## Monday, May 23, 2016

### What is the chemical potential?

I used to find the concept of the chemical potential rather confusing.
Hence, it is not surprising that students struggle too.
I could say the mantra that "the chemical potential is the energy required to add an extra particle to the system" but how it then appeared in different thermodynamic identities and the Fermi-Dirac distribution always seemed a bit mysterious.

However, when I first taught statistical mechanics 15 years ago I used the great text by Daniel Schroeder. He has a very nice discussion that introduces the chemical potential. He considers the composite system shown below, where a moveable membrane connects two systems A and B. Energy and particles can be exchanged between A and B. The whole system is isolated by the environment and so the equilibrium state is the one which maximises the total entropy of whole system.
Mechanical equilibrium (i.e. the membrane does not move) occurs if the pressure of A equals the pressure of B.

Thermal equilibrium (i.e. there is no net exchange of energy between A and B) occurs if the temperature of A equals that of B. Thus, temperature is the thermodynamic state variable that tells us where two systems are in thermal equilibrium.

Diffusive equilibrium (i.e. there no net exchange of particles between A and B) occurs if the chemical potential of particles in A equals that in B, where the chemical potential is defined as

Starting with this one can then derive various useful relations such as those between the Gibbs free energy and the chemical potential (dG= mu dN and G=mu N).
Thus, the chemical potential is the thermodynamic state variable/function that tells us whether or not two systems are in diffusive equilibrium.

Doug Natelson also has a post about this topic. He mentions the American Journal of Physics article on the subject by Ralph Baierlein, drawing heavily from his textbook. However, I did not find that article very helpful, particularly as he mostly uses a microscopic approach, i.e. statistical mechanics. (Aside: the article does have some interesting history in it though).
I prefer to first  use a macroscopic thermodynamic approach before a microscopic one as, I discussed in my post, What is temperature?

#### 1 comment:

1. Hi Ross,
I have always held a similar view that the macroscopic approach was better. But talking with undergrads I see that some don't feel comfortable with defining the chemical potential (or temperature for that matter) as the quantity that tells if the system is in equilibrium.
I think there are to things in this: 1) It really has a lack of heuristical picture and 2) when introducing a porous membrane and diffusive process we're already introducing microscopic ideas too, and so is reasonable to expect a microscopic explanation.
Temperature is equally difficult in this (the average energy is a good mental picture, if untrue), but one can see how the pressure has it good. It does define mechanical equilibrium but it has a nice interpretation in terms of force exerted by the particles when they collide with the walls.
Just saying that although the macroscopic picture is great most students (in my experience) feel uneasy without a microscopic discussion also.