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$${\text{P}}{{\text{V}}^\gamma }$$ = Constant (where, $$\gamma = \frac{{{{\text{C}}_{\text{p}}}}}{{{{\text{C}}_{\text{v}}}}}$$ ) is valid for a/an __________ process.
Answer & Solution
Answer: Option
D
Solution:
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)
\[\begin{array}{l} {C_V}dT = - PdV\\ \Rightarrow {C_V}dT = - \frac{{RT}}{{V\left( {dV} \right)}}\\ \Rightarrow \frac{{dT}}{T} = - \frac{R}{{{C_V}}}\frac{{dV}}{V}\\ \Rightarrow T{V^{\gamma - 1}} = {\rm{CONSTANT}}\\ {\rm{Or,\, }}P{V^\gamma } = {\rm{CONSTANT}} \end{array}\]
(Note : this equation can only be used for ideal gas and when the heat capacities are remaining constant)