Question

A monoatomic gas follows a process TV = constant. The ratio of work done to change in internal energy when the gas goes from one state to another is given by k/3 (in magnitude), where k is an integer. Find k.

Answer 1

Given $TV=\text{constant}$
From ideal gas equation $T\propto PV$
$\Rightarrow P{V}^2=\text{constant}$
Polytropic constant $\eta =2$

Change in internal energy $\Delta U=\frac{nR\Delta T}{(\gamma -1)}$
Work done in polytropic process is
$\Delta W=\frac{nR\Delta T}{(1-\eta )}$
$\frac{\Delta W}{\Delta U}=\frac{\frac{nR\Delta T}{(1-\eta )}}{\frac{nR\Delta T}{(\gamma -1)}}=-\frac{2}{3}$
where $\gamma =\frac{5}{3},\eta =2$