro- trifluoromethane, and ethylenediamine. Antoine coeffi- cients presented by Reid et al. (1977) were used to generate data for propionaldehyde.

Since the Wagner equation requires a knowledge of the critical temperature and pressure of a substance, con- strained generation of coefficients was limited to sub- stances for which these are known; values were taken from the work of Reid et al. (1977).

The resultant Wagner coefficients for 179 substances are given in Table IX. Values of Pc and Tc are also given and these are to be used in the Wagner equation. For com- parison purposes unconstrained regression was also carried out in each case.

In most cases the fitting of the data involved all three constraints and only in the cases of methyl iodide and acetaldehyde were all three active at the solution. How- ever, when the data covered the values of TR relating to one or more constraints, then such constraints were totally relaxed. Furthermore, in five cases (marked with an as- terisk in Table IX) where normal boiling temperatures are not much higher than 273 K, it was considered permissible to improve results by relaxing the constraints to those relating to substances with TB in the range 100 to 273 K.

Fifteen substances in Table M yielded very poor results when compared with those obtained by unconstrained regression and it was concluded that these substances constituted marked exceptions to the constraints. For these substances, therefore, the Wagner coefficients presented are those resulting from unconstrained regres- sion. They are identified by two asterisks in Table E. In all cases of course, the percentage root mean square de- viations given in Table IV can only relate to the actual limited range data. Summary

Liquid vapor pressures of high accuracy are currently available for 14 pure substances over a pressure range extending from about 5 kPa up to the particular critical pressure. The abilities of eight equations to correlate these data are investigated. The Wagner equation is selected for processing the data for a further 58 substances for which vapor pressures up to the particular critical value have been reported. Wagner coefficients presented rep- resent the data for all 72 substances with an average root-mean-square percentage deviation of 0.09.

Vapor pressure data for many more liquids exist but are usually limited to the approximate range 10 to 150 kPa. Analysis of the results obtained by correlation of full-range data indicates that vapor pressures of a substance are usually circumscribed by three constraints selected ac- cording to normal boiling point. Wagner coefficients are then obtained by constrained regression of limited range data for 179 substances. The coefficients in Tables VI1 and IX, therefore, allow correlation and prediction of vapor pressures up to the particular critical value for a total of 251 substances. Nomenclature A, B, C, D = coefficients (constants) in vapor pressure

ao, a, = coefficients in eq 5 g = constant in eq 37

Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 2, 1983

equations

AH = enthalpy of vaporization, J mol-’ AHB = enthalpy of vaporization at the normal boiling point,

P = vapor pressure, kPa P, = critical pressure, kPa PR = reduced pressure, PIPc

R = universal gas constant, 8.3144 J K-’ mol-’ T = absolute temperature, K TB = normal boiling point, K (boiling point at 101.325 kPa) TBR = reduced normal boiling point, TB/Tc T, = critical temperature, K TH = highest data temperature, K TL = lowest data temperature, K TR = reduced temperature, TIT, X = variable defined by eq 4 Z, = compressibility of saturated vapor Z, = compressibility of saturated liquid Greek Letters a, = Riedel parameter, defined by eq 13, K w = Pitzer acentric factor Literature Cited Ambrose. D. J . Chem. Soc. A . 1988, 1381. Ambrose. D. J . Chem. 7hs“@m. 1978, 10, 765. Ambrose, D.; Broderick, 8. E.; Townsend. R. J . Chem. Soc. A 1987, 633. Ambrose, D.; Counseii, J. F.; Davenport, A. J. J . Chem. Thermodyn. 1970,

Ambrose, D.; Counsell, J. F.; Hicks, C. P. J . Chem. ?hrmo@yn. 1978, 10,

Ambrose, D.; Sprake, C. H. S. J . Chem. “ o d y n . 1970, 2 , 631. Ambrose, D.; Sprake, C. H. S.; Townsend, R. J . Chem. Thermodyn. 1972,

4 , 247. Ambrose. D.; Sprake, C. H. S.; Townsend. R. J . Chem. lhmrwdyn. 1974,

6 , 893. Ambrose, D.; Sprake. C. H. S.; Townsend, R. J . Chem. Themrodyn. 1975,

7, 185. Antoine, C. C . R . Acad. Sei. 1888, 107, 681, 836, 1143. “API 44 Tables-Sekcted Values of Properties of Hybocarbons and Related compwnds”, Thermodynamics Research Center, Texas A 8 M Universlty, 1968; Vol IV.

Bender, P.; Furukuwa, (3. T.; Hyndman, J. R. Ind. €ng. Chem. 1952, 44, 387.

Boubk, T.; Fried, V.; Hale, E. “The Vapour Pressure of Pure Substances”; E lsevk New Yo&, 1973; p 7.

Electrical Research Association. “1967 Steam Tables”; Edward Arnold Ltd.: London, 1987; pp 2-9.

Engineering Sciences Data Unk, “Physlcal Data”; London, 1972-1978; Voi. 5.

Frost, A. A.; Kakwarf, D. R. J . Chem. h y s . 1953. 21, 264.

Hugill. J. A.; McQlashan. M. L. J . Chem. Thermodyn. 1978, 10, 95. Kemp, J. D.; Giauque, W. F. J . Am. Chem. SOC. 1937, 59, 79. Kerns, W. J.; Anthony, R. G.; Eubank, P. T. AICM Symp. Ser . No. 140

1974, 70, 14. Mastrdennl, M. J.; Stahi, R. F.; Sheldon, P. N. J . chem. Eng. Data. 1978, 23, 113.

Messeriy, J. F.; Finke, H. L.; Osbwn, A. G.; Douslin, D. R. J . Chem. Thenno- dyn. 1975, 7, 1029.

Miller, D. G. J . Fhys. Chem. 1985, 69, 3209. Oguchi. K.; Tanishita, I.; Watanabe, K.; Yamaguchi. 1.; Sasayama, A. BUN.

JSME 1975, 18, 1456. Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K.. “Properties of Gases and

Liquids”, 3rd ed.; McGraw-Hili: New York, 1977; Appendix A, pp

Rledei, L. Chem. Ing. Tech. 1954, 26, 83. Robinson, D. B.; Senturk, N. H. J . Chem. Thennodyn. 1879, 1 1 . 461. Sheft, I.; Perklns, A. J.; Hymn, H. H. J . Inorg. Nud. Chem. 1973, 35,

Skaates, J. M.; Kay, W. B. Chem. fng. Sci. 1984, 19, 431. Thek, R. E.; Sttel, L. I . A I M J . 1988, 12, 599. momas, L. H. Chem. Eng. J . 1078, 1 1 , 191. WADD Technical Report 60-56; Wright Air Development Division: Ohio,

Wagner, W. Cryogenics 1973, 13, 470. Warlng. W. I n d . Eng. Chem. 1954, 46, 762. Watson, K. M. Ind . Eng. Chem. 1943, 35, 390. WiHimgham, C. J.; Taylor, W. J.; Plgnocco. J. M.; Rossini, F. D. J . Res. Net/.

Receiued for review October 22, 1980 Reuised manuscript received August 9, 1982

Accepted September 10, 1982

Supplementary Material Available: Tables IV, V, and VI (vapor pressure data) are presented as supplementary material (3 pages). Ordering information is given on any current masthead page.

J mol-’

= vapor pressure as calculated by eq 34, kPa

2 . 283.

771.

Oamz-Nieto, M.; Thodos, 0. AIChE J . 1977, 23, 904.

829-665.

3677.

1981; Part I , Chapter 6.

Bur. Stand. 1945, 35. 219.