Question

During an adiabatic process, if the pressure of a gas is found to be proportional to the cube of its absolute temperature, then the ratio of \(\frac{C_{P}}{C_{V}}\) for the gas is :

(1) \(\frac{5}{3}\)

(2) \(\frac{9}{7}\)

(3) \(\frac{3}{2}\)

(4) \(\frac{7}{5}\)    

Answer 1

Correct option is : (3) \( \frac{3}{2}\)

\(\mathrm{P} \propto \mathrm{T}^{3}\)

\(\mathrm{PT}^{-3}=\) constant

\(\because \ \frac{\mathrm{PV}}{\mathrm{T}}=\mathrm{nR}=\) constant from ideal gas equation

(P) \((\mathrm{PV})^{-3}=\) constant

\(\mathrm{P}^{-2} \mathrm{~V}^{-3}=\) cosntant    .......(1)

\(\because\) Process equation for adiabatic process is \(\mathrm{PV}^{\mathrm{y}}=\) constant    ...... (2)

Comparing equation (1) and (2)

\(\frac{\mathrm{C}_{\mathrm{P}}}{\mathrm{C}_{\mathrm{V}}}=\mathrm{y}=\frac{3}{2}\)