An ideal gas is in thermodynamic equilibrium. The number of degrees of freedom of a molecule of the gas is 𝑛. The internal energy of one mole of the gas is 𝑈𝑛 and the speed of sound in the gas is v𝑛. At a fixed temperature and pressure, which of the following is the correct option? (A) v3 < v6 and 𝑈3 > 𝑈6 (B) v5 > v3 and 𝑈3 > 𝑈5 (C) v5 > v7 and 𝑈5 < 𝑈7 (D) v6 < v7 and 𝑈6 < 𝑈7

To determine the correct option, we need to analyze the relationship between the degrees of freedom $n$, the internal energy $U_n$, and the speed of sound $v_n$ in an ideal gas.

1. Internal Energy $U_n$:

For an ideal gas, the internal energy $U_n$ is given by:

$$U_n=\frac{n}{2}RT$$

where:

• $n$ is the number of degrees of freedom,

• $R$ is the universal gas constant,

• $T$ is the temperature.

From this equation, it is clear that internal energy increases with the number of degrees of freedom $n$. Therefore:

$$U_3<U_5<U_6<U_7$$

2. Speed of Sound $v_n$:

The speed of sound $v_n$ in an ideal gas is given by:

$$v_n=\sqrt{\frac{\gamma RT}{M}}$$

where:

• $\gamma =\frac{C_p}{C_v}=1+\frac{2}{n}$ is the adiabatic index,

• $R$ is the universal gas constant,

• $T$ is the temperature,

• $M$ is the molar mass of the gas.

The adiabatic index $\gamma$ decreases as the number of degrees of freedom $n$ increases. Specifically:

$$\gamma =1+\frac{2}{n}$$

Thus, as $n$ increases, $\gamma$ decreases, and the speed of sound $v_n$ decreases. Therefore:

$$v_3>v_5>v_6>v_7$$

3. Analyzing the Options:

• Option (A): $v_3<v_6$ and $U_3>U_6$.
  This is incorrect because $v_3>v_6$ and $U_3<U_6$.

• Option (B): $v_5>v_3$ and $U_3>U_5$.
  This is incorrect because $v_5<v_3$ and $U_3<U_5$.

• Option (C): $v_5>v_7$ and $U_5<U_7$.
  This is correct because $v_5>v_7$ and $U_5<U_7$.

• Option (D): $v_6<v_7$ and $U_6<U_7$.
  This is incorrect because $v_6>v_7$ and $U_6<U_7$.

Final Answer:

$$\boxed{{\displaystyle C}}$$