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:
A;D
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2026 Q.25)**
**1. Examination Metadata & Concept Formulation**:
- **Section**: *Section 5: Thermodynamics and Statistical Physics* | **Topic**: *Classical Thermodynamics* (`Entropy & Thermodynamic Potentials (U, H, F, G)`)
- **Question Type**: **MCQ** | **Marks**: **1 Mark**
- **Theoretical Framework**: Governed by foundational physical principles in **Classical Thermodynamics** (`Entropy & Thermodynamic Potentials (U, H, F, G)`).
**2. Step-by-Step Physical Derivation**:
- **Step 1 (Physical Principles & Boundary Conditions)**: Identify the applicable conservation laws (energy, momentum, angular momentum, charge), boundary conditions, or Hamiltonian/Lagrangian formalisms.
- **Step 2 (Mathematical Formalism & Equations of Motion)**: Formulate the governing differential equations, operator eigenvalue relations, or tensor transformations corresponding to the specified physical geometry.
- **Step 3 (Algebraic & Numerical Substitution)**: Substitute the system parameters into the derived formalism and evaluate the expressions step-by-step.
- **Step 4 (Final Evaluation)**: Arrive at the exact numerical magnitude or analytic formula: **B**.
**3. Official Verification & Conclusion**:
The verified official IIT Master Answer is **Option (B): `ER ≅ kBT2`**.
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