Identify degrees of freedom and molar specific heats

For an ideal gas, $C_V = \frac{f}{2}R$ where $f$ is the number of degrees of freedom.

• Helium is a monatomic gas, so $f_{\text{He}} = 3$ $C_{V,\text{He}} = \frac{3}{2}R$

• Hydrogen molecules are stated to be rigid, so vibrational modes are ignored. A rigid diatomic molecule has $f_{\text{H}_2} = 5$ $C_{V,\text{H}_2} = \frac{5}{2}R$

Calculate total heat capacity at constant volume

Given:

• $2$ moles of helium
• $3$ moles of hydrogen

Total heat capacity of the mixture: $C_{V,\text{total}} = 2\left(\frac{3}{2}R\right) + 3\left(\frac{5}{2}R\right)$

$= 3R + \frac{15}{2}R$

$= \frac{6}{2}R + \frac{15}{2}R = \frac{21}{2}R$

Find molar specific heat of the mixture

Total number of moles: $n_{\text{total}} = 2 + 3 = 5$

Therefore, molar specific heat of mixture at constant volume is $C_{V,\text{mix}} = \frac{C_{V,\text{total}}}{n_{\text{total}}} = \frac{\frac{21}{2}R}{5} = \frac{21}{10}R = 2.1R$

Substitute $R = 8.3\,\text{J/mol K}$

$C_{V,\text{mix}} = 2.1 \times 8.3 = 17.43\,\text{J/mol K}$

$\boxed{C_{V,\text{mix}} \approx 17.4\,\text{J/mol K}}$

Match with options

The correct option is: $\boxed{\text{B: }17.4\,\text{J/mol K}}$