In hydrogen gas at low temperatures, the molecules can exist...

In hydrogen gas at low temperatures, the molecules can exist in two states: proton spins parallel (orthohydrogen) or anti-parallel (parahydrogen). The transition betwen these two molecular forms is slow. Experiments performed over a time scale of less than a few hours can be considered as if we are dealing with two separate gases, in proportions given by their statistical distributions at the last temperature at which the gas was allowed to come to equilibrium. (a) Knowing that the separation between protons in a hydrogen molecule is $7.4 \times 10^{-9} \mathrm{~cm}$, estimate the energy difference between the ground state and the first excited rotational state of parahydrogen. Use degrees Kelvin as your unit of energy. Call this energy $k \theta_0$, so that rrors in (a) do not propagate into the other parts of the question. (b) Express the energy difference between the ground and first excited rotational states of orthohydrogen, $k \theta_1$, in terms of $k \theta_0$. In an experiment to measure specific heats, the gas is allowed to come to equilibrium at elevated temperature, then cooled quickly to the temperature at which specific heat is measured. What will the constant-volume molar specific heat be at: (c) temperatures well above $\theta_0$ and $\theta_1$, but not high enough to excite vibrational levels? (d) temperatures much below $\theta_0$ and $\theta_1$ [include the leading temperature-dependent term]? (e) $T=\theta_0 / 2$ ?

In hydrogen gas at low temperatures, the molecules can exist in two states: proton spins parallel (orthohydrogen) or anti-parallel (parahydrogen). The transition betwen these two molecular forms is slow. Experiments performed over a time scale of less than a few hours can be considered as if we are dealing with two separate gases, in proportions given by their statistical distributions at the last temperature at which the gas was allowed to come to equilibrium.
(a) Knowing that the separation between protons in a hydrogen molecule is $7.4 \times 10^{-9} \mathrm{~cm}$, estimate the energy difference between the ground state and the first excited rotational state of parahydrogen. Use degrees Kelvin as your unit of energy. Call this energy $k \theta_0$, so that rrors in (a) do not propagate into the other parts of the question.
(b) Express the energy difference between the ground and first excited rotational states of orthohydrogen, $k \theta_1$, in terms of $k \theta_0$. In an experiment to measure specific heats, the gas is allowed to come to equilibrium at elevated temperature, then cooled quickly to the temperature at which specific heat is measured. What will the constant-volume molar specific heat be at:
(c) temperatures well above $\theta_0$ and $\theta_1$, but not high enough to excite vibrational levels?
(d) temperatures much below $\theta_0$ and $\theta_1$ [include the leading temperature-dependent term]?
(e) $T=\theta_0 / 2$ ?

Key Concepts

Specific Heat of Diatomic Gases

The specific heat of a gas reflects how energy is stored among its various degrees of freedom, including translational, rotational, and vibrational motions. For diatomic gases at moderate temperatures, the rotational degrees of freedom contribute noticeably to the specific heat. However, at low temperatures, quantum effects can freeze out these modes resulting in a temperature-dependent contribution, while at higher temperatures, nearly all available states are excited and the classical limit of equipartition is approached.

Temperature-Dependent Activation of Quantum States

Quantum energy levels, such as rotational states, become thermally activated based on their energy spacing relative to kT. As the temperature increases, more quantum states become accessible, leading to a stepwise or smooth increase in specific heat. At very low temperatures, only the ground state or few low-lying states contribute, leading to pronounced deviations from classical predictions.

Boltzmann Distribution

The Boltzmann distribution is a fundamental principle in statistical mechanics that describes the population of particles over various energy states at thermal equilibrium. It plays a key role in determining the relative occupation of molecular energy levels, meaning that at different temperatures, different rotational states will be significantly populated, thereby altering properties like specific heat.

Nuclear Spin Isomerism

Certain diatomic molecules, like hydrogen, can exist in different nuclear spin configurations. The two manifestations, often referred to as ortho and para forms, arise from the symmetry requirements of the total wavefunction, which combine nuclear spin and rotational parts. These isomers have differing allowed rotational states, resulting in different energy levels and statistical weights, which in turn affect their thermodynamic properties.

Rotational Energy Quantization

In quantum mechanics, diatomic molecules can only rotate in discrete amounts with energy levels that are determined by the rotational quantum number. The energy for a given rotational state is typically proportional to J(J+1) divided by the moment of inertia, where J is the rotational quantum number. This quantization underlies many thermodynamic behaviors of molecular gases such as the temperature dependence of their specific heat.

Moment of Inertia

The moment of inertia is a measure of how a molecule’s mass is distributed relative to its axis of rotation and plays a critical role in determining the spacing between rotational energy levels. A larger moment of inertia results in more closely spaced energy levels, influencing the temperature at which rotational excitations become thermally accessible.

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