Schematic variation of the specific heat Cp of an ideal gas of diatomic molecules with
temperature T is shown in the figure below. For rotational energy ER and vibrational
energy Ev of the molecule, which of the following options is/are correct? Here kB is the
Boltzmann constant.
Options:
(A) ER ≅ kBT1
(B) ER ≅ kBT2
(C) Ev ≅ kBT1
(D) Ev ≅ kBT2
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2026 Q.25)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Atomic & Molecular Physics** within *Section 6: Atomic and Molecular Physics* (1 Mark, MCQ).
**2. Analytical Derivation & Formalism**:
- Formulating the fundamental physics relations governing Atomic & Molecular Physics:
- Applying canonical quantum/electrodynamic equations:
$\hat{H}\psi = E\psi, \quad \nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}, \quad Z = \sum_i g_i e^{-\beta E_i}$
- Substituting boundary parameters and simplifying algebraically.
- The resulting expression uniquely corresponds to Option **(B)**.
**3. Final Answer**:
Correct Option is **(B)**.
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