Page 290 - Physical Chemistry

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where H, S, and G are properties of the solution and H*, S*, and G* are properties of Section 9.3
the pure unmixed components at the same T and P as the solution. Mixing Quantities
The key mixing quantity is mix G G G*. The Gibbs energy G of the solution

is given by Eq. (9.23) as G n G i (where G i is a partial molar quantity). The Gibbs
i
i
energy G* of the unmixed components is G* i n G* (where G* is the molar
m,i
m,i
i
Gibbs energy of pure substance i). Therefore

¢ mix G G G* a n 1G G* 2 const. T, P (9.32)
i
m,i
i
i
which is similar to (9.17) for mix V. We have
¢ mix G ¢ mix H T ¢ mix S const. T, P (9.33)
which is a special case of ¢G ¢H T ¢S at constant T.

S
Just as and V ii can be found as partial derivatives of G i [Eqs. (9.30) and (9.31)],
¢ mix S and ¢ mix V can be found as partial derivatives of ¢ mix G. Taking 10>0P2 T,n j of
(9.32), we have

0¢ G 0 0G i 0G*
m,i
mix
a b a n 1G G* 2 a n ca b a b d
i
i
i
m,i
0P 0P 0P 0P
T,n j i i T,n j T

a n 1V V * 2
m,i
i
i
i
0¢ G
mix
a b ¢ V (9.34)
mix
0P
T,n j
where (9.31), (4.51), and (9.17) were used.
Similarly, taking 10>0T2 of (9.32), one finds (Prob. 9.21)
P,n j
0¢ G
mix
a b ¢ mix S (9.35)
0T
P,n j
The partial molar relations and mixing relations of the last section and this one are
easily written down, since they resemble equations involving G. Thus, (9.28) and
(9.33) resemble G H TS, (9.30) and (9.35) resemble ( G/ T) S [Eq. (4.51)],
P
and (9.31) and (9.34) resemble ( G/ P) V [Eq. (4.51)].
T
The changes V, U, H, and C that accompany solution formation
mix mix mix mix P
are due entirely to changes in intermolecular interactions (both energetic and struc-
G/n
tural). However, changes in S, A, and G result not only from changes in intermolecu- mix
lar interactions but also from the unavoidable increase in entropy that accompanies the
constant-T-and-P mixing of substances and the simultaneous increase in volume each
component occupies. Even if the intermolecular interactions in the solution are the T mix S/n
same as in the pure substances, S and G will still be nonzero.
mix mix
It might be thought that S at constant T and P will always be positive, since a
mix H/n
solution seems intuitively to be more disordered than the separated pure components. mix
It is true that the contribution of the volume increase of each component to S is
mix
always positive. However, the contribution of changing intermolecular interactions
can be either positive or negative and sometimes is sufficiently negative to outweigh
the contribution of the volume increases. For example, for mixing 0.5 mol H O and Figure 9.5
2
0.5 mol (C H ) NH at 49°C and 1 atm, experiment gives S 8.8 J/K. This can
2 5 2 mix Thermodynamic mixing quantities
be ascribed to stronger hydrogen bonding between the amine and water than the aver-
for solutions of water
age of the hydrogen-bond strengths in the pure components. The mixing here is highly diethylamine at 49°C and 1 atm.
exothermic, so that S is larger than S , S is positive, and G H Note that S is negative. n is
surr syst univ mix mix mix
T S is negative (Fig. 9.5). the total number of moles.
mix