5.7 Problems for This Chapter
253
11. An expression for the entropy S η (T , V ; N) for a classical ideal gas in the
presence of a gravitational field (characterized by the parameter η ≡ βmgH ),
with β ≡ (k B T ) −1 , m the mass of an ideal gas particle, g the acceleration
due to gravity, and H the height of a cylindrical vessel of volume V , has been
given in Example 5.1. Show that lim η→0 S η (T , V ; N) gives the Sackur–Tetrode
equation for the entropy of a classical ideal gas (whose particles undergo only
translational motions) in the absence of a gravitational field. Show also that for a
sufficiently strong gravitational field, the classical expression for S η (T , V ; N)
can become negative, thereby violating the Third Law of Thermodynamics.
Discuss the validity of the classical Sackur–Tetrode equation for this case.
12. The ground electronic level of the F atom has the term symbol 2 P 3
2
, and it
has a very low-lying first excited electronic level (term symbol 2 P 1
2
) lying
404.1 cm −1 above the ground electronic level. What is the probability of finding
a F( 2 P 1
2
) atom for temperatures of 100 K, 500 K, and 2000 K?
13. In Problem 12, we noted that the F atom has both a ground electronic level
and a close-lying excited electronic level lying 404.0 cm −1 above the ground
level. Evaluate the electronic partition function, z el , for the F atom at 298 K,
and determine the temperature for which z el (T ) = 1.
14. Evaluate the electronic partition function for atomic Si at 298 K, given that Si
has a ground electronic term 3 P , with the 3 P 0 level lying lowest in energy (i.e.,
it is the ground level), and the 3 P 1 and 3 P 2 levels lying 77.15 cm −1 and 223.31
cm −1 , respectively, above the ground level. The next highest electronic term,
1 D 2 , lies 6298.0 cm −1 above the ground level. At what temperature will the
1 D 2 term contribute 0.1 to z el (T )?
15. Consider a system made up of one mole of identical, noninteracting, nonlocalized atoms. Each atom can access only three energy levels, with energies
and degeneracies: 1 = 0, ω 1 = 1; 2 /k B = 100 K, ω 2 = 3; and 3 /k B =
300 K, ω 3 = 5. Calculate z plus the average number of atoms in each level at
temperature 200 K. Determine the average number of atoms in each level in the
high-temperature (saturation) limit.
16. Show that Eq. (5.4.15) for the electronic contribution to the chemical potential
for a pure gas indeed provides a logical wrap-up to the normal set of relevant
thermodynamic functions for a thermodynamic system.
17. Suppose that a system consists of N noninteracting, nonlocalized, identical
atoms A and that each atom can access only two (internal) quantum states (i.e.,
translational motion is not being considered here) having energies 1 = 0 and
2 = . Obtain expressions for z int , Z int , U int , C V ,int , and S int , and evaluate
these quantities for one mole of these atoms at 400 K, if = 1.0 × 10 −20
J. Moreover, determine the high-temperature limiting values for U int , C V ,int ,
and S int . Finally, provide physical arguments, in terms of level populations, to
support your results.
253
11. An expression for the entropy S η (T , V ; N) for a classical ideal gas in the
presence of a gravitational field (characterized by the parameter η ≡ βmgH ),
with β ≡ (k B T ) −1 , m the mass of an ideal gas particle, g the acceleration
due to gravity, and H the height of a cylindrical vessel of volume V , has been
given in Example 5.1. Show that lim η→0 S η (T , V ; N) gives the Sackur–Tetrode
equation for the entropy of a classical ideal gas (whose particles undergo only
translational motions) in the absence of a gravitational field. Show also that for a
sufficiently strong gravitational field, the classical expression for S η (T , V ; N)
can become negative, thereby violating the Third Law of Thermodynamics.
Discuss the validity of the classical Sackur–Tetrode equation for this case.
12. The ground electronic level of the F atom has the term symbol 2 P 3
2
, and it
has a very low-lying first excited electronic level (term symbol 2 P 1
2
) lying
404.1 cm −1 above the ground electronic level. What is the probability of finding
a F( 2 P 1
2
) atom for temperatures of 100 K, 500 K, and 2000 K?
13. In Problem 12, we noted that the F atom has both a ground electronic level
and a close-lying excited electronic level lying 404.0 cm −1 above the ground
level. Evaluate the electronic partition function, z el , for the F atom at 298 K,
and determine the temperature for which z el (T ) = 1.
14. Evaluate the electronic partition function for atomic Si at 298 K, given that Si
has a ground electronic term 3 P , with the 3 P 0 level lying lowest in energy (i.e.,
it is the ground level), and the 3 P 1 and 3 P 2 levels lying 77.15 cm −1 and 223.31
cm −1 , respectively, above the ground level. The next highest electronic term,
1 D 2 , lies 6298.0 cm −1 above the ground level. At what temperature will the
1 D 2 term contribute 0.1 to z el (T )?
15. Consider a system made up of one mole of identical, noninteracting, nonlocalized atoms. Each atom can access only three energy levels, with energies
and degeneracies: 1 = 0, ω 1 = 1; 2 /k B = 100 K, ω 2 = 3; and 3 /k B =
300 K, ω 3 = 5. Calculate z plus the average number of atoms in each level at
temperature 200 K. Determine the average number of atoms in each level in the
high-temperature (saturation) limit.
16. Show that Eq. (5.4.15) for the electronic contribution to the chemical potential
for a pure gas indeed provides a logical wrap-up to the normal set of relevant
thermodynamic functions for a thermodynamic system.
17. Suppose that a system consists of N noninteracting, nonlocalized, identical
atoms A and that each atom can access only two (internal) quantum states (i.e.,
translational motion is not being considered here) having energies 1 = 0 and
2 = . Obtain expressions for z int , Z int , U int , C V ,int , and S int , and evaluate
these quantities for one mole of these atoms at 400 K, if = 1.0 × 10 −20
J. Moreover, determine the high-temperature limiting values for U int , C V ,int ,
and S int . Finally, provide physical arguments, in terms of level populations, to
support your results.