Question

A cylinder having a piston contains $1\text{kg}$ of methane gas at $100\text{kPa}$ and ${20}^{\circ }C$. The gas is compressed reversibly to a pressure of $800\text{kPa}$. If the process is polytropic with exponent $n=1.15$, then the work required (in $\text{kJ}$) is (nearest integer value)
[Take $R$ for methane $=0.518\text{kJ/kg K}$ and $8^{0.13}=1.31$]

Answer 1

For a polytropic process,
e know that $P{V}^x$ = constant $..........\left(1\right)$
Given, $x=\mathrm{1.15.}$
To write $\left(1\right)$ in terms of $T$ and $P$ we use ideal gas equation $PV=nRT$
From this we get,
$P^{1-x}{T}^x=$ constant
$\Rightarrow \frac{T_1}{P_1^{\left(\frac{x-1}{x}\right)}}=\frac{T_2}{P_2^{\left(\frac{x-1}{x}\right)}}$
From the data given in the question we get,
$T_2=293\times {\left(\frac{800}{100}\right)}^{\frac{0.15}{1.15}}$
$=293\times 1.31$
$=384\text{K}$
We know that work done during a polytropic process is given by $W=\frac{nR(T_2-T_1)}{1-x}$
Using the given data we get,
$W=\frac{1\times 0.51(384-293)}{1-1.15}=-309.4\text{kJ}\approx -309\text{kJ}$
So, work required is $309\text{kJ}$