6.7 Problems for This Chapter
347
vibrational temperature vib,4 = 415.8 K (obtained from ν 4 = 289 cm −1 ) is shown
as a dashed curve, and several experimental values obtained from heat capacity
measurements by subtracting the translational, end-over-end rotational, and all
vibrational contributions calculated from spectroscopic information are also shown
for comparison. The agreement between calculated and experimental values is, on
the whole, quite acceptable. More recent computations of the eigenenergies and their
employment to compute the hindered rotation contributions to C V (T ), the entropy,
S(T ), and the enthalpy, H (T ), are in excellent agreement [44] with values obtained
using the Pitzer tables.
6.7 Problems for This Chapter
1. Show what happens to P , U , S, A, when a constant value b is added to all
energy levels E r occurring in the definition of z int for a molecule.
2. Most molecules in a gas have tr ≤ 3k B T and tr,x ≤ k B T , where tr,x is the xcomponent of the translational energy. For N 2 in a box of volume 8 cm 3 at 0 ◦ C,
calculate x /(k B T ), with x the spacing between the energies associated
with adjacent x translational states when tr,x is of order k B T .
3. Spectroscopic intensity measurements for a series of vibrational transitions in a
sample of gaseous I 2 give vibrational population ratios N 1 /N 0 = 0.528 and
N 2 /N 0 = 0.279, in which N v represents the number of I 2 molecules in a
vibrational state characterized by vibrational quantum number v. Show that
these results are compatible with the I 2 sample being in thermal equilibrium.
Given that the fundamental vibrational frequency for I 2 is ν = 214.57 cm −1 ,
determine the temperature of the I 2 sample.
4. If a diatomic gas in thermal equilibrium has a v = 1 to v = 0 population ratio
of 0.340, what is the corresponding v = 3 to v = 0 population ratio?
5. If the fundamental vibrational frequency for N 2 is ν osc = 6.9849 × 10 13 s −1 ,
compute the ratio of the v = 1 to v = 0 populations for (a) 25 ◦ C; (b) 800 ◦ C;
(c) 3000 ◦ C.
6. Show, using the SHO approximation for the vibrational motion of a diatomic
molecule, that the fractional population of vibrational level v is given by
N v
N
= e
−vv vib /T (1 − e
− vib /T ) ,
with N the total number of molecules. Given that vib (N 2 ) = 3352 K, plot
the fractional population of the v = 1 vibrational level of N 2 as a function of
temperature from 0 K to 15,000 K. Why might you not expect the high-T part
of your plot to provide an accurate representation of the real behaviour of N 2 ?
7. Show that the SHO (vibrational) partition function, z vib (T ), given in
Eq. (6.2.22) can be rewritten in terms of the hyperbolic cosecant function
as
347
vibrational temperature vib,4 = 415.8 K (obtained from ν 4 = 289 cm −1 ) is shown
as a dashed curve, and several experimental values obtained from heat capacity
measurements by subtracting the translational, end-over-end rotational, and all
vibrational contributions calculated from spectroscopic information are also shown
for comparison. The agreement between calculated and experimental values is, on
the whole, quite acceptable. More recent computations of the eigenenergies and their
employment to compute the hindered rotation contributions to C V (T ), the entropy,
S(T ), and the enthalpy, H (T ), are in excellent agreement [44] with values obtained
using the Pitzer tables.
6.7 Problems for This Chapter
1. Show what happens to P , U , S, A, when a constant value b is added to all
energy levels E r occurring in the definition of z int for a molecule.
2. Most molecules in a gas have tr ≤ 3k B T and tr,x ≤ k B T , where tr,x is the xcomponent of the translational energy. For N 2 in a box of volume 8 cm 3 at 0 ◦ C,
calculate x /(k B T ), with x the spacing between the energies associated
with adjacent x translational states when tr,x is of order k B T .
3. Spectroscopic intensity measurements for a series of vibrational transitions in a
sample of gaseous I 2 give vibrational population ratios N 1 /N 0 = 0.528 and
N 2 /N 0 = 0.279, in which N v represents the number of I 2 molecules in a
vibrational state characterized by vibrational quantum number v. Show that
these results are compatible with the I 2 sample being in thermal equilibrium.
Given that the fundamental vibrational frequency for I 2 is ν = 214.57 cm −1 ,
determine the temperature of the I 2 sample.
4. If a diatomic gas in thermal equilibrium has a v = 1 to v = 0 population ratio
of 0.340, what is the corresponding v = 3 to v = 0 population ratio?
5. If the fundamental vibrational frequency for N 2 is ν osc = 6.9849 × 10 13 s −1 ,
compute the ratio of the v = 1 to v = 0 populations for (a) 25 ◦ C; (b) 800 ◦ C;
(c) 3000 ◦ C.
6. Show, using the SHO approximation for the vibrational motion of a diatomic
molecule, that the fractional population of vibrational level v is given by
N v
N
= e
−vv vib /T (1 − e
− vib /T ) ,
with N the total number of molecules. Given that vib (N 2 ) = 3352 K, plot
the fractional population of the v = 1 vibrational level of N 2 as a function of
temperature from 0 K to 15,000 K. Why might you not expect the high-T part
of your plot to provide an accurate representation of the real behaviour of N 2 ?
7. Show that the SHO (vibrational) partition function, z vib (T ), given in
Eq. (6.2.22) can be rewritten in terms of the hyperbolic cosecant function
as