Question

In a certain thermodynamic process, the pressure of a gas depends on its volume as $k{V}^3$. The work done when the temperature changes from ${100}^{\circ }\text{C}$ to ${300}^{\circ }\text{C}$ will be ______$nR$, where $n$ denotes number of moles of a gas.

Answer 1

Given:
$P=k{V}^3$

$\Rightarrow PV=k{V}^4$

$\Rightarrow nRT=k{V}^4$

$\Rightarrow nRdT=4k{V}^{3}dV$

$\Rightarrow nRdT=4PdV$

$\Rightarrow PdV=\frac{nRdT}{4}$

So, $W=\int PdV={\int }_{T_1}^{T_2}\frac{nRdT}{4}$

$\Rightarrow W=\frac{nR}{4}[T_2-T_1]=\frac{nR}{4}[{300}^{\circ }\text{C}-{100}^{\circ }\text{C}]$

$\Rightarrow W=50nR$