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GATE 2026 PH – Question 45

Thermodynamics and Statistical Mechanics · partition function, free energy, calculation of thermodynamic quantities · 2 marks · Multiple choice

A gas of N classical particles that can occupy energy levels, ε1 and ε2 = ε1 + Δ is in equilibrium with a reservoir at temperature T. From the schematics shown below, choose the correct dependence of the internal energy U on T .

Four candidate temperature dependences of internal energy.
  1. Option A in the figure
  2. Option B in the figure
  3. Option C in the figure
  4. Option D in the figure

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Show answer and explanation

Correct answer: (A) Option A in the figure

Explanation

$U=N\epsilon_1+N\Delta/(1+e^{\Delta/k_BT})$. It starts at $N\epsilon_1$ and approaches $N(\epsilon_1+\Delta/2)$ with a vanishing slope at high temperature, matching A.