Q. The ratio of specific heat of a gas at constant pressure to that at constant volume is $\gamma$. The change in internal energy of one mole of gas when volume change from $V$ to $2V$ at constant pressure $p$ is
A) $R/(\gamma - 1)$
B) $pV$
C) $pV/(\gamma - 1)$
D) $\frac{\gamma V}{\gamma - 1}$
Explanation
(c) As $C_{p}/C_{V} = \gamma$
$\therefore\frac{C_{p} - C_{v}}{C_{v}}\gamma - 1$
or $C_{v} = \frac{C_{p} - C_{v}}{\gamma - 1} = \frac{R}{\gamma - 1}$
$\mathrm{\Delta}U = nC_{v}dT = n\frac{RdT}{(\gamma - 1)} = \frac{npdV}{\gamma - 1}$
$= \frac{np(2V - V)}{\gamma - 1} = \frac{npV}{\gamma - 1}$
As $n = 1$,
$\therefore\ \mathrm{\Delta}U = \frac{pV}{(\gamma - 1)}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.