Question

Pressure P, volume V and temperature T for a certain gas are related by $P=\frac{\alpha T-\beta T^2}{V}$ where $\alpha$ and $\beta$ constants. Find the work done by the gas if the temperature change from $T_1$ to $T_2$ while the pressure remains the constant.

Answer 1

The given relation is $P=\frac{\alpha T-\beta T^2}{V}$

$⟹$ $V=\frac{\alpha T-\beta T^2}{P}$

Differentiating we get, $dV=\frac{\alpha -2\beta T}{P}dT$ ............(1)

Work done by the gas $W=\int PdV$

$\therefore$ $W={\int }_{T_1}^{T_2}(\alpha -2\beta T)dT$

$⟹$ $W=\alpha T{|}_{T_1}^{T_2}-2\beta \times \frac{T^2}{2}{|}_{T_1}^{T_2}=\alpha (T_2-T_1)-\beta (T_2^2-T_1^2)$