P V^ = Constant (where, = C_ p C_ v ) is valid for a/an __________ process.

Answer: Option D

For an adiabatic process, by first law of thermodynamics for an ideal gas under adiabatic conditions we can write :

$$\begin{array}{l}{C}_{V}dT=-PdV\\ \Rightarrow C_{V}dT=-\frac{RT}{V\left(dV\right)}\\ \Rightarrow \frac{dT}{T}=-\frac{R}{C_V}\frac{dV}{V}\\ \Rightarrow T{V}^{\gamma -1}=\mathrm{CONSTANT}\\ \mathrm{Or},P{V}^{\gamma }=\mathrm{CONSTANT}\end{array}$$
(Note : this equation can only be used for ideal gas and when the heat capacities are remaining constant)