Page 48 - Basic physical chemistry for the atmospheric sciences

34 Basic physical chemistry

198 x 1()3
In K = 2478.9 - 79.9
P
or,
3
K = 5 . 0 l x 10 4
P
Since KP is so large, the products of the forward reaction are certainly
favored under equilibrium conditions at a temperature of 25°C and
1 atm.

2.8 Chemical potential; homogeneous nucleation
of water-vapor condensation
If a single molecule is removed from a material in a certain phase, with
temperature and pressure remaining constant, the resulting change in
the Gibbs free energy of the material is called the chemical potential
d
(µ.,) of that phase. In other wor s , the chemical potential is the Gibbs
f r ee energy per molecule at constant temperature and pressure.
Exercise 2 . 8. Show that when a plane surface of a liquid is in
equilibr u m with its vapor, the chemical potentials in the liquid and
i
vapor phases are the same.
i
Solution. If a liquid s in equilibrium with its vapor, molecules may
evaporate and condense without the temperature or pressure of the
system changing. Hence, from Eq. (2.35), dg = 0. But, if g does not
change when a molecule passes from the liquid to the vapor phase (or
vice versa), the Gibbs free energy per molecule (i. e . , the chemical
potential) must be the same in the liquid phase as in the vapor phase.
The pressure exerted by the vapor under these equilibrium conditions
is called the saturation vapor pressure; it depends only on the sub­
stance being considered and its temperature.
We will now derive an expression for the difference in the chemical
potentials of a vapor and its liquid at an arbitrary partial pressure e
and temperature T. Applying Eq. (2.40) to one molecule of the vapor,
we obtain
(2.48)

where � is the chemical potential of the vapor molecule at 1 atm, and
k is the Boltzmann constant. Similarly, the chemical potential (µ.,vsai)
for one molecule of the vapor at its saturation vapor pressure e, at
temperature T is given by