Given that:- the pressure = P
volume = V
Temperature = P
When expanded isothermally to a volume 2V. Pressure = P_I
When expanded adiabatically to a volume 2V. Pressure = P_A.
Ratio of two specific heats = 2
We need to find the ratio P_I/P_A.
Since it is first compressed isothermally ⇒  PV = P_I(2V)
Again when it is compressed adiabatically. ${\mathrm{P}}_{\mathrm{I}}=\frac{\mathrm{P}}{2}$ 

$\Rightarrow {\mathrm{Pv}}^{\mathrm{\gamma }}-{\mathrm{P}}_{\mathrm{A}}(\mathrm{v}{)}^{\mathrm{\gamma }}={\mathrm{P}}_{\mathrm{A}}=\frac{\mathrm{P}}{2^{\mathrm{r}}}$ 

$\frac{{\mathrm{P}}_{\mathrm{I}}}{{\mathrm{P}}_{\mathrm{A}}}=\frac{\frac{\mathrm{P}}{2}}{\frac{\mathrm{P}}{2\mathrm{r}}}=\frac{\mathrm{p}}{2}\times \frac{2^{\mathrm{\gamma }}}{\mathrm{P}}{2}^{2^{\mathrm{\gamma }-1}}$ 

2^γ−1 is the correct.