Internal Energy and Speed of Sound Dependence on Degrees of Freedom

To determine the correct relationship between internal energy $U_n$ and the speed of sound $\text{v}_n$ for an ideal gas with $n$ degrees of freedom at a fixed temperature $T$ and pressure $P$:

1. Internal Energy ($U_n$): The internal energy of one mole of an ideal gas having $n$ degrees of freedom at temperature $T$ is given by: $U_n = \frac{n}{2}RT$ Since the temperature $T$ is fixed, the internal energy $U_n$ is directly proportional to the degrees of freedom $n$: $U_n \propto n$ Therefore, as $n$ increases, $U_n$ increases. Comparing the given values: $U_3 < U_5 < U_6 < U_7$

2. Speed of Sound ($\text{v}_n$): The adiabatic index (ratio of specific heats) $\gamma$ for a gas with $n$ degrees of freedom is: $\gamma = 1 + \frac{2}{n}$ The speed of sound in an ideal gas at a given temperature $T$ is: $\text{v}_n = \sqrt{\frac{\gamma R T}{M}}$ where $M$ is the molar mass of the gas. For a fixed temperature (and considering the standard dependence on degrees of freedom), the speed of sound depends on $\gamma$ as: $\text{v}_n \propto \sqrt{\gamma} = \sqrt{1 + \frac{2}{n}}$ As the number of degrees of freedom $n$ increases, the ratio $\gamma$ decreases, and consequently, the speed of sound $\text{v}_n$ decreases: $\text{v}_3 > \text{v}_5 > \text{v}_6 > \text{v}_7$

3. Evaluation of the Options:

• Option A: $\text{v}_3 < \text{v}_6$ and $U_3 > U_6$ $\implies$ Incorrect (both are false).
• Option B: $\text{v}_5 > \text{v}_3$ and $U_3 > U_5$ $\implies$ Incorrect (both are false).
• Option C: $\text{v}_5 > \text{v}_7$ and $U_5 < U_7$ $\implies$ Correct ($\text{v}_5 > \text{v}_7$ since $\gamma_5 > \gamma_7$, and $U_5 < U_7$ since $5 < 7$).
• Option D: $\text{v}_6 < \text{v}_7$ and $U_6 < U_7$ $\implies$ Incorrect ($\text{v}_6 > \text{v}_7$).

Hence, the correct option is C.