Schematic variation of the specific heat \(C_p\) of an ideal gas of diatomic molecules with temperature \(T\) is shown in the figure below. For rotational energy \(E_R\) and vibrational energy \(E_v\) of the molecule, which of the following options is/are correct? Here \(k_B\) is the Boltzmann constant.
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The lower-temperature step in the Cp graph turns on the closely spaced rotational levels, and the higher-temperature step turns on the widely spaced vibrational levels, so ER matches T1 and Ev matches T2.
Step 1: Compare typical energy scales directly, without appealing to the switch-on rule first. For a diatomic molecule, the rotational energy levels come from $E_{rot} \sim \hbar^2/(2I)$, where $I$ is the moment of inertia, set by the comparatively large bond length and atomic masses. The vibrational energy levels come from $E_{vib} \sim \hbar\sqrt{k/\mu}$, where $k$ is the comparatively stiff bond spring constant. Because real molecular bonds are stiff springs with light reduced mass $\mu$, but the moment of inertia $I$ is large, vibrational level spacing works out to be much larger than rotational level spacing for common diatomic gases.
Step 2: Translate this energy-scale gap into a temperature gap. Since a mode needs $k_BT$ of about its own level spacing before it can absorb heat and add to $C_p$, a mode with small level spacing turns on at low $T$, and a mode with large level spacing needs high $T$. Given that vibrational spacing is much larger than rotational spacing, the vibrational mode must always turn on at a noticeably higher temperature than the rotational mode.
Step 3: Read the two step temperatures off the graph in that light. The graph gives two step temperatures, $T_1 < T_2$. Since rotation has the smaller level spacing, it must be the one that switches on at the lower temperature $T_1$; vibration, having the larger spacing, is forced to be the one switching on at the higher temperature $T_2$. There is no way to assign it the other way round, because that would need rotation to have larger level spacing than vibration, which contradicts the physical picture of a stiff bond vibrating fast but rotating slowly.
Step 4: Write the matching energy estimates. So the rotational spacing is set by the lower step: $E_R \cong k_BT_1$. The vibrational spacing is set by the higher step: $E_v \cong k_BT_2$. Statements (A) and (D) both hold, while (B) and (C) swap the temperatures incorrectly.
Final Answer: The physical size argument for level spacings agrees with the switch-on argument: rotational energy goes with $T_1$, vibrational energy goes with $T_2$.\[ \boxed{E_R \cong k_BT_1,\ E_v \cong k_BT_2} \]
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