Identify degrees of freedom and heat capacities

• Helium is a monoatomic gas. $C_V(\text{He}) = \frac{3}{2}R, \qquad C_P(\text{He}) = \frac{5}{2}R$

• Hydrogen is a diatomic gas (at ordinary temperatures). $C_V(\text{H}_2) = \frac{5}{2}R, \qquad C_P(\text{H}_2) = \frac{7}{2}R$

Write total heat capacities of the mixture

Given:

• helium = $2$ moles
• hydrogen = $n$ moles

So, $C_V^{\text{mix}} = 2\left(\frac{3}{2}R\right) + n\left(\frac{5}{2}R\right)$ $C_V^{\text{mix}} = 3R + \frac{5n}{2}R$

Similarly, $C_P^{\text{mix}} = 2\left(\frac{5}{2}R\right) + n\left(\frac{7}{2}R\right)$ $C_P^{\text{mix}} = 5R + \frac{7n}{2}R$

Use the given ratio

Given, $\frac{C_P^{\text{mix}}}{C_V^{\text{mix}}} = \frac{3}{2}$

Therefore, $\frac{5R + \frac{7n}{2}R}{3R + \frac{5n}{2}R} = \frac{3}{2}$

Cancel $R$: $\frac{5 + \frac{7n}{2}}{3 + \frac{5n}{2}} = \frac{3}{2}$

Solve for $n$

Cross-multiplying: $2\left(5 + \frac{7n}{2}\right) = 3\left(3 + \frac{5n}{2}\right)$

$10 + 7n = 9 + \frac{15n}{2}$

Multiply by $2$: $20 + 14n = 18 + 15n$

$20 - 18 = 15n - 14n$

$n = 2$

Check with options

The correct option is: $\boxed{\text{C: } 2}$