Pressure P, volume V and temperature T of a certain gas are related by $P = \frac{{αT}^{2}}{V}$ . Here, α is a constant. The work done by the gas when temperature changes from T₀ to $2 T_{0}$ while pressure remains constant is

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$2 {αT}_{0}^{2}$

$3 {αT}_{0}^{2}$

$\frac{3}{2} {αT}_{0}^{2}$

$6 {αT}_{0}^{3}$

answer is D.

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Detailed Solution

$\begin{array}{l} p = \frac{{αT}^{2}}{V} \\ ∴ V = \frac{{αT}^{2}}{p} or dV = (\frac{2 αT}{p}) dT \\ W = \int_{T_{0}}^{2 T_{0}} pdV \\ = \int_{T_{0}}^{2 T_{0}} (p) (\frac{2 αT}{p}) dT = 3 {αT}_{0}^{2} \end{array}$

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