Thermodynamics and Chemistry

CHAPTER 9 MIXTURES

9.2 PARTIALMOLARQUANTITIES 236

of species and energy between the phases, and that effects of gravity and other external
force fields are negligible.
The system consists of a reference phase,í^0 , and other phases labeled byí§í^0. Species
are labeled by subscripti. Following the procedure of Sec.8.1.1, we write for the total
differential of the internal energy

dUDdUí

0

C

X

í§í^0

dUí

DTí

0

dSí

0

pí

0

dVí

0

C

X

i

í

0

i dn

í^0

i

C

X

í§í^0

TídSípídVíC

X

i

íidníi

!

(9.2.37)

The conditions of isolation are

dUD 0 (constant internal energy) (9.2.38)

dVí

0

C

X

í§í^0

dVíD 0 (no expansion work) (9.2.39)

For each speciesi:

dní

0

i C

X

í§í^0

dníi D 0 (closed system) (9.2.40)

We use these relations to substitute for dU, dVí
0
, and dní
0
i in Eq.9.2.37. After making the
further substitution dSí
0
DdS

P

í§í^0 dS

íand solving for dS, we obtain

dSD

X

í§í^0

Tí

0

Tí

Tí^0

dSí

X

í§í^0

pí

0

pí

Tí^0

dVí

C

X

i

X

í§í^0

í

0

i 

í

i

Tí^0

dníi (9.2.41)

This equation is like Eq.8.1.6on page 194 with provision for more than one species.
In the equilibrium state of the isolated system,Shas the maximum possible value, dS
is equal to zero for an infinitesimal change of any of the independent variables, and the
coefficient of each term on the right side of Eq.9.2.41is zero. We find that in this state each
phase has the same temperature and the same pressure, and for each species the chemical
potential is the same in each phase.
Suppose the system contains a speciesi^0 that is effectively excluded from a particular
phase,í^00. For instance, sucrose molecules dissolved in an aqueous phase are not accommo-
dated in the crystal structure of an ice phase, and a nonpolar substance may be essentially
insoluble in an aqueous phase. We can treat this kind of situation by setting dníi 000 equal to
zero. Consequently there is no equilibrium condition involving the chemical potential of
this species in phaseí^00.
To summarize these conclusions: In an equilibrium state of a multiphase, multicompo-
nent system without internal partitions, the temperature and pressure are uniform throughout