1. Given data

• Number of moles: $n=1$
• Gas is rigid diatomic? Here “rigid diatomic gas” means a diatomic ideal gas with no vibrational modes active, so $C_V=\frac{5R}{2}$
• Heat supplied: $Q$
• Work done by gas: $W=\frac{Q}{5}$

We need the molar heat capacity during this process.

2. Use the first law of thermodynamics

For the heat supplied, $Q=\Delta U+W$ So, $\Delta U=Q-W=Q-\frac{Q}{5}=\frac{4Q}{5}$

3. Relate internal energy change to temperature change

For 1 mole of a diatomic ideal gas, $\Delta U=nC_V\Delta T=\frac{5R}{2}\Delta T$ Thus, $\frac{5R}{2}\Delta T=\frac{4Q}{5}$

Solve for $\Delta T$: $\Delta T=\frac{4Q}{5}\cdot \frac{2}{5R}=\frac{8Q}{25R}$

4. Find the molar heat capacity for the process

By definition, $C=\frac{Q}{n\Delta T}$ Since $n=1$, $C=\frac{Q}{\Delta T}=\frac{Q}{\frac{8Q}{25R}}=\frac{25R}{8}$

Given that $C=\frac{xR}{8}$ we compare: $\frac{xR}{8}=\frac{25R}{8}$ Hence, $x=25$

5. Comparison with stored answer

Derived answer: $25$

Stored correct answer: $25$

They match.