1076 25 Equilibrium Statistical Mechanics. I. The Probability Distribution for Molecular States
for a volume of 1.00 m^3 at the same temperature. Explain
in words what the difference between the two values
means. Which factors in the partition function change
values when the volume changes and which remain at the
same value?25.27Calculate the molecular partition functions of helium gas
and krypton gas at 298.15 K, assuming each gas is
confined in a volume of 24.45 L. Explain the difference
in the two results in terms of effective number of
accessible states.
25.28Calculate the molecular partition function of argon at
300.0 K and at 500.0 K, assuming the gas is confined in a
volume of 100 L. Explain the difference in the two results
in terms of the effective number of accessible states.
25.29Calculate the molecular partition function of F 2 gas at
300.0 K and at 500.0 K, assuming the gas is confined in a
volume of 100.0 L. Explain the difference in the two
results in terms of the effective number of accessible
states.
25.30Calculate the four factors in the partition function of N 2
at 298.15 K if 1.000 mol is confined at 1.000 atm.
25.31A formula for the rotational partition function of a
diatomic substance that gives corrections to the formula
of Eq. (25.4-13) is^9
zrotσT
Θrot[
1 +1
3
(Θrot/T)+1
15
(Θrot/T)^2+4
315(Θrot/T)^3 +...whereΘrotis called therotational temperature:Θroth^2
8 π^2 IekBhBe
kBCalculate the rotational partition function of H 2 at
298.15 K and compare your answer with that of
Eq. (25.4-13) and with the result of Problem 25.24.25.32Calculate the rotational partition function of Br 2 at 25.0 K
using the formula in Problem 25.31. Compare your result
with that obtained from Eq. (25.4-13).
25.33Calculate the rotational partition function of Cl 2 at
500.0 K using the formula in Problem 25.31. Compare
your result with that obtained using Eq. (25.4-13).
25.34Calculate the rotational partition function of I 2 at 500.0 K
using the formula in Problem 25.31. Compare your result
with that obtained from Eq. (25.4-13).
25.35What fraction of diatomic molecules has rotational
energy greater thankBT? Assume that the sums can be
approximated by integrals.
25.36The bond distances in H 2 O are equal to 95.8 pm, and the
bond angle is equal to 104. 45 ◦. Find the location of the
center of mass and the principal moments of inertia.
Calculate the rotational partition function at 298.15 K.
Don’t forget the symmetry number.
25.37The C–H bond distances in methane are equal to 1.091×
10 −^10 m and the bond angles are equal to the tetrahedral
angle, 109◦ 28 ′ 16. 394 ′′. Calculate the rotational
partition function for methane at 298.15 K. Comment
on the comparison between your value and that for
carbon tetrachloride in Example 25.11. You must
calculate the moments of inertia, all three of which are
equal to each other. Since the molecule is a spherical top,
any perpendicular axes are principal axes. One choice is
to place the hydrogen atoms at alternating vertices of a
cube and to place the axes through the centers of the
faces of the cube. You will have to calculate the size of
the cube using geometry. Another choice is to place the
zaxis on a C–H bond and to place another C–H bond in
thexzplane and to use the fact that all H atoms
contribute equally.
25.38The bond distances in NH 3 are equal to 101.4 pm, and the
bond angles are equal to 107. 3 ◦. Find the location of the
center of mass and the principal moments of inertia. What
is the symmetry number? Calculate the rotational
partition function of NH 3 at 500 K. Don’t forget the
symmetry number.
25.39In calculating rotational and vibrational partition
functions, the upper limit of the sum is taken as infinity.
However, states with very high values of the quantum
numbers will not occur, because the molecule would
dissociate before reaching such a high energy. Explain
why this fact does not produce a serious numerical error
in calculating the partition function.
25.40Carry out the summation of the rotational partition
function separately for theorthoandparaforms of
hydrogen gas. Compare with the result obtained with the
integral approximation using a symmetry number of 2
and a nuclear spin degeneracy factor of 4.(^9) N. Davidson,op. cit., p. 118 (note 2).