The pressure $p$, volume $V$ and temperature $T$ for a certain gas are related by $p=\frac{A T-B T^{2}}{V}$…

The pressure $p$, volume $V$ and temperature $T$ for a certain gas are related by $p=\frac{A T-B T^{2}}{V}$ 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
  1. $A\left(T_{2}-T_{1}\right)+B\left(T_{2}^{2}-T_{1}^{2}\right)$
  2. $\frac{A\left(T_{2}-T_{1}\right)}{V_{2}-V_{1}}-\frac{B\left(T_{2}^{2}-T_{1}^{2}\right)}{V_{2}-V_{1}}$
  3. $ A\left(T_{2}-T_{1}\right)-\frac{B}{2}\left(T_{2}^{2}-T_{1}^{2}\right)$
  4. $\frac{A\left(T_{2}-T_{1}^{2}\right)}{V_{2}-V_{1}}$

Solution

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

Asked in: TEST SERIES MHT-CET Full Test 6

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