3.5 Problems for This Chapter
165
10. Starting from the expression given in Example 3.4 for the density of states for
an ideal structureless gas in a gravitational field such that the single-particle
energy is given by (p, z) =
1
2 p 2 /m + mgz, with g the acceleration due
to the gravitational field that acts in the z-direction, obtain an expression for
the corresponding single-particle partition function z(T , V ) for an ideal gas
of structureless particles. Show that your expression for z(T , V ) becomes
z(T , V ) = V // 3 (T ) for g = 0 (i.e., in the absence of a gravitational field).
11. Particles such as photons, neutrinos, and very high-energy electrons or protons
(i.e., electrons or protons whose relativistic energy greatly exceeds their restmass energies) are referred to as ‘ultrarelativistic’. Show that the relativistic
energy for such particles is related to the magnitude of the (relativistic)
momentum, p, by pc, with c the speed of light, and utilize the result
obtained for z(β, V ) in Problem 8 to obtain an explicit expression for z(β, V )
for ultrarelativistic particles. [Hint: make use of the relativistic momentumenergy relation 2 = p 2 c 2 + 2
0 .]
12. Show that the fractional populations p v (T ) of the SHO vibrational states are
independent of the vibrational zero-point energy associated with the SHO and
evaluate z SHO (T ), accurate to four significant figures, for temperatures T =
100, 250, 500 K, for a SHO characterized by the oscillator frequency ν osc =
1.0418 × 10 13 s −1 . Construct a table of the fractional populations p v (T ) for
this set of temperatures, accurate to three significant figures.
13. The 2 ground electronic term of the NO molecule has two doubly degenerate
electronic levels that correspond to the term symbols 2 3
2
and 2 1
2
, with
the 2 3
2
level lying 121.1 cm −1 above the 2 1
2
level. For the purposes of
this exercise, we shall neglect translational, rotational, and vibrational motions
associated with the NO molecular framework, and focus upon its electronic
nature. In this sense, the ground-term NO molecule thus provides an almost
ideal 2-level system with which to work.
Obtain a value for the characteristic temperature ≡ el for the ground
electronic term NO molecule, and obtain expressions for the partition function
z NO (T ) and the fractional populations p 0 (T ), p 1 (T ) for the NO( 2 ) molecule.
Show that the fractional populations p 0 (T ) and p 1 (T ) are the same as those
for a nondegenerate 2-level (i.e., 2-state) system. Calculate the fractional
populations of the NO ground levels for temperatures such that T // spans
the domain [0,10], and plot them versus the reduced variable T //. What can
you conclude about the fractional populations for such a 2-level system when
the temperature becomes very large (i.e., T )?
14. Evaluate the electronic partition function z el (T ) for NO at temperature T =
298 K and determine the temperature T for which z el (T ) = 3.
15. Show that if the degeneracies associated with the ground and excited levels of
a 2-level system differ, then a 2-level system with degeneracies is no longer
equivalent to a 2-state system. Carry out an explicit calculation for 2-level
system in which the energy splitting between the two levels has the same value
that is found for the NO molecule, i.e., = 121.1 cm −1 , the ground level is
165
10. Starting from the expression given in Example 3.4 for the density of states for
an ideal structureless gas in a gravitational field such that the single-particle
energy is given by (p, z) =
1
2 p 2 /m + mgz, with g the acceleration due
to the gravitational field that acts in the z-direction, obtain an expression for
the corresponding single-particle partition function z(T , V ) for an ideal gas
of structureless particles. Show that your expression for z(T , V ) becomes
z(T , V ) = V // 3 (T ) for g = 0 (i.e., in the absence of a gravitational field).
11. Particles such as photons, neutrinos, and very high-energy electrons or protons
(i.e., electrons or protons whose relativistic energy greatly exceeds their restmass energies) are referred to as ‘ultrarelativistic’. Show that the relativistic
energy for such particles is related to the magnitude of the (relativistic)
momentum, p, by pc, with c the speed of light, and utilize the result
obtained for z(β, V ) in Problem 8 to obtain an explicit expression for z(β, V )
for ultrarelativistic particles. [Hint: make use of the relativistic momentumenergy relation 2 = p 2 c 2 + 2
0 .]
12. Show that the fractional populations p v (T ) of the SHO vibrational states are
independent of the vibrational zero-point energy associated with the SHO and
evaluate z SHO (T ), accurate to four significant figures, for temperatures T =
100, 250, 500 K, for a SHO characterized by the oscillator frequency ν osc =
1.0418 × 10 13 s −1 . Construct a table of the fractional populations p v (T ) for
this set of temperatures, accurate to three significant figures.
13. The 2 ground electronic term of the NO molecule has two doubly degenerate
electronic levels that correspond to the term symbols 2 3
2
and 2 1
2
, with
the 2 3
2
level lying 121.1 cm −1 above the 2 1
2
level. For the purposes of
this exercise, we shall neglect translational, rotational, and vibrational motions
associated with the NO molecular framework, and focus upon its electronic
nature. In this sense, the ground-term NO molecule thus provides an almost
ideal 2-level system with which to work.
Obtain a value for the characteristic temperature ≡ el for the ground
electronic term NO molecule, and obtain expressions for the partition function
z NO (T ) and the fractional populations p 0 (T ), p 1 (T ) for the NO( 2 ) molecule.
Show that the fractional populations p 0 (T ) and p 1 (T ) are the same as those
for a nondegenerate 2-level (i.e., 2-state) system. Calculate the fractional
populations of the NO ground levels for temperatures such that T // spans
the domain [0,10], and plot them versus the reduced variable T //. What can
you conclude about the fractional populations for such a 2-level system when
the temperature becomes very large (i.e., T )?
14. Evaluate the electronic partition function z el (T ) for NO at temperature T =
298 K and determine the temperature T for which z el (T ) = 3.
15. Show that if the degeneracies associated with the ground and excited levels of
a 2-level system differ, then a 2-level system with degeneracies is no longer
equivalent to a 2-state system. Carry out an explicit calculation for 2-level
system in which the energy splitting between the two levels has the same value
that is found for the NO molecule, i.e., = 121.1 cm −1 , the ground level is