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.

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Solution

Given $T V = constant$
From ideal gas equation $T \propto P V$
$\Rightarrow P V^{2} = constant$
Polytropic constant $η = 2$

Change in internal energy $Δ U = \frac{n R Δ T}{(γ - 1)}$
Work done in polytropic process is
$Δ W = \frac{n R Δ T}{(1 - η)}$
$\frac{Δ W}{Δ U} = \frac{\frac{n R Δ T}{(1 - η)}}{\frac{n R Δ T}{(γ - 1)}} = - \frac{2}{3}$
where $γ = \frac{5}{3}, η = 2$

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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.

Given TV=constant From ideal gas equation T∝PV ⇒PV2=constant Polytropic constant η=2 Change in internal energy ΔU=nRΔT(γ−1) Work done in polytropic process is ΔW=nRΔT(1−η) ΔWΔU=nRΔT(1−η)nRΔT(γ−1)=−23 where γ=53,η=2