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 U_n and the speed of sound in the gas is v_n. At a fixed temperature and pressure, which of the following is the correct option?

Correct Answer: v₅ > v₇ and U₅ < U₇

To determine the correct relationship between the speed of sound in the gas (v_n) and the internal energy of one mole of the gas (U_n) for different degrees of freedom (n), we need to analyze their theoretical expressions at a fixed temperature and pressure.

1. Internal Energy of One Mole of Gas (U_n):
According to the law of equipartition of energy, the internal energy of one mole of an ideal gas with n degrees of freedom at temperature T is given by:

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

where R is the universal gas constant and T is the absolute temperature.
Since temperature T is fixed, the internal energy U_n is directly proportional to the number of degrees of freedom n:

$U_n\propto n$

Therefore, as degrees of freedom n increase, internal energy U_n increases.
Comparing for n = 5 and n = 7:

$U_5=\frac{5}{2}RT$

$U_7=\frac{7}{2}RT$

Thus, we clearly have U₅ < U₇.

2. Speed of Sound in Gas (v_n):
The speed of sound in an ideal gas is given by Laplace's formula:

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

where γ is the adiabatic index (ratio of specific heats $\gamma =\frac{C_p}{C_v}$) and M is the molar mass.
The adiabatic index γ in terms of the degrees of freedom n is given by:

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

As n increases, $\frac{2}{n}$ decreases, which means γ decreases.
Since $v_n\propto \sqrt{\gamma }$, a smaller value of γ corresponds to a smaller speed of sound v_n.

Let us calculate γ for n = 5 and n = 7:

For n = 5: ${\gamma }_5=1+\frac{2}{5}=1.4$

For n = 7: ${\gamma }_7=1+\frac{2}{7}\approx 1.286$

Since γ₅ > γ₇, it follows that v₅ > v₇.

Conclusion:
Combining both results, we get:
v₅ > v₇ and U₅ < U₇.