Find the minimum attainable pressure of an ideal gas in the process $T=T_0+\alpha V^2$,  where$T_0$and $\alpha$ are positive constants and V is the volume of one mole of gas.

$p=\frac{RT}{V} (n=1)$

or $p=\frac{R}{V}\left(T_0+\alpha V^2\right)$               ...(i)

For minimum attainable pressure

$\frac{dp}{dV}=0$

or $\frac{-R{T}_0}{V^2}+\alpha R=0$

or

$V=\sqrt{\frac{T_0}{\alpha }}$

At this volume we can see that $\frac{d^{2}p}{d{V}^2}$ is positive or p is minimum.

From Eq. (i)

$p_{\min }=2R\sqrt{\alpha T_0}$