An ideal gas consists of three dimensional polyatomic molecules. The temperature is such that only one vibrational mode is excited. If $R$ denotes the gas constant, then the specific heat at constant volume of one mole of the gas at this temperature is
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- $3R$
- $\dfrac{7}{2}R$
- $4R$
- $\dfrac{9}{2}R$
The Correct Option is C
Solution and Explanation
Step 1: Count degrees of freedom.
A polyatomic molecule has:
• 3 translational DOF,
• 3 rotational DOF (for nonlinear molecules),
• Vibrational mode: each contributes 2 DOF (kinetic + potential).
Given: only one vibrational mode is excited ⇒ 2 extra DOF.
Step 2: Total degrees of freedom.
$3 + 3 + 2 = 8$ DOF.
Step 3: Use equipartition theorem.
Each DOF contributes $\dfrac{1}{2}R$ to $C_V$.
Thus,
$C_V = \dfrac{8}{2}R = 4R$.
Step 4: But vibrational energy counts as full $R$ per mode.
A vibrational mode contributes $R$ (not $R/2$), so:
$C_V = \dfrac{6}{2}R + R = 3R + R = \dfrac{9}{2}R$.
Step 5: Conclusion.
Therefore, $C_V = \dfrac{9}{2}R$.
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