Physical Chemistry Third Edition

6.7 Colligative Properties 297

Osmotic Pressure

This colligative property involves the equilibrium of a liquid solution and the pure

liquid solvent on opposite sides of a semipermeable membrane that allows only the

solvent to equilibrate. The equilibrium is achieved by having different pressures in

the two phases. A simple osmometer is shown in Figure 6.26. The left side of this

apparatus contains a solution containing a solute (component 2) dissolved in a solvent

(component 1), and the right side contains pure solvent. The pressure of the solution is

increased above that of the pure solvent by the gravitational (hydrostatic) force on the

solution in the left column.

PressureP

PressureP

PressureP1P

Solution Pure

solvent

Height of column giving hydrostatic pressure

Semipermeable

membrane with

rigid support

Figure 6.26 An Osmometer

(Schematic).

We denote the pressure on the pure solvent byP, and the pressure on the solution by

P+Π. The differenceΠis called theosmotic pressure. At equilibrium, the value of

the chemical potential of the solvent must be the same on both sides of the membrane.

If the solvent obeys Raoult’s law,

μ∗ 1 (T,P)μ 1 (T,P+Π)μ∗ 1 (T,P+Π)+RTln(x 1 ) (6.7-20)

we must abandon our previous approximation that the chemical potential is independent

of the pressure. From Eq. (5.4-10), if the molar volume of the pure liquid is nearly

independent of pressure (a good approximation),

μ∗ 1 (T,P+Π)−μ∗ 1 (T,P)

∫P+Π

P

Vm,1∗ dP≈ΠVm,1∗ (6.7-21)

which gives

ΠVm,1∗ −RTln(1−x 2 )≈RTx 2 (6.7-22)

It is customary to rewrite Eq. (6.7-22) in another form. For a dilute solution of two

components

x 2 

n 2

n 1 +n 2

≈

n 2

n 1

and we can write

Vn 1 V ̄ 1 +n 2 V ̄ 2 ≈n 1 Vm,1∗

Use of these two equations gives

Π

n 2 RT

V

c 2 RT (6.7-23)

wherec 2 is the molar concentration of the solute. Equation (6.7-23) is one of two

equations known as thevan’t Hoff equation. It is remarkably similar to the ideal gas

equation of state.

The van’t Hoff equation is named for
Jacobus Henricus van’t Hoff,
1852–1911, a Dutch physical chemist
who won the 1901 Nobel Prize in
chemistry for his work on osmotic
pressure. He was also the first person
to propose the tetrahedral carbon atom. EXAMPLE6.20
The osmotic pressure of a moderately dilute solution can be surprisingly large. Find the
osmotic pressure at 25◦C of an aqueous solution of sucrose with a molar concentration of
0.200 mol L−^1.