Question

The pressure $p$, volume $V$ and temperature $T$ for a certain gas are related by $p = \frac{AT-BT^{2}}{V},$ where $A$ and $B$ are constants. The work done by the gas when the temperature changes from $T_1$ to $T_2$ while the pressure remains constant, is given by

• A. $A\left(T_{2}-T_{1}\right)+B\left(T_{2}^{2} - T^{2}_{1}\right)$
• B. $\frac{A\left(T_{2}-T_{1}\right)}{v_{2}-v_{1}}-\frac{B\left(T_{2}^{2} - T^{2}_{1}\right)}{v_{2}-v_{1}}$
• C. $A\left(T_{2}-T_{1}\right)-B\left(T_{2}^{2} - T^{2}_{1}\right)$
• D. $\frac{A\left(T_{2}-T_{2}^{2}\right)}{v_{2}-v_{1}}$

Answer 1

Answer: (A)

Given, $P=\frac{A T-B T^{2}}{V}$
$\Rightarrow P V =A T-B T^{2}$
$\Rightarrow P \Delta V =A \Delta T-B T \Delta T$
On integrating, we get
Work $=\int P d V=A \int\limits_{T_{1}}^{T_{2}} d T-B \int\limits_{T_{1}}^{T_{2}} T d T$
$=A\left(T_{2}-T_{1}\right)-\frac{B}{2}\left[\left(T_{2}\right)^{2}-\left(T_{1}\right)^{2}\right]$