KVPY Sample Paper KVPY Stream-SX Model Paper-14

An ideal gas undergoes an adiabatic process in which internal energy U is \[U=a+b\,(pV),\]where a and b are constants. Value of \[\gamma \] for this process is

A) \[\frac{b}{1+b}\]

B) \[b\]

C) \[\frac{b+1}{b+2}\]

D) \[1+\frac{1}{b}\]

Correct Answer: D

Solution :

As process is adiabatic, \[\Delta Q=0.\]

\[\Rightarrow \]\[dU+pdV=0\]

\[\Rightarrow \]\[d\,(a+bpV)+pdV=0\]

\[\Rightarrow \]\[(b+1)\frac{dV}{V}=-b\frac{dp}{p}\]

Integrating and rearranging, we get

\[\Rightarrow \]\[(b+1)\ln V+b\ln p=\text{constant}\]

\[\Rightarrow \]\[p{{V}^{\left( \frac{b+1}{b} \right)}}=a\,\text{constant}\]

\[\Rightarrow \]\[\gamma =\frac{b+1}{b}=1+\frac{1}{b}\]