25.4 The Calculation of Molecular Partition Functions 1075
Exercise 25.20
Calculate the vibrational partition function of sulfur dioxide at 298.15 K. The vibrational
frequencies are 1. 556 × 1013 s−^1 ,3. 451 × 1013 s−^1 , and 4. 080 × 1013 s−^1.Nuclear Contributions to the Partition Function
The four factors of the molecular partition function that we have discussed are not the
complete partition function, because the nuclear contributions have not been included.
However, excitation of a typical nucleus to its first excited state requires a very large
energy compared with chemical energies (usually millions of electron volts). Excited
nuclear states are therefore nearly completely unpopulated at room temperature, and
only the nuclear ground-states contribute to the partition function. The degeneracy of
nuclear spin states must sometimes be included. For example, if a molecule ofortho-
hydrogen (see Section 22.3) is not in an external magnetic field, the three states with
different projections of the total nuclear angular momentum have the same energy. In
this caseortho-hydrogen must have a factor of 3 included in its partition function for
its nuclear spin degeneracy. There is only one value of the spin projection forpara-
hydrogen, so thatpara-hydrogen requires a factor equal to unity in its partition function.
In the presence of a catalyst that can dissociate the hydrogen molecules,ortho- and
para-hydrogen can interconvert rapidly, and can be considered a single substance that
requires a factor of 4 in its partition function for the nuclear spin degeneracy.PROBLEMS
Section 25.4: The Calculation of Molecular Partition
Functions
25.21a.Calculate the value of the rotational partition function
of CO gas at 298.15 K and at 500.0 K.b.Find theJvalue that has the largest population for CO
at 298.15 K and at 500.0 K.
c.Find the probability of the energy level corresponding
to this value ofJat 298.15 K.25.22a.Calculate the value of the rotational partition function
of I 2 gas at 500.0 K.b.Find theJvalue that has the largest population for I 2
at 500.0 K. Assume that only even values ofJoccur.
c.Find the probability of the energy level corresponding
to this value ofJ.25.23Calculate the value of the molecular partition function of
argon gas at 298.15 K for a volume of 0.02500 m^3 and
also for a volume of 1.00 m^3 at the same temperature.
Explain in words what the difference between the two
values means.
25.24a.Calculate the rotational partition function for H 2 at
298.15 K, using Eq. (25.4-13).
b.Calculate the rotational partition function of H 2 at
298.15 K by adding up terms explicitly, assuming that
the hydrogen is allpara-hydrogen (only even values
ofJoccur). Continue the sum until one additional
term in the sum over levels does not change the first
four digits of the sum. Compare your result with that
of part a.
25.25Calculate the value of the molecular partition function of
Cl 2 gas at 298.15 K and a pressure of 1.000 bar. Find the
Russell–Saunders term symbol for the ground-state of
atomic chlorine and find the degeneracy of this level.
Assume that the electronic partition function can be
approximated byg 0 , the degeneracy of the ground-level.
25.26Calculate the value of the molecular partition function of
Cl 2 gas at 298.15 K for volume of 0.02500 m^3 and also