I've been working on this problem for quite some time trying to figure out the most efficient way of answering it, So here goes the problem:
There is a container, containing a monoatomic gas with a movable piston on the top. The container and piston are all made of perfectly insulating material. Now the container is divided into two compartments by a movable partition that allow slow transfer of heat among the compartments. The adiabatic exponent of gas is $\gamma$.
We are asked to calculate the total work done by the gas till it achieves equilibrium. By equilibrium here I mean that initially after the partition was introduced the gas on each side has different pressure and temperature which with passage of time reached a common temperature and pressure.
Now my approach to the problem is:
We know that in a closed adiabatic container:
$\Delta$$Q = 0$ as there is no net heat change between the compartments.
So, as per first law of thermodynamics we have, $\Delta Q = \Delta U + W$
Now considering the whole container to be the system we can apply the law on the system so,
$\Delta Q = \Delta U + W$ reduces to $0$ = $\Delta U + W$ $\Rightarrow$ $W = - \Delta U$
Now we can calculate $\Delta U$ for both compartments at equilibrium and add them up and the work will be given by $- \Delta U$. And we're done.
But something about my above method doesn't seem right. Are all my above considerations correct? If not how can I improve? I've tried reading up Joule-Thomson effect and other related texts on thermodynamics and expansion. I'll be highly obliged by your help. I've tried to fit this question in with the standards of StackExchange.