The general case

The general case

Now consider the general case in which a change of state is brought about both by doing work and by transferring heat. The change in internal energy for this case is:

For a given change in state, $q$ and $w$ can vary depending on how the change is carried out. However, the sum $q + w = \Delta U$ depends only on the initial and final states. It is independent of the path taken.

If there is no transfer of energy as heat or as work (an isolated system), then $w = 0$ and $q = 0$, so $\Delta U = 0$.