Question

$P-V$ graph for an ideal gas undergoing a polytropic process given by $P{V}^n=k$ is shown below. Find the value of $n$.

Answer 1

Slope of the given $P-V$ curve is $\frac{dP}{dV}=\text{tan}(180-37)=\text{tan}{143}^{\circ }=-0.75$
Given equation of polytropic process is $P{V}^n=k$
Differentiating on both sides, we get
$P\times \left(n{V}^{n-1}\right)dV+V^{n}dP=0$
$\Rightarrow nP{V}^{n-1}dV=-V^{n}dP$
$\Rightarrow \frac{dP}{dV}=-\frac{nP}{V}$
Substituting the data given in the question, we get
$-0.75=-\frac{n\times 2\times {10}^5}{4\times {10}^5}$
$\Rightarrow n=1.5$