Page 341 - Physical Chemistry

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Chapter 10 in a gas mixture is measured by the fugacity coefficient f (phi i) of gas i. The defi-
i
Nonideal Solutions nition of f is f f /P f /x P, so
i i i i i i
f f P f x P (10.99)*
i i
i
i i
Like f , f is a function of T, P, and the mole fractions. (There is no connection
i i
between the fugacity coefficient f and the osmotic coefficient f of Sec. 10.6.)
i
For an ideal gas mixture, f P and f 1 for each component.
i i i
id
id
Subtraction of m m° RT ln (P /P°) from (10.97) gives ln (f /P ) (m m )/RT or
i i i i i i i
id
f P exp[(m m )/RT]. The standard pressure P° does not appear in this equation, so
i
i
i
i
f and f ( f /P ) are independent of the choice of standard-state pressure P°.
i
i
i
i
The formalism of fugacities and fugacity coefficients is of no value unless the f ’s
i
and f ’s can be found from experimental data. We now show how to do this. Equation
i
(9.31) reads (
m /
P) V . Hence dm V dP at constant T and n , where constant
i T,n j i i i j
n means that all mole numbers including n are held fixed. Equation (10.97) gives
j i
dm RT d ln f at constant T and n . Equating these two expressions for dm , we have
i i j i
RT d ln f V dP and
i i

d ln f 1V >RT 2 dP const. T, n j (10.100)
i
i
Since f f x P [Eq. (10.99)], we have ln f ln f ln x ln P and d ln f
i i i i i i i
d ln f d ln P d ln f (1/P) dP, since x is constant at constant composition.
i i i
Thus (10.100) becomes

d ln f 1V >RT2 dP 11>P2 dP
i
i

f i,2 P2 V i 1
ln a b dP const. T, n j
f i,1 RT P
P 1
where we integrated from state 1 to 2. In the limit P → 0, we have f → 1 [Eq.
1 i,1
(10.98)], and our final result is

i,2 V i 1
P 2
ln f a b dP const. T, n j (10.101)
RT P
0
To determine the fugacity coefficient of gas i in a mixture at temperature T, (total)

pressure P , and a certain composition, we measure the partial molar volume V in the
2 i
mixture as a function of pressure. We then plot /RT V i 1/P versus P and measure the
area under the curve from P 0 to P P . Once f is known, f is known from
2 i,2 i,2
(10.99). Once the fugacities f have been found for a range of T, P, and composition,
i
the chemical potentials m in Eq. (10.97) are known for this range. From the m ’s, we
i i
can calculate all the thermodynamic properties of the mixture.
Pure Nonideal Gas
For the special case of a one-component nonideal gas mixture, that is, a pure nonideal

gas, the partial molar volume V i becomes the molar volume V of the gas, and Eqs.
m
(10.97), (10.98), (10.99), and (10.101) become
m m°1T 2 RT ln 1 f>P°2 (10.102)
f fP, f S P as P S 0 (10.103)
2 P 2 V m 1
ln f a b dP const. T (10.104)
0 RT P
where f is a function of T and P; f f (T, P).