Heat and Thermodynamics

Molar heat capacity at constant pressure for a diatomic gas, $C_p=C$We need to find the molar heat capacity at constant volume for a monoatomic gas, $C_V$.Calculations:For a Diatomic Gas:For a diatomic gas, the molar heat capacity at constant volume is:$C_V=\;\frac{5}{2} R$Therefore, the relation between $C_p$ and $C_V$ is:$C_p=C_V+R=\;\frac{5}{2} R+R=\;\frac{7}{2} R$Since $C_p=C$, we have:$\;\frac{7}{2} R=C$Solving for $R$, we get:$R=\;\frac{2C}{7}$For a Monoatomic Gas:The molar heat capacity at constant volume for a monoatomic gas is:$C_V=\;\frac{3}{2} R$Substituting the value of $R$ from Equation (i):$C_V=\;\frac{3}{2} \cdot \;\frac{2C}{7}=\;\frac{3C}{7}$Therefore, the molar specific heat capacity of a monoatomic gas at constant volume is $\;\frac{3C}{7}$.