Physical Chemistry Third Edition

6.5 Thermodynamic Functions of Nonideal Solutions 277

For convention I the standard states are the pure components, so that

G(unmixed)

∑c

i 1

niμ∗i

∑c

i 1

niμ◦i(I) (6.5-7)

where we neglect a small difference betweenμ∗iandμ◦idue to a possible difference

of the pressure fromP◦. The change in Gibbs energy for producing the solution (the

Gibbs energy change of mixing)is

∆Gmix

∑c

i 1

niμ∗i +RT

∑c

i 1

niln

(

a(iI)

)

−

∑c

i 1

niμ∗i

∆GmixRT

∑c

i 1

niln

(

a(iI)

)

RT

∑c

i 1

niln(xi)+RT

∑c

i 1

niln

(

γ(iI)

)

(6.5-8)

The first sum in the right-hand side of the final version of this equation is the same as

for an ideal solution, and the second sum represents a correction for the nonideality of

the solution. This contribution is called theexcess Gibbs energyand is denoted byGE:

∆Gmix∆G

(ideal)

mix +G

E (6.5-9)

so that

GERT

∑c

i 1

niln

(

γi(I)

)

(6.5-10)

Theexcess entropycan be defined for a nonideal solution:

SE∆Smix−∆Smix(ideal) (6.5-11)

Theexcess enthalpyand theexcess volumeare equal to the mixing quantities, since

∆Hmixand∆Vmixboth vanish for an ideal solution.

Exercise 6.24

a.Show that

SE−R

∑c

i 1

nilnγi(I)−RT

∑c

i 1

ni

⎛

⎝

∂ln

(

γi(I)

)

∂T

⎞

⎠

P,n

(6.5-12)

b.Show that

HE∆Hmix−RT^2

∑c

i 1

ni

⎛

⎝

∂ln

(

γ(iI)

)

∂T

⎞

⎠

P,n

(6.5-13)

c.Show that

VE∆VmixRT

∑c

i 1

ni

⎛

⎝

∂ln

(

γi(I)

)

∂P

⎞

⎠

T,n

(6.5-14)