Question

$n$ moles of monoatomic gas is undergoing a polytropic process $P{T}^3=$ constant. Initial and final temperatures of gas are $T_0\&3T_0$ respectively. Find the work done by the gas.

• A. $nR{T}_0$
• B. $2.5nR{T}_0$
• C. $8nR{T}_0$
• D. $6nR{T}_0$

Answer 1

The correct option is C $8nR{T}_0$

For an ideal gas, The relation between $P$ and $T$ is given by $T=\frac{PV}{nR}..........\left(1\right)$
Given that,
$P{T}^3=$ constant$..........\left(2\right)$
From $\left(1\right)$ and $\left(2\right)$ we get,
$P\times {\left(\frac{PV}{nR}\right)}^3=$ constant.
$\Rightarrow P{V}^{3/4}=$ constant
This is a polytropic process with exponent $x=\frac{3}{4}$
Workdone during a polytropic process is given by
$\Rightarrow W=\frac{nR(T_2-T_1)}{1-x}$
From the data given in the question,
$W=\frac{nR(3T_0-T_0)}{1-\frac{3}{4}}=8nR{T}_0$
Thus, option (c) is the correct answer.