116 Fugacity and Fugacity Coeficient: Species in Solution
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In addition, from Eq (1 152),
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This equation demonstrates that In
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aiis a partial property with respect to G
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R / ~ ~
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Fugacity Coefficients from the Virial Equation of State
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Values of )i for species i in solution are readily found from equations of state The simplest form of the virial equation provides a useful example Written for a gas mixture it is exactly the same as for a pure species:
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CHAPTER 1 I Solution Thermodynamics: Theory
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The mixture second virial coefficient B is a function of temperature and composition Its exact composition dependence is given by statistical mechanics, and this makes the virial equation preeminent among equations of state where it is applicable, ie, to gases at low to moderate pressures The equation giving this composition dependence is:
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where y represents mole fractions in a gas mixture The indices i and j identify species, and both run over all species present in the mixture The virial coefficient Bij characterizes a bimolecular interaction between molecule i and molecule j , and therefore Bij = B j i The summations account for all possible bimolecular interactions For a binary mixture i = l , 2 and j = l , 2 ; the expansion of Eq (1157) then gives:
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Two types of virial coefficients have appeared: Bll and B22,for which the successive subscripts are the same, and B I 2 ,for which the two subscripts are different The first type is a purespecies virial coefficient; the second is a mixture property, known as a cross coeficient Both are functions of temperature only Expressions such as Eqs (1157) and (1158) relate mixture coefficients to pure-species and cross coefficients They are called mixing rules Equation ( 1158) allows derivation of expressions for In $1 and In $2 for a binary gas mixture that obeys Eq (337) Written for n mol of gas mixture, it becomes:
Differentiation with respect to nl gives:
in Eq (1156) yields:
where the integration is elementary, because B is not a function of pressure All that remains is evaluation of the derivative Equation (1158) for the second virial coefficient may be written:
116 Fugacity and Fugacity CoefJicient: Species in Solution
where Since yi = n i / n ,
-- 2B12 - B l l - B22
nB = ~ ~ B I I n2B22
+ 121122 -al2 n
P In41 = -(&I RT
( 1 159) (1 160)
P In42 = -(B22 RT
Equations (1159) and ( 1 160) are readily extended for application to multicomponent gas mixtures; the general equation is:9
where the dummy indices i and j run over all species, and
Bii - Bkk
&J - 2 B 'J - B 1 - B J J ' = 1
Ski = aik,etc
Sii = 0 , 8kk = 0, etc,
9 ~ C Van Ness and M M Abbott, Classical Thermodynamics of Nonelectrolyte Solutions: With Applications to Phase Equilibria, pp 135-140, McGraw-Hill, New York, 1982
CHAPTER 11 Solution Thermodvnamics: Theory
117 GENERALIZED CORRELATIONS FOR THE FUGACITY COEFFICIENT
The generalized methods developed in Sec 36 for the compressibility factor Z and in Sec 67 for the residual enthalpy and entropy of pure gases are applied here to the fugacity coefficient Equation (1134) is put into generalized form by substitution of the relations,
d P, 1)P, where integration is at constant T,Substitution for Ziby Eq (354) yields: ln4i = (z'
where for simplicity we have dropped subscript i This equation may be written in alternative form: I n 4 = ln@O w l n 4 '
(1163) lnd =
1 n 4 O - - l (Z -1)-
The integrals in these equations may be evaluated numerically or graphically for various values of T, and P, from the data for Z0 and Z' given in Tables El through E4 (App E) Another method, and the one adopted by Lee and Kesler to extend their correlation to fugacity coefficients, is based on an equation of state Since Eq (1163) may also be written,
rather than for their logarithms This is the choice made here, and Tables E13 through E16 present values for these quantities
4 = (4O)(4'>" we have the option of providing correlations for 4' and 4'