Question w/ chemical potential, equivalence to Energy or Potential?

I have a quick question, is the chemical potential $$\mu=\partial F /\partial N$$ where F is the free energy physically equivalent to a potential or energy?

For example, in electrostatics, $$V=U/q$$

Does $$\mu_{ext}= U \text{ or } V$$

Also, same thing could be asked about gravity, $$V_g=U_{g}/m$$

Does $$\mu_{ext}= U_g \text{ or } V_g$$

Thanks ahead of time.