Question

A cylinder having a piston contains $1\text{mole}$ 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 $x=1.15$, then the work required (in $\text{kJ}$) is (nearest integer value)
[Take $R$ for methane $=8.314\text{J/mol K}$ and $8^{0.13}=1.31$]

Answer 1

For a polytropic process,
we 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 the 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
$T_2=293\times {\left(\frac{800}{100}\right)}^{\frac{0.15}{1.15}}$
$=293\times 8^{0.13}=293\times 1.31$
$=384\text{K}$

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
$W=\frac{1\times 8.314(384-293)}{1-1.15}=-5043.83\text{J}\approx -5\text{kJ}$
So, work required is $5\text{kJ}$