Page 376 - Physical Chemistry

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left and right. Since the membrane is thermally conducting, thermal equilibrium is Section 12.4
maintained: T T T. The chemical potential of A on the left is m*. With equal T Osmotic Pressure
A
R
L
and P in the two liquids, the presence of solute B in the solution on the right makes
m on the right less than m* (Sec. 12.1). Substances flow from high to low chemical
A
A
potential (Sec. 4.7), and we have m* m A,L m A,R . Therefore substance A will flow
A
through the membrane from left (the pure solvent) to right (the solution). The liquid
in the right tube rises, thereby increasing the pressure in the right chamber. We have

( m / P) V A [Eq. (9.31)]. Since V A is positive in a dilute solution (Prob. 9.55),
T
A
the increase in pressure increases m A,R until eventually equilibrium is reached with
m A,R m A,L . Since the membrane is impermeable to B, there is no equilibrium rela-
tion for m . If the membrane were permeable to both A and B, the equilibrium condi-
B
tion would have equal concentrations of B and equal pressures in the two chambers.
Let the equilibrium pressures in the left and right chambers be P and P ß, re-
spectively. We call ß the osmotic pressure. It is the extra pressure that must be ap-
plied to the solution to make m in the solution equal to m* so as to achieve membrane
A
A
equilibrium for species A between the solution and pure A. In the solution, we have
m m* RT ln g x [Eqs. (10.6) and (10.9)], and at equilibrium
A
A
A A
m A,L m A,R (12.20)
m*1P, T 2 m*1P ß, T 2 RT ln g x (12.21)
A
A
A A
where we do not assume an ideally dilute solution. Note that g in (12.21) is the value
A
at P ß of the solution. From dm* dG* S* dT V* dP, we have
m,A
m,A
A
m,A
dm* V* dP at constant T. Integration from P to P ß gives
m,A
A
P ß
m*1P ß, T 2 m*1P, T 2 V* dP¿ const. T (12.22)
m,A
A
A
P
where a prime was added to the dummy integration variable to avoid the use of the
symbol P with two different meanings. Substitution of (12.22) into (12.21) gives
P ß
RT ln g x V* dP¿ const. T (12.23)
A A
m,A
P
V* of a liquid varies very slowly with pressure and can be taken as constant unless
m,A
very high osmotic pressures are involved. The right side of (12.23) then becomes
V* (P ß P) V* ß, and (12.23) becomes RT ln g x V* ß, or
m,A m,A A A m,A
ß 1RT>V* 2 ln g x (12.24)
A A
m,A
For an ideally dilute solution of a solute B that is neither associated nor dissoci-
ated, g is 1 and ln g x x [Eq. (12.11)]. Hence
A A A B
ß 1RT>V* 2x ideally dil. soln. (12.25)
B
m,A
Since the solution is quite dilute, we have x n /(n n ) n /n and
B B A B B A
RT n B
ß ideally dil. soln. (12.26)
V* m,A n A
where n and n are the moles of solvent and solute in the solution that is in mem-
B
A
brane equilibrium with pure solvent A. Since the solution is very dilute, its volume V
is very nearly equal to n V* , and (12.26) becomes ß RTn /V, or
B
A m,A
ß c RT ideally dil. soln. (12.27)*
B
where the molar concentration c equals n /V. Note the formal resemblance to the
B B
equation of state for an ideal gas, P cRT, where c n/V. Equation (12.27), which
is called van’t Hoff’s law, is valid in the limit of infinite dilution.