Activity Coefficients of Gases

law. Similarly, the use of the partial fugacity in place of the partial pressure avoids errors which are due to the deviation from the ideal gas law. ...
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Activity Coefficients of Gases ROGERH. NEWTON Yale University, New Haven, Conn.

T

HE fugacity is a special function, so defined that the simple form of the thermodynamic r e l a t i o n s h i p s which apply to an ideal gasmay be preserved for actual gases by the substitution of the fugacity for the Pressure. Thus the use

bitrary constants such as the van der Waals, Dieterici, or Berthelot equations, is known to be only an approximation, but the agreemerit is sufficiently good o v e r of calculating the fugacity. This ratio is shown surprisingly wide ranges to make it very convenient for engineerto be a function of the reduced temperature and ing calculations. reduced pressure only, for a large number of subAny two of the reduced magstances, and curces of the acticiiy coejicient us. ~e~~~~~~~~ nitudes-for instance, p , a n d PTat w ~ i o u sl’dues Of Tr are git‘en for the enequations dealing with equilibria T,-are sufficient to define the makes them valid for real gases tire known range of T,. and P,.. state of a gas and t h e r e f o r e which deviate the i d e a l - g a s to fix its thermodynamic proplaw. S i m i l a r l y , t h e use of erties. Consequently, j / P , or the partial fugacity in place of the partial pressure avoids y, may be represented as a function of P, and T,. Furthererrors which are due to the deviation from the ideal gas law. more, it appears that y is not very sensitive to slight differFor these reasons the ratio of the fugacity to the pressure is ences in the surfaces of state, except in certain limited teman important function in dealing with high-pressure equilibria, perature and pressure ranges, and better agreement is obincluding both phase and chemical equilibria. From an tained for this function than for some other thermodynamic analogy to the similar term used in dealing with liquid solu- functions. A number of workers (6,19,dO, d f ) have recently tions, the ratio of the fugacity to the pressure might be called correlated the properties of gases in this manner. Lewis the “activity coefficient.” Furthermore, the problems con- and co-workers, in particular, have applied this law to the cerning gaseous solutions arising in engineering work have calculation of ‘the f / P ratio, or the activity coefficient, but frequently been simplified by the use of the “partial fugacity have restricted it to a rather limited set of conditionsrule” which states that the fugacity of a gas in a mixture namely, hydrocarbons having more than three carbon atoms is equal to its mole fraction times the fugacity of the pure per molecule-a pressure range up to ten times the critical gas at the temperature and total pressure of the mixture. This pressure, and a temperature range up to 1.5 times the critical was originally stated as an empirical rule but has since been temperature. shown to rest on the- assumption that the volume of the It is the purpose of this paper to show that this correlation mixture is the sumo a n be greatly exm a t i o n of t h e tended and applied volumes of the comto the existing data ponents a t the presfor a large number sure and temperaof gases. With a ture of the mixture graph of y vs. P, (the (11). T h i s rule reduced pressure or m a y be restated pressure divided by thus: The fugacity t h e c r i t i c a l presk of a gas in a mixture sure) a t various is equal to the acvalues of T,(the retivity c o e f f i c i e n t duced temperature or absolute temperatimes i t s p a r t i a l pressure. For the ture divided by the sake of simplicity, a b s o l u t e critical RfDUC€D PRESSURE-pR temperature), it now the ratio f / P will be termed y. FIQURE 1. ACTIVITY COEFFICIENT OF GASESBELOW THE CRITICAL TEMPERATURE becomes possible to The activity coeffiestimate the activity cient may be calculated from an equation of state or by a coefficient for any gas or vapor whose critical constants are graphical treatment of the pressure volume-temperature known. This paper will deal only_with the .calculation of data. It is well known that no equation of state will ac- these graphs; their application to the solution of actual probcurately represent the P-V-T behavior of gases over wide lems will be given in another paper. ranges; therefore it seemed best to calculate the fugacities METHOD OF CALCULATION from the actual experimental data by graphical methods. I n order to present the results in a simple and compact The calculations of Deming and Shupe (7, 8 , 9 ) ,g!ving the form and also to permit predictions where compressibility fugacities of nitrogen, hydrogen, and carbon monoxide, were measurements are lacking, the law of corresponding states used as a starting point. The values of fwere divided by the has been used. This law says that all gases should obey a pressure to give y, and these were plotted as functions of universal equation of state when the pressure, volume, and T,and P,. temperature are expressed in “reduced” units-that is, In order to present values for the activity coefficient over ratios to the critical values. The law, which can readily be a wider range of temperature and pressure and also to ascerdeduced from any of the equations of state with only two ar- tain the applicability of the law of corresponding states to

f

r

~

I n dealing with high-pressure equilibria, the

ratioof thefugacity to thepressure of a gas (herein called the “activity coeficient”) is a n important funcfion in that furni-yhes a convenient means

~

~

~

302

~

~

~

~

March, 1935

INDUSTRIAL AND ENGINEERING CHEMISTRY

303

REDUCED PRESSUR€ - PR OF GASESIN FIGURE 2. ACTIVITYCOEFFICIENT

this function, y was calculated from actual P-T7-Tdata for all substances for which such data were available over a sufficient range of temperature and pressure. The method of Lewis and Randall (26) for the calculation of the fugacity was used. This is briefly: the difference between the volume of the gas if it mere an ideal one and that actually measured for the gas is calculated; this is called a.

THE

INTERMEDIATE TEMPERATURE RANGE

was made to calculate the activity coefficient for methyl chloride from the data of the Roessler and Hasslacher Company (SI), but this was a t a wide variance with that calculated from the data of Holst (1.2) and of Kuenen (17'). TABLE I. SOURCESOF P-V-T DATA PRESSURE

GAB

y is then calculated from a by the following equation:

The value of the integral was obtained by determining the area under a curve of CY vs. P , by means of a planimeter. Inasmuch as the y is thus obtained directly from the P-V-T data, no error is introduced by any assumptions, such as an equation of state of imperfect accuracy. Such calculations were made for argon, nitrogen, oxygen, methane, carbon dioxide, ethylene, ammonia, hydrogen, neon, helium, dichlorodifluoromethane, methyl chloride, water, sulfur dioxide, n-hexane, acetylene, cyclohexane, isopentane, nitric oxide, krypton, methanol, ethanol, and ethyl ether. A brief summary of the P-V-T data from which these calculations were made is given in Table I. The first part of the above mentioned data covers a low enough range of pressure to permit the evaluation of a a t low pressures so that the calculated values of the activity coefficient should not be subject to any calculation error. For ethyl ether, methyl chloride, and dichlorodifluoromethane the data did not cover a low enough range of pressure to allow as much faith to be placed in the calculation. An attempt

RANQE 80 kg./sq. EO kg./sq. 80 kg./sq. 80 kg./sq. 17 atm. EO kg./sq. 1000 atm.

TEMP. CITARANQE TION -183' to 400' C. (67) -183' to 400' C. (67) -183' to -100' C. (67) -130' to -50' C. (67) -252.8' C. ($3) om. -183' to -150' C. (67) -70' to 200' C. (56)

Argon Neon Hydrogen Nitrogen Helium Helium Helium

0 to 0 to 0 to 0 to 0 to 0 to 0 to

em. cm. om. cm.

Oxygen Ethylene Carbon dioxide n-Hexane Methane Methane

0 to 1000 atm. 0 to 1000 atm. 0 to 1000 atm. 0 to 18 atm. 32 to 255 atm. 36 to 321 atm.

0' to 199.5O C. 100' to 200' C. 30' to 200' C. 170' to 200' C. 0' to 200' c. 0' to 200' C.

Methane Ammonia Ammonia Ammonia Steam Acetylene Sulfur dioxide Isopentane Cyclohexane Ethyl ether Methyl chloride

100 to 1000 atm. 0 to 30 atm. 15 to 130 atm. 100 to 1100 atm. 0 to 240 atm. 0 to 12 atm. 0 to 9 atm. 1 to .-72 .~ stm. 4 to 105 atm. 10 to 200 atm. 15 to 40 atrn.

-70' to 200' C. -35' to 300' C. 75' to 325' C. 0" to 2000 c. 100' to 1000° F. 0' to 25' C. -20' t o 300' F. 30" to 280' C. 140' to 300' C. 150' to 325' C. 70" to 115' C.

Dichlorodifluoromethane Nitric oxide Krypton Methanol Ethanol

5 to 18 atm. 30 to 160 atm. 26 to 104 atm. 2 to 78 atm. 4 to 58 atm.

30' to 126' C. (8) -78.6' to 9' C . (4) 11.2' and 237.5' C. ( d 8 120' to 240' C. 130' to 246' C.

(1) (1) (1) (34)

(16) (16)

[id

These data were correlated by plotting y for each gas a t a certain reduced temperature (given by the isotherm calculated) vs. the reduced pressure. The values of y for all substances were then taken from these plots a t even values of the reduced pressure and plotted against the reduced

INDUSTRIAL AND ENGINEERING CHEMISTRY

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TABLE 11. AGREEMENTOF OBSERVED AND CALCULATED VALUES OF ACTIVITY COEFFICIENT TI Pr Calcd. Obsvd. Argon ( T c = 151' K.; P c = 48.0 atm.) 1.145 1.0 0.78 0.79 1.145 2.0 0.60 0.60 1.478 1.0 0.915 0.91 1.478 3.0 0.795 0.79 Oxygen (Tc= 190.6° K.; 45.8 atm.l 1.0 0.955 10.0 0.87 20.0 1.14 3.0 1.00 12.0 1.135 21.0 1.43 3.0 1.04 12.0 1.20 21.0 1.48

Pc =

1.76 1.76 1.76 2.41 2.41 2.41 3.06 3.06 3.06

0.955

Methane (TO= 190.13~K.; 45.8 atm.) 1.43 1.0 0.91 1.43 10.0 0.67 1.43 20.0 0.93 1.17 1.0 0.80 1.17 10.0 0.39 1.17 20.0 0.59 1.96 1.0 0.975 1.96 0.95 10.0 1.96 20.0 1.23 2.48 3.0 1.005 2.48 1.145 12.0 2.48 21.0 1.44

Pe =

Nitrogen (To = 126' K.; 33.5 atm.) 1.14 1.0 0.77 1.14 2.0 0.59 1.77 0.955 1.0 1.77 10.0 0.88 1.77 20.0 1.16 2.56 1.01 3.0 2.56 1.24 15.0 2.56 21.0 1.455 4.55 3.0 1.05 4.55 12.0 1.23 4.55 21.0 1.455 6.93 21.0 1.33

-

0.86

1.13 1.00 1.135 1.43 1.04 1.20 1.485

0.91 0.62 0.85 0.80 0.395 0.59 0.985 0.93 1.21 1.005 1.10 1.37

Carhon dioxide ( T e = 304.1O K.; PC = 73.0 atrn.) 0.997 2.0 0.385 0.38 0.997 6.0 0.23 0.23 0.997 10.0 0.205 0.21 1.227 1.0 0.835 0.84 1.227 5.0 0.495 0.49 1.227 10.0 0.47 0.47 1.55 1 . 0 0.93 0.93 1.55 5.0 0.775 0.785 1.55 10.0 0.775 0.785 Ethylene ( T o = 282.8' K.; 50.9 atm.) 1.32 1.0 0.875 1.32 10.0 0.565 1.32 20.0 0.795 1.45 1.0 0.91 1.45 10.0 0.68 1.45 20.0 0.955 1.67 1.0 0.945 1.67 10.0 0.845 1.67 20.0 1.13

ACTIVITY COEFFICIENT T7 PI Calcd. Obsvd. Ammonia (Te = 405.5' K.; P C= 111.5O atrn.) 0.796 0.02 0.979 0.979 0.796 0.05 0.952 0.952 0.796 0.15 0.868 0.868 0.990 0.15 0.939 0.940 0.99 10.0 0.180 0.185 1.166 1.0 0.79 0.79 1.166 5.0 0.415 0.405 1.166 10.0 0.385 0.32

Pc = 0.87 0.56 0.80 0.91 0.675 0.95 0.95 0,845 1.14 Po

-

0.78 0.59 0.955 0.86 1.16

Carbon monoxide ( T O 134.1° K.: Pe = 35.0 atm.) 1.51 6.0 0.74 0.735 1.61 10.0 0.73 0.73 1.51 20.0 1.02 1.02 2.09 3.0 0.955 0.955 2.09 12.0 1.035 1.035 2.09 21.0 1.35 1.355 3.16 3.0 1.04 1.04 3.16 12.0 1.22 1.22 3.16 21.0 1.50 1.505 5.03 12.0 1.22 1.24 5.03 21.0 1.43 1.46

-

Dichlorodifluoromethane ( T C 384.8O K.: P C = 39.5 atm.) 0.778 0 . 0 2 0.976 0 .'980 0.778 0.910 0.10 0.920 0.807 0.959 0.05 0.968 0.807 0.880 0.15 0.910 0.852 0.980 0.984 0.02 0.852 0.30 0.815 0,820 Methyl chloride (Tc= 415.9' K.; P. = 61.0 atrn.) 0.825 0.02 0.980 0.825 0.884 0.15 0.861 0.05 0.964 0,810 0.861 0.30 0.02 0,898 0.984 0.941 0.10 0,898 0.833 0.30 0.898

Sulfur dioxide (Tc= 430.5" K.; PC = 77.7 atrn.) 0.05 0.935 0.938 0.10 0.909 0.902

0.714 0.778

Water vapor ( T c = 647' K.; P c = 217.7 atm.) 0.652 0.02 0.960 0.962 0.737 0.02 0.971 0.971 0.737 0.05 0.940 0.940 0.823 0.10 0.921 0.919 0.909 0.30 0.840 0.840 1.080 1.0 0.70 0.68

t e m p e r a t u r e . Four such plots were made covering the i.oi 1.24 reduced temperature ranges 1.455 0 to 1.0,l.O to 2.0, 2.0 to 3.5, 1.05 1.23 and 3.5 to 36.0 because the 1.45 1.32 shawe of the curves was such that all the points could not be adequately s h o w n on a single plot. Through the points thus obtained, "average" curves were drawn which were smooth and, for the most part, passed directly through all points for all substances. The agreement between the average curves, so drawn, and the individual points is shown in Table 11. The values for the activity coefficient for hydrogen were found to be lower than those for nitrogen in the region of overlap, and likewise those for helium were found to be lower than those for hydrogen. It appeared that the addition of a constant to the critical pressure and critical temperature in calculating the reduced pressure and temperature would correct this deviation, as it would affect the values for the substances having very low critical constants more than it would those having higher critical constants. By trial and error it was found that, when T,mas expressed in OK. and Po in atmospheres, if the constant 8 n-ere added t o each, the desired correlation was obtained. The same method was applied to neon, the only other gas having very low critical constants and the agreement with the standard

Vol. 21, No. 3

y

ACTIVITY

COEFFICIENT Pr Calcd. Ohsvd. n-Hexane ( T c = 507.9' K.; Pc 29.5 atrn.) 0.873 0.02 0.982 0.985 0.873 0.30 0.829 0.831 0.932 0.05 0.973 0,971 0.932 0.50 0.761 0.760

-

TI

-

Acetylene (Tc= 309O K . ; P e 62 atrn.) 0.02 0.984 0.984 0.15 0.910 0.921

0.883 0.883

-

Ethyl ether (Te = 466.9O K.; Pc 35.5 atm ) 0.02 0.10 0.30 0.05 0.15 0.50 1.0 3.0 2.0 6.0 Nitric oxide ( T c = 179' K.; P c = 65 atm.) 1.082 1.0 0.71 0.79

t::

:::; 0":;

2.0

0.88

;:::: ::: t.;: 1.575

!:;

l:; ::;: 8:;:

2.44 2.44

1.0 2.0

-

0.85

Krypton (Tc= 210" K.; P c 54 atrn.) 1.00 1.00

0.995 0.995

Isopentane ( T c = 460.9' K.; P o = 32.9 atm.) 0.02 0.658 0.960 0.962 0.02 0.81 0.981 0.981 0.15 0.81 0.880 0.87 1.025 1.00 0.64 0.66 1,025 0.43 2.0 0.42 1.0 1.20 0.815 0.81 1.20 2.0 0.66 0.66

COEFFICIENT ACTIVITY

Tr PT Calod. Obavd. Cyclohexane (Tc = 554.1' K.; PC = 40.6 atm.) 0.855 0.30 0.815 0.83 0.983 0.20 0.93 0.93 0.983 0.70 0.72 0.74 1.035 1.0 0.67 0.71 1.035 2.0 0.44 0.43 Methanol ( T e = 513O K.; 78.7 atm.) 0.847 0.05 0.961 0.847 0.15 0.896 0.923 0.10 0,958 0.923 0.30 0,850 1.00 0.50 0,825 1.00 1.0 0.61

0.961 0.892 0.953 0.846 0.89 0.69

Ethanol (Tc = 516.2' K.; 63.1 atrn.) 0.972 0.05 0.30 0.846 0.05 0.980 0.30 0.881 0.60 0.806

0.980 0.880 0,990 0.948 0.890

0.918 0.918 0.986 0.986 0.986

Po

Pc

=

-

curve8 was improved. Thus for helium, hydrogen, and neon an arbitrary function of the reduced wressure and reduced temperature is to be used instead of Trand Pr; these are defined as follows: -

T L

+8 and P , = Pc + 8 ?'"

To

n r

The agreement of the values of the activity coefficients for these substances is shown in Table 111.

TABLE111. AGREEMENTOF CALCULATED AND OBSERVED HAVINQVERY Low CRITICAL VALUESOF y FOR SUBSTANCES CONSTANTS Tr 3.91 17.3 23.7 39.0 39.0 39.0 71.4 71.4 91.0 91.0 2.04 2.04 3.90 10.56

Pr

-

Neon (Tc 1.73 1.73 3.31 8.95

Pn

ACTIVITY COEFFICIENT Calcd. Obavd. 0.910 1.135 1.10 1.07 1.24 2.03 1.03 1.32 1.04 1.33

44.4O K.: P C = 25.9 atm.) 0.92 2.61 2.0 0.89 3.92 3.0 3.92 3.0 1.05 3.0 1.03 3.92

Hvdroaen ( T e = 33.2' K.: Pe 12.8 atm.) 1.41 3.25 2.0 0.83 1.75 2.19 0.985 3.25 2.0 2.71 2.19 4.0 0.985 6.50 2.71 9.0 1.13 8.25 6.65 14.6 2.05 60.0 6.65 8.25 81.2 1.07 14.6 9.0 14.2 17.6 1.455 50.0 81.2 14.2 17.6 1.05 23.4 18.8 14.6 9.0 1.33 50.0 18.8 81.2 23.4 a The calculation of this value is doubtful since the a valuei a smooth curve.

0.94 0.92 1.05 1.03 0.835 0.98 0.975 1.13 2.05 1.07 1,455 1.05 1.33 do not give

From the "average" curves drawn as described, another set was prepared. The values of y a t even values of T, were plotted against P,. These are given in Figure 1 for the region up to the critical temperature, in Figure 2 for the region between T , of 1.0 to 3.5, and in Figure 3 for the higher region. For any reduced pressure, the function y shows a

March, 1935

INDUSTRIAL AND ENGINEERING

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5

10

CHEMISTRY

305

15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 REDUCED PPESSURE -6

FIGURE3. ACTIVITY COEFFICIENT

OF

G4SES IN

maximum near a reduced temperature of 3.5. These curves were drawn through the values obtained from the first set and no further smoothing was necessary. DISCUSSION OF RESULTS From the nature o f this graphical method for calculating the activity coefficient, it is seen that accurate, low-pressure P-V-T data are necessary in certain ranges of Tr. For values of T,greater than 2.5, the curves of a vs. P a r e nearly straight lines, and the extrapolation from the critical pressure down to zero pressure is relatively certain. For values of T, below 2.5, however, these curves show maxima near the critical pressure, and it is necessary that accurate data be available to plot the curve below this point. This may introduce a large uncertainty, because frequently the area of the curve up to the critical pressure is equal to the whole area up to a P r of 8 or 10. The lack of complete agreement in the cases of dichlorodifluoromethane, methyl chloride, ethyl ether, nitric oxide, and krypton may be attributed to lack of data from which to calculate a! in the low-pressure range. For these substances no data were available below a pressure of about 0.5.

THE

HIGHTEMPERATURE RANGE

In order to present the activity coefficient for oxygen and nitrogen over a wider range than the data of the experimental P-V-T tables covered, an attempt was made to calculate y from the P-V-T tables of Millar and Sullivan (25) in a region for which the values were extrapolated by means of an equation of state. The values of y thus calculated do not agree well with the standard curves and hence are not included. As stated by Deming and Shupe (7) “from inspection of the curves of cy vs. P, it was decided that it was expecting too much of any equation of state to faithfully reproduce trends in these curves.” This is particularly true below the critical pressure a t temperatures near and below the critical. This brings out the danger of the use of an equation of state of imperfect accuracy for calculation of such functions as y, because in the low-pressure region the volume is large and a very small percentage error in the volume will cause a very large error in a because it is the difference between two large volumes. The P-V-T data for water vapor were taken from the Keenan Steam Tables (IS) because it was thought that they represented the best critical summary of a large amount of scattered data.

Itviil.; 40; 848 (1932). Fiake. Urbana. Ill. (1925). Gillcsiiie, J . Am. Chwn. Soc.. 48, 28 (1926). Hoist. Bull. ((swc. intern. froid, 6, 48 (1915). (13) Keonim. "Steam Tkblee and MoIlier Diarram." Am. SOC. Mech. . ;Sj (10) (11) (12)

Eng., 1930.

Keyes. J . Am. Chem. Soc.. 53, 965 (1931). Keyes and Burka. Ibid., 49, 1403 (1827). Keves. Smith. and Joubcrt. J . M d h . Phus.. . . Mass. I w t . Tech.. 1. i91 '(1922j.' Kuenen, Amh. ngcrland., 76,354 (1893). Kvdnes and Geddy. 3. Am. Chem. Soc., 53, 394 (1931). Lcwis and Kav. ".O i l Gaa 3... 32.. No.45. 40 (1934). Lewis and Luke. IND.ENU.C n ~ x .25, , 725 (1933). Lewis and Luke. Trans. Am. Soc. A4ech. Engre., 54. 55 (l93W. Lewia and Randall, "Thermodynamics." New York. MeGrawllill Book C o , 1923. Levden Laboratory. unwblished data, Mdyerr and J c s s u ~ , ~ R & Enq.. ~ Q ~ 11, T ~345 ~ ~(1925). Millar and Sullivan. Bm. Mines, Cere. 424 (1928). Nsgornov w d Rotinjens. Ann. inst. anal. phw. chim. (Leningmd), 2, 371 (1924); Z. physik. Chem., A169, 20 (1934). Otto and Rolborn, 2. Physib. 33, 1 (19261, Rairmsay and Trslvers, Proc. Roy. Soc. (London), A197.47 (1901). Rarnsay and Y o u r , Ihid.. A177, 123 (1886). fhid., A178, 314 (1887). Hoessler and Hasdadher Go.. "Artio-the Refrigerant:' 4th. ed., pub. by E. I. du Pmt de Nemours & Co ,1932. Snmeshirna, Bull. Chem. Soc. Japon, 1, 41 (1926). Selheimer, Souders, Smith. and Brown, IND. ENG.CEEM.,24.

Wiebe. Gaddy, and €bins, J . Am. CBenr. Sac.. 53, 1721 (1931). Young, Z. phgeik. Chrm., 29, 193 (1890). RBCEZYBU September lS, 1034.