ChE 101 2012
Problem Set 6
Read Schmidt, Chapter 7
1.
Derivation of the Langmuir isotherm:
In this problem, we will use the tools of statistical
mechanics to derive the Langmuir adsorption isotherm and understand how it changes with
temperature.
All of the theory we will need was developed in Ph 2B and can be found in
Chapter 5 of
Thermal Physics
by Kittel if you need a refresher.
We will model our heterogenous catalyst as a collection of
N
possible adsorption sites on
the surface of a solid, onto which gas molecules can adsorb. Each site has binding energy
and can accommodate at most one molecule. The adsorbed phase is in equilibrium with a
gas surrounding the solid.
(a) What three intensive properties of the two phases must be equal in order for equilibrium
to be established?
(b) If there are
M
adsorbed molecules, what is the energy
E
(
M
) of the system? What is
the degeneracy Γ(
M
) of the energy, treating molecules as indistinguishable?
(c) In statistical mechanics, any system that is allowed to exchange both energy and particles
with a larger reservoir (in our case the surrounding gas) is called a “grand canonical
ensemble”. All of the thermodynamic properties of such an ensemble are encoded in its
Gibbs sum
Z
, which is an extension of the partition function to systems with fluctuating
particle numbers. Letting
{
s
i
}
denote the set of all microstates of the system, the Gibbs
sum for a system at temperature
T
and chemical potential
μ
can be expressed as follows:
Z
=
X
{
s
i
}
e
β
(
N
i
μ

E
i
)
Here,
N
i
and
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 Winter '11
 ARNOLD
 Gibbs, Langmuir Adsorption Isotherm, Gibbs Sum

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