Physical Chemistry , 1st ed.

different pressure than the column on the right side. The difference in the two

pressures, represented by the difference in column heights, is called the osmotic

pressure,which is given the symbol . Therefore, at equilibrium the left side is

exerting a total pressure Pand the right side is now exerting pressure P.

Therefore, the equality of the two chemical potentials can be written as

(P) °(P) (7.51)

where a capital Pis used to differentiate this variable from the lowercase pused

for gas pressure. The chemical potential of the solution that has a mole frac-

tion of solute,xsolute, is related to the standard chemical potential as given by

equation 7.35, but in slightly different notation:

(P) °(P) RTln xsolute (7.52)

To determine an expression for , we begin with the natural variable expres-

sion for d:

dSdTVdp

At constant temperature:

dVdp

To find , we integrate both sides of the equation from one pressure extreme

to the other. In this case, the pressure extremes are Pand P.We get

P+

P

d

P+

P

Vdp

If we actually perform the integration on the left side of this expression, we get

solvent,solution(P) °solvent,pure(P) 

P+

P

Vdp (7.53)

We have embellished the ’s with subscripts: the side where the total liquid

pressure is Phas the solvent combined with a solute, whereas the side

where the total liquid pressure is Phas the pure solvent (hence the superscript °).

Using equation 7.52 to substitute for (P) in equation 7.51:

°(P) RTln xsolute°(P)

Next, rearrange:

°(P) °(P) RTln xsolute

The left side of this equation is the same as the left side of equation 7.53 (but

without the subscripts). We can substitute and get:

RTln xsolvent,solution

P+

P

Vdp (7.54)

If we assume that the molar volume remains constant between pure solvent and

solution,Vcan be removed from the integral and the answer is straightforward:

RTln xsolvent,solutionV

P+

P

dp

VpPP

V(PP)

RTln xsolvent,solutionV (7.55)

7.8 Colligative Properties 197

Dilute

solution

Pure

solvent

P

P

Semipermeable

membrane

(b)

P  



P

P

Semipermeable

membrane

(a)

P

Figure 7.28 The two-part system is filled with
pure solvent on one side and a dilute solution on
the other. (a) Initially, the liquid levels are even
with each other. However, it is not at equilibrium.
Solvent will pass through the semipermeable
membrane in a preferential direction. (b) At equi-
librium, the two levels are uneven. The difference
between the two levels is defined as the osmotic
pressure .