$${\text{P}}{{\text{V}}^\gamma }$$  = Constant (where, $$\gamma = \frac{{{{\text{C}}_{\text{p}}}}}{{{{\text{C}}_{\text{v}}}}}$$  ) is valid for a/an __________ process.

Solution (By Examveda Team)

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

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