A gas of molecular hydrogen H2, is originally in equilibrium at a temperature of 1,000 K. It is

step 1 To start out, this problem gives us the following situation. We have a hydrogen molecule with a

A gas of molecular hydrogen H2, is originally in equilibrium...

A gas of molecular hydrogen $\mathrm{H}_2$, is originally in equilibrium at a temperature of $1,000 \mathrm{~K}$. It is cooled to 20 K so quickly that the nuclear spin states of the molecules do not change, although the translational and rotational degrees of freedom do readjust through collisions. What is the approximate internal energy per molecule in terms of temperature units K ? Note that the rotational part of the energy for a diatomic molecule is $A l(l+1)$ where $l$ is the rotational quantum number and $A \sim 90 \mathrm{~K}$ for $\mathrm{H}_2$. Vibrational motion can be neglected.

A gas of molecular hydrogen $\mathrm{H}_2$, is originally in equilibrium at a temperature of $1,000 \mathrm{~K}$. It is cooled to 20 K so quickly that the nuclear spin states of the molecules do not change, although the translational and rotational degrees of freedom do readjust through collisions. What is the approximate internal energy per molecule in terms of temperature units K ?

Note that the rotational part of the energy for a diatomic molecule is $A l(l+1)$ where $l$ is the rotational quantum number and $A \sim 90 \mathrm{~K}$ for $\mathrm{H}_2$. Vibrational motion can be neglected.

Key Concepts

Degrees of Freedom in Molecular Gases

Molecular gases possess multiple types of motion including translational, rotational, and vibrational degrees of freedom. In thermodynamic processes, the internal energy of a molecule is partitioned among these degrees of freedom. The contribution from each mode depends on the temperature and the energy spacing between quantized states, influencing how energy is stored and transferred in the system.

Quantum Rotational Energy Levels

For diatomic molecules, rotational motion is quantized, with energy levels given by expressions proportional to l(l+1) where l is the rotational quantum number. The separation between these discrete levels means that at low temperatures only the lowest few rotational states are populated, modifying the expected energy contribution from rotational motion compared to the classical continuous prediction.

Non-equilibrium Effects and Frozen States

In rapid thermodynamic processes, some degrees of freedom may not have time to equilibrate. When certain internal states, such as nuclear spin configurations, are frozen during rapid cooling, they can remain in a non-equilibrium distribution. This decoupling means that the corresponding energy contributions do not adjust in the same way as other degrees of freedom, impacting the overall internal energy of the system.

Energy-Temperature Units Conversion

Energy in statistical mechanics is often expressed in temperature units by using the Boltzmann constant (k). This conversion allows one to compare energy directly to temperature, simplifying calculations and providing a useful scale for understanding thermal excitations and the relative occupancy of quantum states.

Equipartition Theorem

The equipartition theorem states that, at thermal equilibrium, energy is equally distributed among all quadratic degrees of freedom, with each contributing (1/2)kT to the internal energy. However, this classical result applies only when the thermal energy is significantly larger than the spacing between quantized energy levels. In cases where quantum effects are important, such as in low-temperature rotational states, the theorem must be applied with caution.

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