RE-NEET 2026 Physics Degrees of Freedom MCQ Question

To solve the problem involving an ideal gas composed of polyatomic molecules and the ratio of heat capacities $\frac{C_p}{C_v} = \frac{8}{7}$, we start by recalling some key concepts from thermodynamics, particularly the degrees of freedom and the relationship between heat capacities.

Step 1: Understanding Degrees of Freedom

For a polyatomic ideal gas, the degrees of freedom can be categorized as:

• Translational degrees of freedom (3): Each molecule can move in three dimensions (x, y, z).
• Rotational degrees of freedom (3): A non-linear polyatomic molecule can rotate about three perpendicular axes.
• Vibrational degrees of freedom: For a polyatomic molecule with $f$ vibrational modes, each mode contributes an additional two degrees of freedom (one for potential energy and one for kinetic energy).

Thus, the total degrees of freedom $D$ for a polyatomic molecule can be expressed as: $D = 3 + 3 + 2f = 6 + 2f$

Step 2: Relating Degrees of Freedom to Heat Capacities

For an ideal gas, the molar heat capacities are related to the degrees of freedom by the following equations, derived from the equipartition theorem:

• The heat capacity at constant volume $C_v$ is given by: $C_v = \frac{D}{2} R$
• The heat capacity at constant pressure $C_p$ is given by: $C_p = C_v + R$

Using the above relationships:

1. Substitute $C_v$: $C_p = \frac{D}{2} R + R = \left( \frac{D}{2} + 1 \right) R$

2. The ratio of heat capacities can then be expressed as: $\frac{C_p}{C_v} = \frac{\left( \frac{D}{2} + 1 \right) R}{\frac{D}{2} R} = \frac{\frac{D}{2} + 1}{\frac{D}{2}} = \frac{D + 2}{D}$

Step 3: Solving for $f$

Given that $\frac{C_p}{C_v} = \frac{8}{7}$: $\frac{D + 2}{D} = \frac{8}{7}$

Cross multiplying to eliminate the fraction: $7(D + 2) = 8D$ Expanding the left side: $7D + 14 = 8D$ Rearranging the equation: $14 = 8D - 7D$ $14 = D$

Step 4: Finding $f$

Now, we can substitute back for $D$: $D = 6 + 2f$ Setting this equal to 14, we have: $6 + 2f = 14$ Solving for $f$: $2f = 14 - 6$ $2f = 8$ $f = 4$

Thus, the value of $f$ is $4$.

Conclusion

The correct answer is D) 4.

Clarifying Incorrect Options

• Option A) 3: This would imply $D = 12$, leading to a different (incorrect) ratio of heat capacities.
• Option B) 2: This would imply $D = 10$, also incorrect.
• Option C) 1: This would imply $D = 8$, leading to yet another incorrect ratio.

Therefore, $f = 4$ is indeed the correct answer, as it satisfies the given condition of the heat capacities ratio $\frac{C_p}{C_v} = \frac{8}{7}$.