Understanding why we have these four different energies is crucial to
saving yourself time doing thermodynamics problems. In a given problem,
identify the variables that are remaining constant. Then choose the energy that has
those variables in the differential, so that when you are to calculate the energy,
you aren't left with many non-zero terms.
For example, in a process where the volume and number remain constant but the entropy
is changing, we should use U, because the Thermodynamic Identity
simplifies to dU = τdσ. Becoming adept at these choices will be
critical to your success at solving problems quickly.