3.5 Problems for This Chapter
163
so that our expression for p 1 (( 1 , V 1 )d 1 becomes
p 1 (( 1 , V 1 )d 1 =
1 (( 1 , V 1 )) 2 (( 2m , V 2m )e
ββ 1m +ζ 1m e
−ββ 1 −ζ V 1
2 (( 2m , V 2m )e
ββ 1m +ζ V 1m
∞
0
∞
0
1 (( 1 , V 1 )e
−ββ 1 −ζ V 1 d 1 dV 1
d 1
=
1 (( 1 , V 1 )e
−ββ 1 −ζ V 1
(β, ζ )
d 1 ,
(3.4.8)
with the partition function ζ ) defined as
(β, ζ ) ≡
∞
0
dV
∞
0
dE E(E, V ) e
−βE−ζ V
=
∞
0
z(β, V )e
−ζ V dV .
(3.4.9)
The final equality in Eq. (3.4.9) relates ζ ) to the single-particle canonical
partition function z(β, V ) of Eq. (3.2.21). We may now examine the connections
between these ensembles and thermodynamics.
3.5 Problems for This Chapter
1. Consider two systems of particles, A and B, such that the particles in both
systems have the same ground state energy, 0 , and possess a single excited
energy level lying 208.51 cm −1 above 0 . The two particle types differ in that
the excited energy levels for system A are nondegenerate, while those for
system B are doubly degenerate. Assuming that the Boltzmann distribution
law applies to each system in turn, calculate the corresponding (absolute)
temperature for which the particles in that system have probability 0.15 of
occupying the excited state.
2. Consider now two systems of particles, A and B , having the same ground
and first excited states as systems A and B of Problem 1, respectively, but
with the addition in each case of a nondegenerate second excited level lying
208.51 cm −1 above the first excited level. Carry out the same calculations as
made in Problem 1 for systems A and B. How do your results for the addition
of a second excited level differ from those obtained in Problem 1?
3. How many translational states with energies less than 3k B T are available to
neon atoms in thermal equilibrium at temperature T = 300 K and contained in
a box of volume 10 cm 3 ?
4. Consider particles that possess a ground state together with three excited energy
states separated by 100 cm −1 from adjacent states, and consider three sets of
60 particles each populating these four equally-spaced energy levels as shown
in the table below. Check that each set corresponds to the same total energy,
163
so that our expression for p 1 (( 1 , V 1 )d 1 becomes
p 1 (( 1 , V 1 )d 1 =
1 (( 1 , V 1 )) 2 (( 2m , V 2m )e
ββ 1m +ζ 1m e
−ββ 1 −ζ V 1
2 (( 2m , V 2m )e
ββ 1m +ζ V 1m
∞
0
∞
0
1 (( 1 , V 1 )e
−ββ 1 −ζ V 1 d 1 dV 1
d 1
=
1 (( 1 , V 1 )e
−ββ 1 −ζ V 1
(β, ζ )
d 1 ,
(3.4.8)
with the partition function ζ ) defined as
(β, ζ ) ≡
∞
0
dV
∞
0
dE E(E, V ) e
−βE−ζ V
=
∞
0
z(β, V )e
−ζ V dV .
(3.4.9)
The final equality in Eq. (3.4.9) relates ζ ) to the single-particle canonical
partition function z(β, V ) of Eq. (3.2.21). We may now examine the connections
between these ensembles and thermodynamics.
3.5 Problems for This Chapter
1. Consider two systems of particles, A and B, such that the particles in both
systems have the same ground state energy, 0 , and possess a single excited
energy level lying 208.51 cm −1 above 0 . The two particle types differ in that
the excited energy levels for system A are nondegenerate, while those for
system B are doubly degenerate. Assuming that the Boltzmann distribution
law applies to each system in turn, calculate the corresponding (absolute)
temperature for which the particles in that system have probability 0.15 of
occupying the excited state.
2. Consider now two systems of particles, A and B , having the same ground
and first excited states as systems A and B of Problem 1, respectively, but
with the addition in each case of a nondegenerate second excited level lying
208.51 cm −1 above the first excited level. Carry out the same calculations as
made in Problem 1 for systems A and B. How do your results for the addition
of a second excited level differ from those obtained in Problem 1?
3. How many translational states with energies less than 3k B T are available to
neon atoms in thermal equilibrium at temperature T = 300 K and contained in
a box of volume 10 cm 3 ?
4. Consider particles that possess a ground state together with three excited energy
states separated by 100 cm −1 from adjacent states, and consider three sets of
60 particles each populating these four equally-spaced energy levels as shown
in the table below. Check that each set corresponds to the same total energy,