Problems and Solutions on Thermodynamics and Statistical Mechanics

324 Problem8 d Solutions on Thermodynamics d Statisticd Mechanks

chemical potential p (determined by the pressure p and the temperature
T). Assuming that the adsorbed molecule has energy -EO compared to one
in a free state.

(a) Find the grand canonical partition function (sometimes called the

(b) calculate the covering ratio 6, i.e., the ratio of adsorbed molecules

grand sum) and

to adsorbing sites on the surface.

[A useful relation is (1 + z)N = CN!~”/N~!(N - N~)!J.
Ni
(UC, Berkeley)

Solution:
(a) With Nl molecules adsorbed on the surface, there are
N!
cN”l = N1!(N- Nl)!

different configurations. The grand partition function is therefore

N

= C c”, , eNl(~+c~)/kT = (1 + e(P+cO)/kT 1. N

N1=O

• a
  aa^11

(b) N = __ lIlz = Ne(P+cO)/kT/(l + e(M+cO)/kT

P

kT

where a = --, so that the covering ratio is

• 6=-= N^1
  N l+e-(P+ao)/kT ’

The chemical potential of the adsorbed molecules is equal to that of gas
molecules. For an ideal gas,

Hence

1

6=

-6olkT

P