Question

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

• A. $3R$
• B. $\dfrac{7}{2}R$
• C. $4R$
• D. $\dfrac{9}{2}R$

Hint:

Remember: each vibrational mode adds *two* energy contributions—kinetic and potential—equivalent to an extra $R$ in heat capacity.

Answer 1

The Correct Option is C

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, $C_V = \dfrac{9}{2}R$.