Set 4
1. An ideal gas of classical charged particles with mass m is confined between
two capacitor plates of area A, separated by distance L. The capacitors produce
a force f perpendicular to the plates which pushes the particles to the lower
plate. The particles can be absorbed on either plate, with an absorption potential
  (   0 ); on each plate the particles can move freely. The system is in
equilibrium at temperature T. The average density of the non-adsorbed
particles, i.e. those between the plates, is n 0 .
a. Evaluate the density n(x,y,z), choosing z perpendicular to the plates.
b. Evaluate the densities on the lower and upper plates.
c. Evaluate the forces acting on the top and bottom plates. Why is there a
net force on the capacitor?
2. a. Consider ideal gases of atoms A, atoms B and atoms C undergoing the reaction
C is an integer). If nA, nB and nC denote the respective densities show
that in equilibrium
(nC)
(f )
2 C
=V
nAnB
fAfB = K(T)
(law of mass action).
Here V is the volume while the f's are the respective single particle partition
functions. The quantity K(T) is known as the equilibrium constant of the reaction.
b. Derive the equilibrium constant of the reaction H2+D2 D in terms of the
masses mH, mD and 0 the vibrational frequency of HD. Assume temperature is
high enough to allow classical approximation for the rotational motion. Show tha
K(T) = 4.
3. The partition functions of a diatomic molecules AB or A2 (within an ideal gas)
has the form
f AB  g AB (T )(mAmB )3/ 2
f A2 
1
g A2 (T )m 3A
2
mA, mB are atomic masses and B is an isotope of A. gAB and gA2 are are independent
of the isotope masses.
a) Explain the origin of the factor ½.
b) In the reaction H 2  Cl2
2 HCl the Cl atom has two isotopes, Cl35 and Cl37.
Write the relevant reactions and their laws of mass action.
c) Show that the relative abundance of Cl35 and Cl37 in Cl2 is the same as in HCl,
i.e. the various densities n satisfy
2nCl 37  nCl 35Cl 37
2
2nCl 35  nCl 35Cl 37
2

nHCl 37
nHCl 35
4. A cylinder of radius R and height H rotates about its axis with a constant angular
velocity 
a. Consider the Hamiltonian H'(p,q; )=H(p,q;L(p,q))L(p,q) with L(p,q) the
angular velocity. Show that the ensemble average of H' is an energy E'(S,V,)
which is relevant for given  and find its derivatives.
b. Derive the density distribution as function of the distance r from the axis for an
ideal classical gas at temperature T (effects of gravitation are negligible).