The P-V relation for monoatomic ideal gas undergoing an adiabatic change is

short cut    

$P{V}^{\mathrm{γ}}=CONSTANT$

ALTERNATE METHOD: We know that 
$\mathrm{E}=\frac{3}{2}\mathrm{NkT} \mathrm{or} \mathrm{dE}=\frac{3}{2}\mathrm{NkdT}$
Further, $\mathrm{dW}=\mathrm{P}⋠\mathrm{dV}=\frac{\mathrm{NkT}}{\mathrm{V}}⋠\mathrm{dV}$
For Adiabatic change dE = - dW. Hence
$\frac{3}{2}\mathrm{NkdT}=−\frac{\mathrm{NkT}}{\mathrm{V}}⋠\mathrm{dV}$
or     $\frac{\mathrm{dT}}{\mathrm{T}}=−\frac{2}{3}\frac{\mathrm{dV}}{\mathrm{V}}$     or    $\frac{\mathrm{dT}}{\mathrm{T}}+\frac{2}{3}\frac{\mathrm{dV}}{\mathrm{V}}=0$
$∴\log ⁡\mathrm{T}+\frac{2}{3}\log ⁡\mathrm{V}= \mathrm{constant}$
or    ${\mathrm{TV}}^{2/3}= \mathrm{constant}$
or     ${\mathrm{PV}}^{5/3}= \mathrm{constant} (\mathrm{Using} \mathrm{ideal} \mathrm{gas} \mathrm{law})$