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Section 12.4
3
where the volume unit in R was converted to dm to match that in c . Osmotic Pressure
B
Alternatively, Eq. (12.25) or (12.26) can be used to find ß. These equations give
answers very close to that found from (12.27).
Exercise
At 25°C, a solution prepared by dissolving 82.7 mg of a nonelectrolyte in water
and diluting to 100.0 mL has an osmotic pressure of 83.2 torr. Find the molecu-
lar weight of the nonelectrolyte. (Answer: 185.)

Note the substantial value of ß for the very dilute 0.01 mol/kg glucose solution
in this example. Since the density of water is 1/13.6 times that of mercury, an osmotic
pressure of 186 torr (186 mmHg) corresponds to a height of 18.6 cm 13.6 250
cm 2.5 m 8.2 ft of liquid in the right-hand tube in Fig. 12.5. In contrast, an aque-
ous 0.01 mol/kg solution will show a freezing-point depression of only 0.02 K. The
large value of ß results from the fact (noted many times previously) that the chemical
potential of a component of a condensed phase is rather insensitive to pressure. Hence
it takes a large value of ß to change the chemical potential of A in the solution so that
it equals the chemical potential of pure A at pressure P.
The mechanism of osmotic flow is not the business of thermodynamics, but we
shall mention three commonly cited mechanisms: (1) The size of the membrane’s
pores may allow small solvent molecules to pass through but not allow large solute
molecules to pass. (2) The volatile solvent may vaporize into the pores of the mem-
brane and condense out on the other side, but the nonvolatile solute does not do so.
(3) The solvent may dissolve in the membrane.

Polymer Molecular Weights
The substantial value of ß given by dilute solutions makes osmotic-pressure mea-
surements valuable in finding molecular weights of high-molecular-weight substances
such as polymers. For such substances, the freezing-point depression is too small to
4
be useful. For example, if M 10 g/mol, a solution of 1.0 g of B in 100 g of water
B
has T 0.002°C and has ß 19 torr at 25°C.
f
Polymer solutions show large deviations from ideally dilute behavior even at very
low molalities. The large size of the molecules causes substantial solute–solute inter-
actions in dilute polymer solutions, so it is essential to measure ß at several dilute
concentrations and extrapolate to infinite dilution to find the polymer’s true molecular
weight. ß is given by the McMillan–Mayer expression (12.28). In dilute solutions
it is often adequate to terminate the series after the A term. Thus, ß/RT r /M
2
B
B
2
A r , or
2 B
ß>r RT>M RTA r dil. soln. (12.29)
B
B
2 B
A plot of ß/r versus r gives a straight line with intercept RT/M at r 0. In some
B
B
B
B
cases, the A term is not negligible in dilute solutions (see Prob. 12.22).
3
A synthetic polymer usually consists of molecules of varying chain length, since
chain termination in a polymerization reaction is a random process. We now find
the expression for the apparent molecular weight of such a solute as determined by
osmotic-pressure measurements. If there are several solute species in the solution, the
solvent mole fraction x equals 1
i A x , where the sum goes over the various
i
A
solute species. Use of the Taylor series (8.36) gives
1
ln x ln a 1 a x b a x a n i (12.30)
A
i
i
i A i A n A i A