Work done in Thermo!

For a process $PV^{x}=constant$

molar specific heat capacity $\large{C=\frac{R}{\gamma-1}+\frac{R}{1-x}}$

$W=\frac{nRdT}{1-x}$

where $\gamma$ is ratio of specific heat capacities

$H=nCdT$ and here $x=\frac{1}{k}$ and $\gamma=\frac{5}{3}$

Given in the question $\frac{H}{2}=W$

on solving

$\frac{k}{k-1}=\frac{3}{2}$

$k=3$

$50 \text{\%}$ of heat is used up as work done by gas. $\Rightarrow 50 \text{\%}$ of heat is used in increasing the internal energy of the gas.

$\Rightarrow \Delta U = \frac{1}{2} \Delta Q$

$\Delta U = n C_v \Delta T = n \frac{R}{\gamma - 1} \Delta T$ and $\Delta Q = n C \Delta T , C = \frac{R}{\gamma - 1} + \frac{R}{1 - x}$ where $x$ is polytropic factor.

$n \frac{R}{\gamma - 1} \Delta T = \frac{1}{2} \times n( \frac{R}{\gamma - 1} + \frac{R}{1 - x} )\Delta T$

$\frac{R}{\gamma - 1} = \frac{R}{1-x}$

$\gamma -1 = 1 - x$

$x = 2 - \gamma = 2 - \frac{5}{3} = \frac{1}{3}$

The process is $P V^x = k$ and $x = \frac{1}{3} \Rightarrow P^3 V = k^3 = const.$