Thermodynamics and Chemistry

CHAPTER 12 EQUILIBRIUM CONDITIONS IN MULTICOMPONENT SYSTEMS

12.2 SOLVENTCHEMICALPOTENTIALS FROMPHASEEQUILIBRIA 372

’

A(l)

p^0

“

(A + B)(sln)

p^00

Figure 12.2 Apparatus to measure osmotic pressure (schematic). The dashed line

represents a membrane permeable only to the solvent A. The cross-hatched rectangles

represent moveable pistons.

This quantity can be measured calorimetrically at any temperature higher thanTf. Making
this substitution in Eq.12.2.3together with that of Eq.12.2.4, carrying out the integration
of the first integral and rearranging, we obtain finally

A.l; T^0 /A.sln; T^0 /DT^0



Åsol,AH.Tf/TfÅsol,ACp

 1

Tf

1

Tf



CT^0 Åsol,ACpln

Tf

Tf

CT^0

ZT 0

Tf

ÅdilH

T^2

dT (12.2.5)

12.2.2 Osmotic-pressure measurements

A second method for evaluatingAAuses the solution property calledosmotic pressure.
A simple apparatus to measure the osmotic pressure of a binary solution is shown schemat-
ically in Fig.12.2. The system consists of two liquid phases separated by a semipermeable
membrane. Phaseíis pure solvent and phaseìis a solution with the same solvent at the
same temperature. The semipermeable membrane is permeable to the solvent and imper-
meable to the solute.
The presence of the membrane makes this system different from the multiphase, mul-
ticomponent system of Sec.9.2.7, used there to derive conditions for transfer equilibrium.
By a modification of that procedure, we can derive the conditions of equilibrium for the
present system. We take phaseìas the reference phase because it includes both solvent and
solute. In order to prevent expansion work in the isolated system, both pistons shown in the
figure must be fixed in stationary positions. This keeps the volume of each phase constant:
dVí DdVìD 0. Equation9.2.41on page 236 , expressing the total differential of the
entropy in an isolated multiphase, multicomponent system, becomes

dSD

TìTí

Tì

dSíC

ìAíA

Tì

dníA (12.2.6)

In an equilibrium state, the coefficients.TìTí/=Tìand.ìAíA/=Tìmust be zero.
Therefore, in an equilibrium state the temperature is the same in both phases and the solvent
has the same chemical potential in both phases. The presence of the membrane, however,
allows the pressures of the two phases to be unequal in the equilibrium state.