Page 68 - Bruno Linder Elementary Physical Chemistry

August 18, 2010 11:36 9in x 6in b985-ch06 Elementary Physical Chemistry

Phase and Chemical Equilibria 53

The piston on the solution compartment will rise. To prevent that piston
to rise, extra pressure, say Π, must be applied. Osmotic pressure is the
extra pressure that must be applied to the solution to prevent the flow of
solvent.
It turns out that, to good approximation, Π = [B]RT ,where [B] =
n B/V .

∗
Proof. At equilibrium, the chemical potential of the pure solvent is µ ,
A
and of the solvent component A in solution is µ A. These two chemical
potentials differ not only because the concentrations differ, but also because
their pressures are different. For the pure solvent, x A =1 and the
pressure is P. For the solution the mole fraction is x A and the pressure
is P +Π.
More explicitly, what is required is

∗
µ A (P +Π; x A )= µ (P; x A = 1) (6.15)
A
First consider the non-starred µ. Assuming the solution to be sufficiently
dilute, so that Raoult’s Law is applicable, we have

∗
µ A (P +Π; x A )= µ (P +Π)+ RT ln x A (6.16)
A
Next consider the variation of the chemical potential with pressure: dµ =
V mdP,where V m is the molar volume. This is similar to the variation of G
with P,namely, dG = V dP. Assuming that V ∗ is constant, we get

P +Π

∗
µ (P +Π) − µ (P)= V ∗ dP (6.17a)
∗
A
A
P
∗
= V (P +Π − P)= ΠV ∗ (6.17b)
Finally, since µ A (solution) = µ A (pure solvent), so
µ A (P +Π,x A)= µ ∗ A
µ (P +Π)+ RT ln x A = µ (P)
∗
∗
A
A
∗
∗
µ (P)+ ΠV + RT ln x A = µ (P) (6.18)
∗
A
A