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GATE 2026 PH – Question 25

Thermodynamics and Statistical Mechanics · partition function, free energy, calculation of thermodynamic quantities · 1 mark · Multiple select

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.

Specific heat rises in two steps near T1 and T2.
  1. $E_R\simeq k_BT_1$
  2. $E_R\simeq k_BT_2$
  3. $E_v\simeq k_BT_1$
  4. $E_v\simeq k_BT_2$

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Correct answer: (A) $E_R\simeq k_BT_1$; (D) $E_v\simeq k_BT_2$

Explanation

Rotational modes become thermally accessible at the lower temperature T1, while vibrational modes require the higher T2. Each onset occurs when kBT is comparable to the relevant level spacing.