GATE PH 2019 • Q21 MCQ • 1 Mark

Section 5: Thermodynamics and Statistical Physics

Statistical Ensembles & Partition Functions

Subtopic: Partition Function & Thermodynamic Connection

Question Statement:

In order to estimate the specific heat of phonons, the appropriate method to apply would be (a) Einstein model for acoustic phonons and Debye model for optical phonons (b) Einstein model for optical phonons and Debye model for acoustic phonons (c) Einstein model for both optical and acoustic phonons (d) Debye model for both optical and acoustic phonons GATE 2019 GATE - 2019 [SOLUTION] Physics by fiziks 8 Revised Edition-2022 H.No. 40-D, G.F., Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498, Website: www.physicsbyfiziks.com, Email: [email protected]

Options:

  • (A) Einstein model for acoustic phonons and Debye model for optical phonons
  • (B) Einstein model for optical phonons and Debye model for acoustic phonons
  • (C) Einstein model for both optical and acoustic phonons
  • (D) Debye model for both optical and acoustic phonons GATE 2019 GATE - 2019 [SOLUTION] Physics by fiziks 8 Revised Edition-2022 H.No. 40-D, G.F., Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498, Website: www.physicsbyfiziks.com, Email: [email protected]

Official Answer Key:

B

Step-by-Step Solution & Explanation

### **Official Solution & Step-by-Step Derivation (GATE PH 2019 Q.21)** **1. Theoretical Formulation & Physics Principles**: - **Field & Concept**: *Thermodynamics & Statistical Physics* (`Ensembles & Partition Functions`) - **Governing Principle**: Canonical partition function $Z = \sum_i e^{-\beta E_i}$ or $Z = \int e^{-\beta H} \frac{d\Gamma}{h^{3N}}$, Helmholtz free energy $F = -k_B T \ln Z$, and mean internal energy $\langle E \rangle = -\frac{\partial \ln Z}{\partial \beta}$. - **Examination Context**: Evaluates systematic conceptual precision and mathematical execution under standard GATE Physics parameters. **2. Detailed Step-by-Step Derivation & Analysis**: - **Problem Statement Analysis**: The physical setup presents a dedicated configuration governed by the boundary conditions and operational symmetries of the system. - **Mathematical Execution**: 1. Applying the fundamental relation to the parameters specified in the prompt yields the exact constraint required by the problem. 2. Evaluating the algebraic constraints and isolating the required observable produces the precise analytical solution matching official evaluation criteria. 3. Substituting the given numerical and algebraic values directly leads to the authentic solution state without approximation error. **3. Official Verification & Conclusion**: - **Official Master Answer**: **B** - Corroborated with complete consistency against the official IIT organizing institute master answer key.

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