Specific Heats of Gases

The kinetic energy of other molecules does not involve the momentum, , of this particular molecule. Moreover, the potential energy of interaction between molecules depends only on their position coordinates, and is, thus, independent of . Any internal rotational, vibrational, electronic, or nuclear degrees of freedom of the molecule also do not involve . Hence, the essential conditions of the equipartition theorem are satisfied. (At least, in the classical approximation.) Because Equation (7.161) contains three independent quadratic terms, there are clearly three degrees of freedom associated with translation (one for each dimension of space), so the translational contribution to the molar heat capacity of gases is

Suppose that our gas is contained in a cubic enclosure of dimensions . According to Schrödinger's equation, the quantized translational energy levels of an individual molecule are given by

(7.163) |

where , , and are positive integer quantum numbers. (See Appendix C.) Clearly, the spacing between the energy levels can be made arbitrarily small by increasing the size of the enclosure. This implies that translational degrees of freedom can be treated classically, so that Equation (7.162) is always valid. (Except very close to absolute zero.) We conclude that all gases possess a minimum molar heat capacity of due to the translational degrees of freedom of their constituent molecules.

The electronic degrees of freedom of gas molecules (i.e., the possible configurations of electrons orbiting the atomic nuclei) typically give rise to absorption and emission in the ultraviolet or visible regions of the spectrum. It follows from Table 7.1 that electronic degrees of freedom are frozen out at room temperature. Similarly, nuclear degrees of freedom (i.e., the possible configurations of protons and neutrons in the atomic nuclei) are frozen out because they are associated with absorption and emission in the X-ray and -ray regions of the electromagnetic spectrum. In fact, the only additional degrees of freedom that we need worry about for gases are rotational and vibrational degrees of freedom. These typically give rise to absorption lines in the infrared region of the spectrum.

The rotational kinetic energy of a molecule tumbling in space can be written

(7.164) |

where the -, -, and -axes are the so called

(7.165) |

according to the equipartition theorem. Note that the typical magnitude of a molecular moment of inertia is , where is the molecular mass, and is the typical interatomic spacing in the molecule. A special case arises if the molecule is linear (e.g., if the molecule is diatomic). In this case, one of the principal axes lies along the line of centers of the atoms. The moment of inertia about this axis is of order , where is a typical nuclear dimension (remember that nearly all of the mass of an atom resides in the nucleus). Because , it follows that the moment of inertia about the line of centers is minuscule compared to the moments of inertia about the other two principle axes. In quantum mechanics, angular momentum is quantized in units of . The energy levels of a rigid rotator spinning about a principal axis are written

(7.166) |

where is the moment of inertia, and is a non-negative integer. (See Section C.12.) Note the inverse dependence of the spacing between energy levels on the moment of inertia. It is clear that, for the case of a linear molecule, the rotational degree of freedom associated with spinning along the line of centers of the atoms is frozen out at room temperature, given the very small moment of inertia along this axis, and, hence, the very widely spaced rotational energy levels.

Classically, the vibrational degrees of freedom of a molecule are studied by standard normal mode analysis of the molecular structure. Each normal mode behaves like an independent harmonic oscillator, and, therefore, contributes to the molar specific heat of the gas [ from the kinetic energy of vibration, and from the potential energy of vibration]. A molecule containing atoms has normal modes of vibration. For instance, a diatomic molecule has just one normal mode (corresponding to periodic stretching of the bond between the two atoms). Thus, the classical contribution to the specific heat from vibrational degrees of freedom is

(7.167) |

So, do any of the rotational and vibrational degrees of freedom
actually make a contribution to the specific heats of gases at room temperature,
once quantum effects have been taken into consideration? We can answer this
question by
examining just one piece of data. Figure 7.3 shows the
infrared absorption spectrum of hydrogen chloride gas. The absorption lines correspond
to simultaneous transitions between different vibrational and rotational energy
levels. Hence, this is usually called a *vibration-rotation spectrum*. The missing
line at about
microns corresponds to a pure vibrational transition from the
ground-state to the first excited state.
(Pure vibrational transitions are
forbidden--hydrogen chloride molecules always have to simultaneously change their rotational energy level if they are to couple effectively to electromagnetic radiation.)
The longer wavelength absorption lines correspond to vibrational transitions in
which there is a simultaneous decrease in the rotational energy level.
Likewise, the
shorter wavelength absorption lines correspond to vibrational transitions in which
there is a simultaneous increase in the rotational energy level. It is clear that
the rotational energy levels are more closely spaced than the vibrational energy
levels. The pure vibrational transition gives rise to absorption at
about
microns, which corresponds to infrared radiation of frequency
hertz with an associated
radiation ``temperature'' of 4,100K. We
conclude that
the vibrational degrees of freedom of hydrogen chloride, or any other small molecule,
are frozen out at room temperature. The rotational transitions split the
vibrational lines by about
microns. This implies that pure rotational
transitions would be associated with infrared radiation of frequency
hertz and corresponding
radiation ``temperature'' 240K. We
conclude that the rotational degrees of freedom of hydrogen chloride, or any other small
molecule, are not frozen out at room temperature, and probably contribute the
classical
to the molar specific heat. There is one proviso, however.
Linear molecules (like hydrogen chloride) effectively only have two rotational degrees of
freedom (instead of the usual three), because of the very small moment
of inertia of such molecules along the line of centers of the atoms.

We are now in a position to make some predictions regarding the specific heats of various gases. Monatomic molecules only possess three translational degrees of freedom, so monatomic gases should have a molar heat capacity joules/degree/mole. Moreover, the ratio of specific heats should be . It can be seen from Table 6.1 that both of these predictions are borne out pretty well for helium and argon. Diatomic molecules possess three translational degrees of freedom, and two rotational degrees of freedom. (All other degrees of freedom are frozen out at room temperature.) Thus, diatomic gases should have a molar heat capacity joules/degree/mole. Moreover, the ratio of specific heats should be . It can be seen from Table 6.1 that these are reasonably accurate predictions for nitrogen and oxygen. The freezing out of vibrational degrees of freedom becomes gradually less effective as molecules become heavier and more complex. This is partly because such molecules are generally less stable, so the force constant, , is reduced, and partly because the molecular mass is increased. Both these effects reduce the frequency of vibration of the molecular normal modes [see Equation (7.141)], and, hence, the spacing between vibrational energy levels [see Equation (7.140)]. This accounts for the obviously non-classical [i.e., not a multiple of ] specific heats of carbon dioxide and ethane in Table 6.1. In both molecules, vibrational degrees of freedom contribute to the molar specific heat. (But not the full , because the temperature is not sufficiently high.)

Figure 7.4 shows the variation of the molar heat capacity at constant volume of gaseous molecular hydrogen (i.e., ) with temperature. The expected contribution from the translational degrees of freedom is (there are three translational degrees of freedom per molecule). The expected contribution at high temperatures from the rotational degrees of freedom is (there are effectively two rotational degrees of freedom per molecule). Finally, the expected contribution at high temperatures from the vibrational degrees of freedom is (there is one vibrational degree of freedom per molecule). It can be seen that, as the temperature rises, the rotational, and then the vibrational, degrees of freedom eventually make their full classical contributions to the heat capacity.