1. Internal energy of one mole of an ideal gas

For an ideal gas with $n$ degrees of freedom, the internal energy of $1$ mole is

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

At fixed temperature $T$, we have

$U_n \propto n$

So,

$U_3 < U_5 < U_6 < U_7$

2. Speed of sound in an ideal gas

The speed of sound is

$v = \sqrt{\gamma \frac{P}{\rho}}$

Using $\rho = \frac{PM}{RT}$,

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

At fixed temperature and pressure, for the same gas family comparison here, dependence is through $\gamma$.

For an ideal gas with $n$ degrees of freedom,

$C_V = \frac{n}{2}R, \qquad C_P = C_V + R = \frac{n+2}{2}R$

Hence,

$\gamma = \frac{C_P}{C_V} = \frac{n+2}{n}$

Therefore,

$v_n \propto \sqrt{\gamma} = \sqrt{\frac{n+2}{n}}$

Now check values:

$\gamma_3 = \frac{5}{3}, \qquad \gamma_5 = \frac{7}{5}, \qquad \gamma_6 = \frac{8}{6} = \frac{4}{3}, \qquad \gamma_7 = \frac{9}{7}$

As $n$ increases, $\gamma$ decreases. Therefore,

$v_3 > v_5 > v_6 > v_7$

3. Check each option

Option A: $v_3 U_6$

This is not a complete relation/equality/inequality, so it is not a meaningful correct option.

Option B: $v_5 > v_3$ and $U_3 > U_5$

But we found

$v_5 < v_3$

and

$U_3 < U_5$

So option B is false.

Option C: $v_5 > v_7$ and $U_5 < U_7$

From above,

$v_5 > v_7$

and

$U_5 < U_7$

So option C is true.

Option D: $v_6 < v_7$ and $U_6 < U_7$

We found

$v_6 > v_7$

though n$U_6 < U_7$

Since the first part is false, option D is false.

4. Final answer

The correct option is

$\boxed{\text{C}}$

This matches the stored correct answer.