Page 144 - Modern physical chemistry

6.17 Equilibria among Phases 735

For the final step, n B is the number of moles of B, n A the number of moles of A, in the
solution. If we also set
nAV A =V' when [6.92]

formula (6.90) reduces to
V'fJJ=RT. [6.93]
Interestingly, this has the form of the ideal gas law. Here V' is the volume of solvent
containing one mole solute, fJJ is the osmotic pressure, R the gas constant, and T the
temperature.

Example 6.8

Calculate the osmotic pressure for a 0.100 m sucrose solution at 30° C.
The weight of solvent containing one mole solute is

W = 1000 g kg- l = 10 000 g mo}"l
0.100 mol kg-l ' ,
whence the corresponding solvent volume is

l
V'= 10,000 gmol- =10 Imo}"l.
1000 g 1-1
Solve (6.93) for fJJ. Then substitute in the temperature and the solvent volume:
RT (0.082061 atm K- mOl-lX303.2 K)
l
fJJ = - = = 2.49 atm
V' 10 I mol- l
If the pressure on the pure solvent is 1.00 atm, then a pressure of 3.49 atm must be exerted
on the 0.100 m sucrose solution to stop osmosis of water from a reservoir of pure water
into the solution.

6. 17 Equilibria among Phases
Let us now summarize the conditions governing the equilibria among differing homo-
geneous mixtures. We assume that the entire system is free from outside influences, par-
ticularly heat and work So inequality (5.114) applies. In the inunediate neighborhood of
equilibrium, the system exhibits a given entropy, volume, and energy.
For simplicity, consider two uniform, composite phases separated by an interface. Start-
ing from equilibrium, let us introduce a small fluctuation. During this, the internal energy
of each phase changes following formula (5.94). But for bookkeeping purposes, we add
the numerical superscripts (1) and (2) to the symbols to identify the pertinent phase.
For the energy change in an infinitesimal fluctuation, we now have

However, the total moles of constituent i are conserved:

[6.95]

Also, the total volume is constant,
[6.96]