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.
Show Hint
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: Understanding the Concept: A diatomic molecule stores heat in translational, rotational and vibrational motion, but each type of motion only switches on once the thermal energy \(k_BT\) becomes comparable to the spacing between that motion's quantum energy levels. Translational levels are essentially continuous, so translation contributes at all temperatures. Rotational levels are closely spaced, so they switch on at a fairly low temperature. Vibrational levels are widely spaced, so they need a much higher temperature to switch on. This is exactly why the graph shows two separate step-rises in \(C_p\) rather than one smooth rise.
Step 2: Key Formula or Approach: The rule of thumb from statistical mechanics is that a degree of freedom becomes thermally active once \(k_BT\) grows to about the size of its energy level spacing. So the temperature at which \(C_p\) starts to rise for a given motion marks \(k_BT \cong (\text{energy spacing for that motion})\).
Step 3: Assign the first step to rotation. Since \(T_1\) is the lower temperature (the first step-rise happens near \(T_1\)), and rotational levels are always more closely spaced than vibrational levels for a real diatomic molecule, the first rise in \(C_p\) must be rotational motion switching on. This means the rotational energy spacing satisfies \(E_R \cong k_BT_1\). So (A) is TRUE, and by the same reasoning (B), which pairs \(E_R\) with the higher temperature \(T_2\), is FALSE.
Step 4: Assign the second step to vibration. The second, higher-temperature step-rise near \(T_2\) must be vibrational motion switching on, since vibration needs much more thermal energy to activate than rotation does. This means the vibrational energy spacing satisfies \(E_v \cong k_BT_2\). So (D) is TRUE, and (C), which pairs \(E_v\) with the lower temperature \(T_1\), is FALSE, since that would wrongly put vibration at a lower activation temperature than rotation.
Final Answer: Rotational motion switches on near \(T_1\) and vibrational motion switches on near \(T_2\), so \(E_R \cong k_BT_1\) and \(E_v \cong k_BT_2\).\[ \boxed{E_R \cong k_BT_1,\ E_v \cong k_BT_2} \]
