The potential energy V between the two atoms ( .mH=1.672 ×10^-24 g) in a hydrogen molecule is g

Step 1: **Determine the rotational temperature u005C( T_R u005C)**u000Au000AThe rotational temperature u005C( T_R u005C)

The potential energy V between the two atoms ( .mH=1.672...

The potential energy $V$ between the two atoms ( $\left.m_{\mathrm{H}}=1.672 \times 10^{-24} \mathrm{~g}\right)$ in a hydrogen molecule is given by the empirical expression $$ V=D\left\{e^{-2 a\left(r-r_0\right)}-2 e^{-a\left(r-r_0\right)}\right\} . $$ where $r$ is the distance between the atoms. $$ D=7 \times 10^{-12} \mathrm{erg} \text {, } $$ $$ \begin{aligned} & a=2 \times 10^8 \mathrm{~cm}^{-1} \\ & r_0=8 \times 10^{-9} \mathrm{~cm} \end{aligned} $$ Estimate the temperatures at which rotation $\left(T_{\mathrm{R}}\right)$ and vibration $\left(T_{\mathrm{V}}\right)$ begin to contribute to the specific heat of hydrogen gas. Give the approximate values of $C_v$ and $C_p$ (the molar specific heats at constant volume and at constant pressure) for the following temperatures: $$ T_1=25 \mathrm{~K}, T_2=250 \mathrm{~K}, T_3=2500 \mathrm{~K}, T_4=10000 \mathrm{~K} . $$ Neglect ionization and dissociation.

The potential energy $V$ between the two atoms ( $\left.m_{\mathrm{H}}=1.672 \times 10^{-24} \mathrm{~g}\right)$ in a hydrogen molecule is given by the empirical expression
$$
V=D\left\{e^{-2 a\left(r-r_0\right)}-2 e^{-a\left(r-r_0\right)}\right\} .
$$
where $r$ is the distance between the atoms.
$$
D=7 \times 10^{-12} \mathrm{erg} \text {, }
$$
$$
\begin{aligned}
& a=2 \times 10^8 \mathrm{~cm}^{-1} \\
& r_0=8 \times 10^{-9} \mathrm{~cm}
\end{aligned}
$$

Estimate the temperatures at which rotation $\left(T_{\mathrm{R}}\right)$ and vibration $\left(T_{\mathrm{V}}\right)$ begin to contribute to the specific heat of hydrogen gas. Give the approximate values of $C_v$ and $C_p$ (the molar specific heats at constant volume and at constant pressure) for the following temperatures:
$$
T_1=25 \mathrm{~K}, T_2=250 \mathrm{~K}, T_3=2500 \mathrm{~K}, T_4=10000 \mathrm{~K} .
$$
Neglect ionization and dissociation.

Key Concepts

Specific Heat in Diatomic Gases

The specific heat capacity of a diatomic gas varies with temperature as different degrees of freedom become active. At very low temperatures, only translational motion contributes. As the temperature increases past the rotational threshold (T_R), rotational degrees of freedom add to the molar specific heat. Further increase beyond the vibrational excitation threshold (T_V) results in vibrational contributions, leading to a higher overall specific heat at constant volume (C_v) and constant pressure (C_p).

Equipartition Theorem

This principle states that at thermal equilibrium, each quadratic degree of freedom contributes (1/2)kBT per molecule to the thermal energy. In the context of molecular gases, it explains how translational, rotational, and vibrational motions contribute to the specific heat, but only when the thermal energy is large compared to the spacing between quantized energy levels. At low temperatures, certain degrees of freedom remain “frozen out” because kBT is insufficient to excite them.

Rotational Energy Quantization

In diatomic molecules, rotational motion is quantized, meaning that the molecule can only occupy discrete energy levels. The energy spacing between these rotational levels depends on the moment of inertia of the molecule. A characteristic temperature, often denoted T_R, can be defined as the energy difference between levels divided by Boltzmann's constant. When the temperature exceeds T_R, rotational states become thermally populated and contribute to the specific heat.

Vibrational Energy Quantization

Vibrational motion in molecules is modeled as a quantized harmonic oscillator. The energy levels of the vibrational mode are spaced by an amount proportional to ??, where ? is the vibrational frequency. The corresponding vibrational temperature T_V is given by this energy spacing divided by Boltzmann’s constant. At temperatures below T_V, vibrational levels are not significantly excited; however, as T approaches T_V, vibrational contributions begin to significantly raise the heat capacity.

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