Charles ~ a r z z a c c o ' and Marvin ~ a l d m a n ~ New York Universitv Bronx, 10453
Heat Capacity Calculations
I
of Diatomic Molecules A n undergraduate computer calculation
For several years now we have given students the option of doing one or two of several "dry" exercises as part of their physical chemistry laboratory course. One of these is a statistical mechanical calculation of the heat capacities of several diatomic molecules as a function of temperature. The students are encouraged to take a one week course i n Fortran IV given a t N.Y.U. between the fall and spring semesters. The students find this to he a very exciting and interesting exercise. The programming makes use of most of the features of Fortran IV and the results lead to a depth of understanding of statistical mechanics. The results are discussed in class so that all students henefit from the exercise. R[(,ae-~ihr)(2gj@)e-~J/~~)
Cm '
Translational Energy and Heat Capacity The average translational enerm for a mole of pas is given by ET,,,~ = %RT. The translational contribution to the heat cauacitv is therefore
and is independent of temperature and molecular structure. Rotational Energy and Heat Capacity I t is assumed that the molecule is a rigid rotor with remz) and internuclear duced mass fi = (ml . mz)/(ml distance r. The rotational energy levels, EJ, and degeneracies, gJ, are given by
+
=
=
CJ N=~J
~ N ~ ~ - ~ J ~ E J ' " T
and the average rotational energy per mole is L2NJEJ ~ N J
Exot= where L is Avogadro's number. The rotational heat capacity is determined by differentiating EROf with respect to T. The result can he expressed in the followina . manner which is ideal for programming
- (2g.(g)"e-~~~>(zg~(g)e-'"'")]
(~#.,e-~~hr)~
We present below a sketch of the procedure we use in these calculations and a discussion of the results. We would be glad t o send a copy of the program and the output to any interested party.
E.,
N
hZJ(J + 0 ,,&. = & , 8*2pr2
The rotational contribution to the heat capacity involves infinite sums hut since the exponential terms go to zero as J gets large we find that we can truncate the summations a t 25 terms for most molecules below room temperature. The results of these calculations for several heteronuclear diatomic molecules is shown in Figure 1. We see that all heat ca~acitiesexhihit a maximum and then level off a t the classi'cal value of R. The existence of this maximum has been observed and is onlv realistic for HD --- which exists as a gas over most of this temperature range.3 Calculations by some students using just the first two levels of HD indicate that the maximum is almost entirely due to these levels. I t occurs because of the three-fold change in degeneracy on going from the first to the second level. We also notice that the arrival of classical behavior occurs a t lower temperatures as the moment of inertia of the molecule increases. Anomalous behavior will occur with a molecule like CIF because the use of only 25 rotational levels is not a good approximation for this molecule above 250°K.
+1
where J = 0, 1, 2 . . . . The rotational constant, h2/8rZrrZ, may be obtained from Hertzherg's hook or from analysis of the rotational vibrational spectrum of the molecule.2 From the Boltzmann law, the ratio of the number of molecules with energy EJ to the number with energy Eo is given by N , / N , = g,e-"'*'
c *
2.c
b
a
3
where k is Boltzmann's constant. The total number of molecules is then 'To whom program requests should be sent. ZF'resent address: Department of Chemistry, Harvard University, Cambridge, Mass. 444 1 Journal of Chemical Education
TLPLRLTURL X Figure 1. The rotational heat capacities of CN. HCi, and HD as a function of temperature.
The Vibrational Heal Capacity
We assume the molecule is a harmonic oscillator with force constant k and frequency
The vibrational energy levels are given by E,
= (u +
4)b
where v
=
1,2,3....
and all levels are nondegenerate. The vibrational heat capacity for harmonic oscillators is given by4
IlWlRLTURE 'K
The students use either their experimental u's or get them from Hertzberg. The results for several molecules are shown in Figure 2. As expected molecules with large vibrational frequencies require a higher temperature to reach the classical value of R.
Figure 2. The vibrational heat camcities of CIF. CN. HCi. and function of IemDerature
HD as a
Ortho and Para Hydrogen
Heat capacities of homonuclear diatomic molecules present an interestine extension of these calculations. The kxp~anationof the h;drogen gas heat capacity as a function of temperature came in the late 1920's. In 1927 Hund extended Heisenberg's work on the symmetry of the helium wavefunctions and showed that the rotational wavefunctions belonging to even values of J are symmetric to the interchange of the two hydrogen nuclei and those of odd J are antisymmetric.3 The electronic and vibrational wavefunctions were both shown to be symmetric. Hori examined the electronic spectrum of hydrogen and found that transitions between states of odd J were three times more intense than those between states of even J . 3 This result was rationalized by assuming that the hydrogen nucleus like the electron has a nuclear spin of $%. I t was therefore shown that symmetric (triplet) and antisymmetric (singlet) nuclear spin functions exist for the hydrogen molecule. I t was also necessary to assume that the total wavefunction must be antisymmetric t o interchange of the nuclei. This meant that triplet nuclear spin states could only exist with odd J values and that singlet nuclear spin states could only exist with even J values. These requirements meant that the degeneracies of the energy levels of hydrogen were 1,9,5,21,9. . . insteadof 1,3,5,7,9. . . . The calculated heat capacity versus temperature assuming a rapid rate of conversion between singlet and triplet nuclear spin states is shown in Figure 3a. This curve shows oscillatory behavior with maxima occurring a t 51 and 322°K and manima a t 147 and 1050°K (not shown in the figure). After the latter minimum the heat capacity slowly rises toward the classical value of R. This oscillation is presumably due tb the large alternation in the magnitudes of the degeneracies. The first maximum is on -going mainly due to the large change in degeneracy from the first to the second level: Experimentally the heat capacity of hydrogen shows no .maximum but only a slow approach to the classical value of R. The explanation of this puzzle is due to Denn i s ~ n In . ~ 1927 he showed that normal hydrogen is a 3 to 1 mixture of triplet (ortho) and singlet (para) hydrogen. The rate of conversion between these two forms is very slow so that the ratio of ortho to para does not change with temperature. The rotational energy levels of para hy-
Figure 3. The rotational heat capacity of rapid equilibrium hydrogen para hydrogen (bj, and ortho hydrogen (cj.
(a),
drogen are given by
where J can only be an even integer. For ortho hydrogen the energy levels are given by
where J can only be an odd integer. The rotational heat capacities of ortho and para hydrogen as a function of temperature are shown in Figure 3b and 3c. The maximum for para hydrogen is due to the fivefold increase in degeneracy on going from its first to its second level. Ortho hydrogen does not show such a maximum because its degeneracy only increases by a factor of 2119 on going from the first to the second level. Dennison reproduced the experimental curve by taking a 3 to 1 average of the ortho and para curves. Literature Cited (11 Hertzbcm, G.. "Molecular Spetra and Molecular Structure." D. Van Noatrand Compsny,Ine.,Plinceton,N.J. 1966. (21 Shocmskar, D. P., and Garland. C. W., "Experiments in Phyoiesl Chemistri;' 2nd Ed.,MeGraw-HillBmkCo..NewYork. 1962.p.3W. (31 Gopal, E. S. R.. "Specific Heats at Loui Temperatures." Plenum Plesa. New York. 1966. Rurhbmoks, G. S.. '%tmduction to Statlatical Mechanics." Oxford University Press, Oxford. 1964. (41 Mmre. W. J., "Physical Chemistry." 3rd Ed., Prentire-Hall, Ine. Englewmd Cliffi, N. J., 1962. (51 Dcnnixm, D.M.,Roe.Rw.Soc., A
115.182(1927).
Volume 50, Number 6, June 1973 / 445
