26.5 Miscellaneous Topics in Statistical Thermodynamics 1119
ADDITIONAL PROBLEMS
26.35 a.Find the value of the statistical entropy of the model
system of four oscillators in Section 25.1 for the
given case thatE 4 hν.
b.Find the value ofΩfor the example system of four
oscillators if the system energy equals 5hν.
c.Find the value ofΩfor the example system of four
oscillators if the system energy equals 3hν.
d.Find the value of the statistical entropy of the system
corresponding to parts b and c.
e.Consider a combined system made up of two
systems, each of which consists of four
distinguishable oscillators. The value ofΩfor the
combined system is equal toΩaΩb, where we denote
the two subsystems by a and b. Find the statistical
entropy of the combined system if each subsystem
has an energy of 4hν.
f. Find the statistical entropy of the combined system if
one subsystem has an energy of 5hνand the other has
an energy of 3hν. Show that the entropy of the
combined system increases if one quantum of energy
is transferred from the system with energy equal to
5 hνto the system with an energy of 3hν.
g.Find the temperature of each subsystem prior to the
transfer of energy given thatν 2. 00 × 1013 s−^1.
26.36 a.Choose a diatomic gas and compute its translational,
rotational, vibrational, and electronic partition
functions at 298.15 K and 1.000 bar, looking up
parameters as needed in Table A.2 in Appendix A or
in some more complete table.^11 Unless you choose
NO or a similar molecule with an unpaired electron,
assume that only the ground electronic state needs to
be included.
b.Compute the standard-state energy, entropy,
Helmholtz energy, Gibbs energy, and heat capacity of
your gas at 298.15 K. It might be interesting for
members of a class to choose different substances and
to make a table of their collective results.
26.37 a.Calculate the difference between the Helmholtz
energy, A of 1.000 mol of^36 Ar and 1.000 mol of^40 Ar
at 298.15 K and 1.000 bar. The molar masses are
35.9675 g mol−^1 and 39.9624 g mol−^1.
b.Calculate the same difference forU.
(^11) K. P. Huber and G. Herzberg,op. cit.(note 3).
c.Calculate the same difference forS.
d.Calculate the entropy of mixing for 0.0040 mol of
(^36) Ar and 99.60 mol of (^40) Ar, which approximates
100.00 mol of ordinary argon. Explain why this
entropy of mixing and similar entropies of isotopic
mixing can be ignored in ordinary chemical
reactions.
26.38The ground electronic level of NO is a^2 Π 1 / 2 term. The
first excited level is a^2 Π 3 / 2 term with an energy
2. 380 × 10 −^21 J above the ground level. Both these states
have degeneracy equal to 2. All other electronic states are
more than 7× 10 −^19 J above the ground level and can be
neglected. The value of the vibrational constant,v ̃e,is
1904.20 cm−^1 , and the value of the rotational constant,
B ̃e, is 1.67195 cm−^1.
a.Find the electronic factor in the molecular partition
function of NO at 500.0 K and 1.000 atm.
b. Find the fraction of NO molecules in the first excited
electronic state at 500.0 K.
c.Find the rotational factor in the molecular partition
function of NO at 500.0 K and 1.000 atm.
d.Find the translational factor in the molecular partition
function of NO at 298.15 K and 1.000 atm.
e.Find the vibrational factor in the molecular partition
function of NO at 500.0 K and 1.000 atm.
f. Find the value ofA◦mfor 1.000 mol of NO at 500.0 K
and find the value of each contribution to it. Does the
vibrational contribution need to be included?
g.Find the value ofUm◦for 1.000 mol of NO at 500.0.
Take the energy of the ground electronic state as zero.
26.39 a.Using the C−C and CC bond energies, estimate the
activation energy for acis transisomerization around
aCC bond.
b. Estimate the ratio of the rate of acis trans
isomerization at 310.0 K to the rate at 300.0 K.
26.40The rotational states ofpara-hydrogen are restricted to
even values ofJ(see Chapter 22). The rotational
contribution to the heat capacity ofpara-hydrogen
exhibits an interesting effect similar to the Schottky effect
(see Problem 26.18).^12 Assume that in solid and
(^12) R. L. Scott,J. Chem. Educ., 83 , 1071 (2006).