Chemistry - Thermodynamics - I Question with Solution | TestHub

Question Explanation:

Calculate the work performed by gas in a polytropic process given the relation $\mathrm{U}=\alpha \sqrt{\mathrm{V}}$. Concept:

Work done in a polytropic process

Solution:

$\mathrm{U}=\alpha \sqrt{\mathrm{V}}\mathrm{U}={\mathrm{nC}}_{\mathrm{V}}\mathrm{T}=\alpha \sqrt{\mathrm{V}}$

$\mathrm{T}=\frac{\mathrm{PV}}{\mathrm{nR}}$

$\frac{{\mathrm{C}}_{\mathrm{V}}}{\mathrm{R}}\cdot \mathrm{PV}=\alpha \sqrt{\mathrm{V}}$

$\mathrm{P}=\left[\frac{\alpha \mathrm{R}}{{\mathrm{C}}_{\mathrm{V}}}\right]\frac{1}{\sqrt{\mathrm{V}}}$

$\mathrm{W}=-\int \mathrm{PdV}=-\left(\frac{\alpha \mathrm{R}}{{\mathrm{C}}_{\mathrm{V}}}\right)\int \frac{\mathrm{dV}}{\sqrt{\mathrm{V}}}$

$=2\alpha (\gamma -1)\left[\sqrt{{\mathrm{V}}_2}-\sqrt{{\mathrm{V}}_1}\right]$

Final Answer: Option (A)