TD5 Statistical Physics (M1)
Sublimation of a Solid and Surface Adsorption
Exercise 1: Sublimation.
In this exercise, we will study the sublimation of a crystalline solid, with a simple microscopic model.
In a box with volume V, and temperature T, coexist a crystalline solid with its monoatomic gas. The
mass of the atoms is m. They have no spin. The volume of the solid is negligible in comparison with
the volume occupied by the gas. The vapor is considered as a classical perfect gas.
The total number of atoms is N, with Ng atoms in the gas phase, and Ns=N-Ng atoms in the solid
phase.
1) Gas. Give the expression of the partition function Zg(Ng) of the gas, its free energy, its
pressure, its chemical potential, its entropy.
2) Solid. In the solid, the atoms are positioned at the nodes of a crystalline lattice. The atoms
are supposed to be independent from each other and behave as a 3D harmonic oscillator, as
in the Einstein’s model. The energy of each atom is thus given by three quantum numbers nx,
ny and nz:
n
x ,n y ,n z
3

  o    n x  n y  n z  
2

where o and  are characteristic constants of the crystal.
2-a) What do represent o and  ? Give an order of magnitude for these constants.
2-b) Give the expression of the partition function Zs(Ns) of the solid, its free energy and its
chemical potential.
3) Equilibrium condition between solid and vapor.
The number of atoms in the gas phase, Ng, can take a priori every value between 0 and N. In
this question, we will determine its value at a given temperature.
3-a) Give the partition function of the total system (solid + gas) as a function of Zg and Zs.
3-b) Give the probability P(Ng=Ngo), where Ngo is a given value of Ng, as a function of Zg and
Zs.
3-c) Check that the most probable value Ngm of Ng is given by the equality of the chemical
potentials of the gas phase and of the solid phase.
3-d) At a given temperature, how does Ngm depend on the volume V? At a given N and T,
prove that the solid-vapor equilibrium is possible only if V<Vo. Determine the volume Vo.
What happens if V>Vo?
We will now consider only the case where V<Vo.
4) Equilibrium pressure.
4-a) Compute the pressure for a given volume V<Vo and at a given temperature T. Prove that
it is proportional to the average number <Ng> of atoms in the gas phase.
4-b) Considering that for a large system <Ng>≈Ngm, prove that the pressure P depends only
on T, and on parameters characteristics of the solid phase, but is independent on V.
Exercise 2: Adsorption of a gas on a solid surface.
We consider in this exercise a gas in contact with a solid surface containing M sites that can get gas
molecules. The gas quantity is sufficiently large for the gas not to be disturbed by adsorption of some
molecules on the solid surface.
I.
We consider first that each of the M sites can get only one molecule at maximum.
The energy of each site is =0 when it is empty, and =-1 (1>0) when it gets one molecule.
The distance between the sites is supposed to be sufficiently large for the adsorbed molecules not to
interact. When n molecules are adsorbed, the energy of adsorption is thus E=-n1.
1) Chemical potential of the gas. The gas is a classical perfect gas. The mass of the
molecules in the gas, is m. The molecules have no internal degree of freedom. They are
confined in a volume V.
1-a) Compute the grand-canonical partition function, and the grand-potential of the gas.
1-b) Find the equation of state PV=<N>kBT
1-c) Prove that the chemical potential of a molecule in the gas can be written:
   o T   k B T ln P / Po 
P is the pressure of the gas, Po is a reference pressure, for example Po=1 bar, o(T) is
called “standard chemical potential”.  o T   k B T. ln f T  with f(T) a function of
temperature T.
2) Solid. What imposes the equilibrium condition for the chemical potential ’ of the
molecules adsorbed at the solid surface?
3) Adsorption sites. Write first the grand-canonical partition function for a given site on the
solid surface, and then the grand-canonical function for the M adsorption sites. What is
the corresponding grand-potential?
4) Coverage ratio. Compute the average number <n> of adsorbed molecules and the
coverage ratio =<n>/M as a function of 1, T, ’. Compute and draw (T,P) as a function
of the temperature T and of the pressure P in the gas. These curves are called
“Langmuir’s isotherms”.
II.
Let’s now consider the case where each site can get many molecules.
This case corresponds to the case of successive adsorption layers.
On each site, the energy of adsorption for the first molecule is still equal to –1. The energy of
adsorption for the following molecules are equal to -2 (0<2<1).
We suppose now that the number of molecules that can be adsorbed on a given site is unlimited. As
previously, the chemical potential of the adsorbed molecules is given by the equilibrium condition
with the surrounding gas.
1) Compute the grand-canonical partition function for the adsorption sites.
2) Compute the coverage ratio =<n>/M as a function of the temperature T and of the
pressure P in the gas. Prove that

cx
(1  x  cx )(1  x )
where x=P/P1(T) and c=exp[(1-2)] with P1(T)=Po.exp[-(2+o(T))].
3) When c>>1, (T,P) is shown in Figure 1. What does represent the “elbow” in the
beginning of the figure?
(T,P) is called the “Brunauer-Emmett-Teller adsorption’s isotherms” (B.E.T. isotherms).
It is used to measure the effective surface of divided solids (like powders).