An isolated box with volume V is initially split into two halves by a partition. One side has gas at temperature T with pressure P and number of molecules N, and the other side has nothing (it is a void). The partition instantaneously disappears and the gas is free to occupy the entire box. During this process how much work does the gas do?

The gas does not do any work  due to free expansion.

Explanation:

Free expansion is an irreversible process in which a gas expands into an evacuated chamber. During this expansion, the external pressure is zero. The work done in this case is PΔV. Since the pressure applied is zero, the work done by the gas is zero.

We have ∆Q=∆U+∆W where ∆Q is the change in internal energy, ∆U is the heat input and ∆W the work done.

Now in our case, once the partition disappears instantaneously, the gas molecules gradually occupies the entire box. Since there is no applied heat and the no external pressure applied. ∆U and ∆W are zero.  ∆U = 0, since there is no change in internal energy.

Therefore, during this process the gas does not do any work as it freely expands into the void and occupies the entire box.

a.   mass number from atomic number