Q. Pressure $p$, volume $V$ and temperature $T$ of a certain material are related by $p = \alpha T^{2}/V$, where $\alpha$ is constant. Work done by the material when temperature changes from $T_{2}$ to $2T_{0}$ and pressure remains constant is
A) $3\alpha T_{0}^{2}$
B) $5\alpha T_{0}^{2}$
C) $\frac{3}{2}\alpha T_{0}^{2}$
D) $7\alpha T_{0}^{2}$
Explanation
(a) The given relation is $p = \frac{\alpha T^{2}}{V}\ \ \ \therefore\ \ V = \frac{\alpha T^{2}}{P}$
As pressure is kept constant, $dV = \left( \frac{2\alpha T}{P} \right)dT$
$W = \int_{}^{}{p\ dV = \int_{T_{0}}^{2T_{0}}{p\left( \frac{2\alpha T}{p} \right)dT}}$
$= 2\alpha\left\lbrack \frac{T^{2}}{2} \right\rbrack_{T_{0}}^{2T_{0}} = 3\alpha T_{0}^{2}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.