Generalized Equation for Activity Coefficients of Gases SAMUEL 1-1. MARON AND DAVID TURNBULL Case School of Applied Science, Cleveland, Ohio
to pressures of 300400 atmospheres a t reduced temperatures above 1.25. The expression obtained from Equation 1 for loglo y, where y is the activity coefficient of the gas, was found to be:
A general equation for all gases is derived, giving the activity coefficient of the gas as a function of the reduced temperature, T,, and reduced pressure, P,, only. Activity coefficients calculated from this equation are compared with activity coefficients read from Newton's generalized curves at various reduced temperatures and pressures. The agreement between the calculated activity coefficients and those read from Newton's curves is good at values of T , = 1.3 and above and at values of P , = 12 and below. At values of T , below 1.3 the agreement is good only at low values of P,.
log10 y = k P
+ ZP2 + mP3
(2)
where k , 1, and m were functions of the temperature only, and were defined by the expressions:
P =
2.303 ( R T ) 2 YB
= 2 X 2.303(RT)S 6 m =
3 X 2.303(RT)'
(24
(2B)
(2C)
The coefficientsp, y B ,and 6 are in turn given by the equations:
p EWTON (5) showed that if the experimentally determined activity coefficients of various gases are plotted against the reduced pressure, P,, then a smooth average curve can be drawn through all the points a t any given reduced temperature, T,. He showed, further, that the activity coefficients of the twenty-three gases studied agreed, with a few exceptions, to within 2 per cent with the values read from these averaged curves. The whole correlation was empirical and graphical in nature, and no attempt was made to deduce an equation which would reproduce the variation of the activity coefficients of these gases with T , and P,. The significance of Newton's correlation lies in the fact that i t shows the activity coefficients of the gases studied t o be the same function of T, and P,, and further, that a t any given values of T , and P, the activity coefficients of all gases should be the same. If this is so, then it should be possible to deduce the functional relation between the activity coefficients of gases and T , and P, as soon as we have an equation relating the dependence of the activity coefficients on both temperature and pressure. The purpose of this paper is to show how such a generalized equation may be deduced from the Beattie-Bridgeman equation of state, and to compare the activity coefficients calculated from the derived equation with the activity coefficients read from Newton's generalized curves.
YB
N
= RTBo
- Ao - RC
- RTBob +
A& - RBoc 7 T
a = - - RBobc T2
(3-4) (3W
(3C)
where A,,, Bo,c, a, and b are the Beattie-Bridgeman constants for a given gas and are independent of pressure and temperature. If Equations 3A, 3B, and 3C are substituted for p, y B ,and 6 in Equations 2A, 2B, and 2C, respectively, and the expressions for L, I , and m thus obtained are substituted into Equation 2, we arrive a t the expression for log,, y :
[m
log'' [2
x
BQ - 2.303A.( R T ) 2- 2.303 RT4] P + -Bob A oa BOC 2.303 ( E T ) *+ 2 x 2.303 (RT)s 2 x 2 . 3 0 3 ~ " ~ 1 Bobc [3 X 2.303 R3T6
However,
P = P,P, T = T,T,
where P, and Toare the critical pressure and temperature, respectively, while P, and T , are the reduced pressure and temperature corresponding to P and T . Substituting the expressions for P and T from Equations 5A and 5B into Equation 4, we have:
Derivation of Equation Recently the authors (4) showed that an approximate form of the Beattie-Bridgeman equation (I)-namely,
(2
where p, y B ,and 6 are functions of the temperature onlymay be used to calculate the activity coefficients of various gases, for which Beattie-Bridgeman constants are known, up 246
x
A~xIP~~ 1 2.303(RTJ3)
INDUSTRIAL AND ENGINEERING CHEMISTRY
February, 1941
Letting now BoPc
A =2.303RTC
B =
c
(64
-AoPc
(6B)
2.303(RTC)*
=---
-CP,
(6C)
2.303RTe4 -BobPo$ D = 2 X 2.303(RTJP AoaP,2 E = 2 X 2.303(RTJ8
(6D)
(6E)
BocP.2 F = 2 X-2.303ReTo6 B = 3 X BobcPc8 2.303R8T.6
(6F) (6G)
Equation 6 becomes: loglo 7
e
D E F ($ + + &) + (n + + p)Pi* + (&) (7) pr
pr8
Equation 7 gives the variation of loglo y of a gas with reduced temperature and pressure. The coefficients A , B, C, D, E, F , and G are constants independent of T, and P,, and appear to depend only on the nature of the gas in question. Since, however, according to Newton's generalization the activity coefficients of all gases must be the same a t any given value of T, and P,, it must follow, if this generalization is correct, that an equation for logloy for any one gas in terms of T, and Pr,must be obeyed also by any other gas. Hence the constants A , B, C, D, E, F, and G, once evaluated for some one gas, should be applicable also to any other gas, and consequently Equation 7 should represent the variation of logloy for any gas with T, and P,; i. e., Equation 7 should be a generalized equation for the activity coefficients of all gases.
247
Y~ as read from Newton's curves. These comparisons are summarized for values of T , = 1.00 and below in Table I, and for values of T , = 1.2 and above in Table 11. At T, = 1.00, ycalod. agrees with y N within 5 per cent for P, = 0.6 and below, the agreement becoming better a t lower pressures. In all other cases for T, less than unity, the deviation is no greater than 5 per cent up to the highest value of P, read from Newton's curves, with the exception of T, = 0.95, where the deviation becomes greater than 5 per cent a t P, = 0.5 or higher. Table I1 shows that a t T,= 1.2 the deviation of ycalod. from y N up to P, = 13 is within 7 per cent except for two cases, where the deviation is 9 per cent. These rather large deviations a t this low reduced temperature are not surprising in view of the fact that few equations of state agree well with experimental data a t temperatures near the critical. Moreover, in obtaining quation 1 the actual volume was assumed equal to the i ea1 volume in the higher order terms (I), and this assumption should become more serious, the lower the value of T,. At T, = 1.3 and above, the agreement is much better. A t T , = 1.3 the deviation is no greater than 4 per cent up to P, = 13. At T , = 1.5 the deviation is no greater than 3 per cent up to P, = 12, and with two exceptions it is no larger than 2 per cent a t values below 12. At T , = 3.5 the deviation is 4 per cent a t P, = 13.0 and less than 1 per cent a t P, = 10 and below. At the highest temperature used (TI= 10) which is far outside of the temperature range in which the Beattie-Bridgeman constants were evaluated, the agreement is still very good, the deviation being less than 4 per cent a t Pr = 15. From these results we may conclude that Equation 8, which gives logto y of any gas in terms of P, and T,only, reproduces values of the activity coefficients in good agreement with the values obtained from Newton's average curves for values of T, = 1.3 or greater and values of P, = 12 or less. At values of T, = 1.5 and higher, the agreemenbis within 3 per cent or less up to P, = 12. However, a t values of T, = 1.2 or lower, the agreement is not so good but improves as the value of P, is reduced. For T , below unity the deviation is in no case greater than 5 per cent a t values of P, = 0.5 or below,
?
Test of Generalized Equation The authors selected nitrogen as a reference gas to evaluate the constants A, B, C, D, E, F,and G. This gas was chosen because Deming and Shupe (2) showed that its Beattie-Bridgeman constants, as evaluated by them, reproduce the compressibility data with considerable accuracy for pressures up to 1000 OF CALCULATED AND NEWTON'S ACTIVITY TABLEI. COMPARISON COEFFICIENTS AT Low T , atmospheres, over a temperature range of -50' to 600' C. The Beattie-Bridgeman constants Tr = 0.70 Tr = 0.80 TI sa 0.90 Tr = 0.95 Tr 1.00 for nitrogen as determined by Deming and Shupe Pr Yaalad. YN Yoalod. YN Yadod. YN Yodcd. YN Yodod. YN 0.05 0.95 0.93 for pressures in atmospheres and volume in liters o.lo ,, .. 0:94 0192 0195 0194 0:96 0195 0:67 0:96 per mole are: A0 = 1.2539, BO = 0.04603, c = 0.20 .. . . 0.87 0.84 0.91 0.88 0 . 9 2 0.90 0.93 0.92 0.30 .. .. . . . . 0.87 0.84 0.89 0 . 8 6 0.90 0.89 6.16 x 104, a = 0.01868, and b = -0.02588. 0.40 .. .. 0.83 0.79 0.85 0.82 0.87 0.85 0.50 .. .. . . 0.82 0.78 0.85 0.82 Using these constants and Pc = 33.5 atmos0.60 .. .. .. .. 0.79 0 . 7 4 0.82 0.78 pheres and T,,= 126" K. as given by Interna0.70 ., .. ,. .. *. .. 0.76 0.69 0.79 0.74 0.80 .. .. .. .. .. 0.77 0.70 tional Critical Tables (S), the constants in 0.90 ,. .. .. .. .. 0.74 0.66 Equation 7 were evaluated by Equations 6A to 1.00 .. .. .. .. .. . . 0.72 0.61 6G and substituted in Equation 7. The following expression was obtained: TABLE11. COMPARISON OF CALCULATED AND NEWTON'S ACTIVITY COEFFICIENTS AT T,ABOM 1.00 Tr -10.0 Tv = 1.2 Tr 1.3 TI = 1.5 Tr 3.5
-
log10 Y =
(
T,
(0.000$669)
p,3
In order to test Equation 8, activity coefficients were calculated a t various values of P, for T , between 0.7 and 10, and the calculated activity coefficients, yodod.; were compared with
-
Pr 1.0 2.0 3.0 4.0 5.0 6.0
yoalad.
YN
7oslod.
YN
0.83 0.70 0.61
0.82
0.87 0.76 0.67
8.0
0.41 0.41
0.87 0.77 0.69 0.62 0.58 0.55 0.53 0.52 0.52
7.0
9.0 10.0 12.0 13.0 15.0
0.67
0.54
0.56 0.50
0.43
0.44 0.43
0.49 0.45
0.41 0.44
0.47 0.56
0.46
0.43 0.43 0.44 0.46 0.47 0.51
0.52 0.57 0.60
..
0.60 0.56 0.54 0.53 0.53 0.54 0.54 0.56 0.58
..
-
Ycslad.
YN
Yaelod.
yN
yodod.
YN
0.91
0.92 0.86 0.81
1.01
1.01
1.01
1.11 1.14 1.17
1.03 1.05 1.06 1.07 1.08
1.03
0.71 0.71 0.72 0.73 0.76
1.01 1.02 1.04 1.06 1.08 1.10 1.12 1.14 1,16 1,18 I .23 1.25
1.04 1.05 1.06 1.07
1.11 1.12 1.15 1.16
1.09 1.10 1.12 1.14 1.16
0.85 0.80 0.76 0.73 0.71 0.70 0.69
0.70 0.72 0.78 0.82 0.94
1.02
0.76 0.74 0.72
1.04 1.05 1.07 1.09
0.78
1.19 1.26 1.30
0.82
..
..
1.02
1.02
1.10 1.08 1.20
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
248
Nomenclature a, b, c, Ao,Bo =
Beattie-Bridgeman constants, for P in atmospheres and Ti in liters/mole IC, E, m = coefficients in Equation 2, dependent only on temperature for a given gas A , B , C, D,E , P , G = coefficients in Equaiion 6, independent of T , and P , P = pressure, atmospheres P , = critical pressure P, = reduced pressure R = molar gas conEtant, liter-atmospheres/’ K. T = temperature, K. T o = critical temperature T, = reduced temperature V = volume, liters/mole
p,
Vol. 33, No. 2
virial coefficients of Beattie-Bridgeman equation which are functions of T only for a given gas y = activity coefficient of a gas = fugacity/pressure Yoalcd. = calculated activity coefficients y . ~ = activity coefficients read from Newton’s curves YB, 6 =
Literature Cited (1) Beattie, Proc. N a t l . Acad. Sci., 16, 14 (1930). (2) Deming and Shupe, J . A m . Chem. Soc., 52, 1382 (1930). (3) International Critical Tables, Vol. 111, p. 248, New York, Mc-
Graw-Hill Book Co., 1928. (4) Maron and Turnbull, IND.EKG.CHEM.,33, 69 (1941). (5) Newton, Ibid., 27, 302 (1035).
PRESENTED before t h e Division of Industrial and Engineering Chemistry a t the 100th Meeting of the American Chemical Society, Detroit, hIich.
NOMOGRAPH FOR EQUIVALENT DIAMETERS OF ANNULI
I
N CONNECTION with the design of double-pipe coolers, calculation of the film coefficient of the fluid flowing in the annular space is made by means of the Dittus-Boelter equation1,
-’
h = 0.0225 ;(D;
)o,a(%)Q.4
intended for turbulent flow inside clean round pipes where D is the inner diameter of the pipe, in feet. I n the case of an annulus D should be the equivalent diameter, defined as 4 times the hydraulic radius, which is the area of the annulus divided by the perimeter of the heating surface. The outer surface of the inner pipe is usually the heating surface and so the equivalent diameter, D , in feet, is given by Di - Df -~ D = 4 lr (Di - 0;) 4 T Di (12) 12 Di
where Dz and D1 are the inner diameter of the outer pipe and the outer diameter of the inner pipe, respectively, in inches. Since standard iron pipe is usually used in consbructing double-pipe coolers, it is convenient to calculate the equivalent diameter in terms of the nominal rather than the actual diameter. The nomograph facilitates this computation, and its use is illustrated by the broken lines. What is the equivalent diameter of the annulus between I- and 2-inch standard iron pipes? Following the key, connect 2 on the D2 scale with 1 on the left-hand D lscale and note the intersection with the a axis. Follow the guide lines t o the p axis and connect the point so found with 1 on the right-hand Dl scale, reading the equivalent diameter as 0.162 foot on the D scale. Dittus and Boeiter, Unzv Calif. Pub. Eng , 2, 443 McAdams, “Heat Tiansmission”, p 169 (1933) and “Elements of Chemical Engineering“, p. 134 (1936),New York, MoGraw-Hill Book Company 1
(1930) ;
D. S. DAVIS Wayne University, Detroit, Mich.
