iii$)($From an old Thermo $1$ exam$)$: A horizontal cylinder contains an explosive gas and an inert

gas separated by a piston held by a spring as shown in the diagram below. The relaxed

spring length is $L_0$ and the spring constant is $k.$ The piston, spring and cylinder have

negligible mass. The initial temperature and pressure of the explosive gas are $T_0$ and $p_0,$

respectively. At time $t=0,$ the gas is ignited and a total amount of energy $Q$ is deposited in

the gas, bringing the gas to a higher temperature prior to the piston moving $($i$.$e$.,$ the

explosion process is so rapid that the energy addition can be considered to be at constant

volume$).$ The gas then expands and the piston moves to a final equilibrium position after the

oscillations stop. Neglect friction holding the piston in the final equilibrium position.

Assuming that the piston is a good heat conductor and that there is friction between the

piston and the cylinder, find an equation involving the final position of the piston, L $,$ in

terms of the initial conditions $($don$\text{'}$t try to find an explicit equation for L $).$

Note: Define your system and clearly show the governing equations used.