The internal energy change for isothermal processes is known to be equal to zero. But in the case of liquid vaporization (e.g. liquid water to steam) (boiling at constant temperature), the internal energy change is not zero. Why??
that is because of the latent heat of phase transition. For a first order phase transition like boiling of water, the energy added to the isothermal process goes into the entropy change. This is explained in entry level textbooks on thermodynamics, there is probably little left to discuss on the topic.
The internal energy change for isothermal processes is zero for ideal gases, but can be different from zero for other systems. A battery discharged at constant temperature certainly changes its internal energy.