The molecule of a perfect gas consists of two atoms, of mass...

The molecule of a perfect gas consists of two atoms, of mass $m$, rigidly separated by a distance $d$. The atoms of each molecule carry charges $q$ and $-q$ respectively, and the gas is placed in an electric field $\varepsilon$. Find the mean polarization, and the specific heat per molecule, if quantum effects can be neglected. State the condition for this last assumption to be true.

The molecule of a perfect gas consists of two atoms, of mass $m$, rigidly separated by a distance $d$. The atoms of each molecule carry charges $q$ and $-q$ respectively, and the gas is placed in an electric field $\varepsilon$. Find the mean polarization, and the specific heat per molecule, if quantum effects can be neglected.
State the condition for this last assumption to be true.

Key Concepts

Rigid Rotor Model

This concept models a diatomic molecule as two point masses connected by a fixed distance, allowing rotation without vibration. In statistical mechanics, such a system is studied by considering its rotational degrees of freedom, which are described by a moment of inertia. The rigid rotor is a fundamental model in molecular physics, used to derive properties like the rotational partition function and associated thermodynamic quantities.

Dipole Moment

A dipole moment is a measure of the separation of positive and negative charges in a system. For a diatomic molecule where the atoms carry opposite charges, the dipole moment is given by the product of the charge magnitude and the separation distance. This intrinsic feature underlies the molecule’s response to an external electric field, leading to polarization phenomena.

Orientational Polarization

In an ensemble of polar molecules exposed to an external electric field, the molecules tend to align with the field. This alignment, however, is counteracted by thermal motion, which randomizes orientations. The net effect is an average or mean polarization, which can be computed by statistically averaging the molecular dipole moments using Boltzmann factors in classical statistical mechanics.

Specific Heat of Molecular Gases

Specific heat in the context of molecular gases accounts for contributions from translational motion, rotational motion, and possibly vibrational motion. For a rigid diatomic molecule, rotational degrees of freedom contribute to the specific heat. Typically, each quadratically dependent degree of freedom contributes (1/2)k_B per molecule (in the classical limit), making the analysis of specific heat a direct application of the equipartition theorem.

Classical Treatment and the Negligibility of Quantum Effects

The assumption that quantum effects can be neglected is valid when the thermal energy (k_B T) is large compared to the spacing between quantized energy levels, such as the rotational energy levels of a molecule. For a rigid rotor, this means that k_B T must be much greater than ?²/(2I), where I is the moment of inertia. This condition ensures that the energy level spacing is small relative to the thermal energy, justifying the use of classical Boltzmann statistics.

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