For an ideal gas $(\frac{C_P}{C_v}=\gamma )$; of molar mass M, its specific heat capacity at constant volume is:

The correct option is not given in the solution

Given that$C_p-C_v=R$ and $\frac{C_p}{C_v}=\text{γ}$

Let's calculate the specific heat capacity

$\frac{C_p}{C_v}=\text{γ}$

$C_p-C_v=R⟶\left(1\right)$

$C_v=\frac{C_p}{\text{γ}}⟶\left(2\right)$

$Put\left(2\right)in\left(1\right)$

$C_p-\frac{C_p}{\text{γ}}=R$

$\frac{\text{γ}{C}_p-C_p}{\text{γ}}=R$

$\frac{\text{C\_p}(\gamma -\text{1})}{\text{γ}}=R$

$C_p=\frac{\text{γ}R}{\text{γ}-1}$

$C_p=\text{γ}{C}_v=\frac{\text{γ}R}{\text{γ}-1}\therefore ⟹C_v=\frac{R}{\text{γ}-1}$

We also have an equation:

$(C_p-C_v)M=R$

$C_p-C_v=\frac{R}{M}$

or we can simply written the specific heat capacity at constant volume as:

$C_v=\frac{R}{M(\text{γ}-1)}$