Specific Heats of Gases

Let us now investigate the specific heats of gases. Consider, first of all, translational degrees of freedom. Every molecule in a gas is free to move in three dimensions. If one particular molecule has mass $m$ and momentum ${\bf p} = m\,{\bf v}$ then its kinetic energy of translation is

$$K = \frac{1}{2\,m}\left(p_x^{\,2}+ p_y^{\,2}+p_z^{\,2}\right).$$ (5.400)

The kinetic energy of other molecules does not involve the momentum, ${\bf p}$, of this particular molecule. Moreover, the potential energy of interaction between molecules depends only on their position coordinates, and is, thus, independent of ${\bf p}$. Any internal rotational, vibrational, electronic, or nuclear degrees of freedom of the molecule also do not involve ${\bf p}$. Hence, the essential conditions of the equipartition theorem are satisfied. (At least, in the classical approximation.) Because Equation (5.400) 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

$$(c_V)_{\rm translation} = \frac{3}{2}\,R.$$ (5.401)

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

$$E = \frac{\hbar^{2}\, \pi^{2}}{2\, m\, a^{2}} \left(n_x^{\,2}+n_y^{\,2}+ n_z^{\,2}\right),$$ (5.402)

where $n_x$, $n_y$, and $n_z$ are positive-integer quantum numbers. (See Section 4.4.2.) 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 (5.401) is always valid. (Except very close to absolute zero.) We conclude that all gases possess a minimum molar heat capacity of $(3/2)\,R$ 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 5.1 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 $\gamma$-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

$$K = \frac{1}{2}\, \frac{L_x^{\,2}}{I_{xx}}+\frac{1}{2}\, \frac{L_y^{\,2}}{I_{yy}}+ \frac{1}{2} \,\frac{L_z^{\,2}}{I_{zz}},$$ (5.403)

where the $x$-, $y$-, and $z$-axes are the so called principal axes of rotation of the molecule (these are mutually perpendicular), $L_x$, $L_y$, and $L_z$ are the angular momenta about these axes, and $I_{xx}$, $I_{yy}$, and $I_{zz}$ are the principal moments of inertia about these axes. (See Sections 1.7.2 and 1.7.3.) No other degrees of freedom depend on the angular momenta. Because the kinetic energy of rotation is the sum of three quadratic terms, the rotational contribution to the molar heat capacity of gases is

$$(c_V)_{\rm rotation} = \frac{3}{2}\, R,$$ (5.404)

according to the equipartition theorem. Note that the typical magnitude of a molecular moment of inertia is $m \,d^{\,2}$, where $m$ is the molecular mass, and $d$ 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 $m_e\,d^{\,2}$, where $m_e$ is the electron mass. (See Section 5.3.6.) Because $m_e\ll m$, it follows that the moment of inertia about the line of centers is minuscule compared to the moments of inertia about the other two principal axes. In quantum mechanics, angular momentum is quantized in units of $\hbar$. The energy levels of a rigid rotator spinning about a principal axis are written

$$E = \frac{\hbar^{2}}{2 \,I}\, J\,(J+1),$$ (5.405)

where $I$ is the moment of inertia, and $J$ is a non-negative integer. 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. Thus, the rotational contribution to the molar heat capacity of a diatomic gas is

$$(c_V)_{\rm rotation} = R.$$ (5.406)

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 $R$ to the molar specific heat of the gas [$(1/2)\,R$ from the kinetic energy of vibration, and $(1/2)\,R$ from the potential energy of vibration]. A molecule containing $n$ atoms has $n-1$ 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

$$(c_V)_{\rm vibration} = (n-1)\, R.$$ (5.407)

Figure 5.4: The infrared vibration-absorption spectrum of hydrogen chloride gas.

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 5.4 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 $3.47$ 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 $3.47$ microns, which corresponds to infrared radiation of frequency $8.5\times 10^{\,13}$ hertz with an associated radiation “temperature” of 4,100 K. 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 $0.2$ microns. This implies that pure rotational transitions would be associated with infrared radiation of frequency $5\times 10^{\,12}$ hertz and corresponding radiation “temperature” 240 K. 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 $(1/2)\,R$ 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.

Figure 5.5: The molar heat capacity at constant volume of gaseous molecular hydrogen versus temperature.

Figure 5.5 shows the variation of the molar heat capacity at constant volume of gaseous molecular hydrogen (i.e., ${\rm H}_2$) with temperature. The expected contribution from the translational degrees of freedom is $(3/2)\,R$ (there are three translational degrees of freedom per molecule). The expected contribution at high temperatures from the rotational degrees of freedom is $R$ (there are effectively two rotational degrees of freedom per molecule). Finally, the expected contribution at high temperatures from the vibrational degrees of freedom is $R$ (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.