# What is chemical potential?

I have seen various treatments of the chemical potential. Basically it all amounted to a definition of it as a quantity maximizing the entropy for systems allowed to exchange particles.
Now this quantity has become quite important in some models I am studying for current flow, specifically an equation I am looking at right now, which relates the current flowing between a source and a drain to the chemical potentials of the leads as well as the temperature.
Now, a question in the book is what happens when the leads have different temperatures - so basically what happens to the expression for the current when I vary the temperature but not the chemical potential. But my question is, are temperature and chemical potential independent of each other? Applying a voltage keeps the leads at different chemical potentials because I raise the energy of the electrons in the source wrt the drain lead. But if I raise the temperature in one lead, does this not raise the energy of the electrons in the lead wrt the electrons in the other lead? I guess this all amounts to me not being completely confident with the distinction between energy and temperature. Temperature is a stastical quantity which maximizes energy given a known mean energy of the system. But still I can't see how you can raise the temperature and not the energy - and how exactly does it all relate to the chemical potential?

Simon Bridge
Homework Helper
The "chemical potential" is just a generalization of the overall concept of a potential in physics.
The exact details will depend on the situation which is why there is no overall definition you can get a grasp on ... it'll sound very vague.

When you maintain a constant voltage - you are maintaining the electrostatic potential.
When you make a temperature difference, you change the thermal potential - heat will flow along the wire and free charges pick up some thermal drift maybe.

Both would, strictly, contribute to the "chemical" potential, so you have to be careful to read between the lines.

if I raise the temperature in one lead, does this not raise the energy of the electrons in the lead wrt the electrons in the other lead?
yep. But, reading between the lines, the thermal drift is not a very big effect in the situation.
If you put one end of a wire in ice and the other in boiling water, the change in the resistivity of the wire will have a much bigger effect on the current.

It's a bit like how they don't always tell you it's OK to ignore air resistance (or not) when they give you a ballistics problem.

Chestermiller
Mentor
The chemical potential of a chemical species in a mixture is precisely defined as the partial derivative of the Gibbs free energy with respect to number of moles of the species at constant temperature, pressure, and number of moles of all other species.

Chet

Simon Bridge
Homework Helper
The chemical potential of a chemical species in a mixture is precisely defined as the partial derivative of the Gibbs free energy with respect to number of moles of the species at constant temperature, pressure, and number of moles of all other species.

Chet
Yep - context is everything :)

berkeman
Mentor
And with an increase in temperature, I was mainly thinking of an increase in the reverse leakage current across the junction. Not sure how that works when there is a temperature gradient across the junction...

Simon Bridge