# What is a chemical potential good for?

I read that the definition of the chemical potential is, that it is the partial derivative of the Free energy with respect to the number of particles, $$\mu=\frac{\partial F}{\partial N}.$$ Unfortunately, I don't get this definition, wherefore I wanted to choose an example: Let me think of a system for example a water in oil emulsion. How would one evaluate the chemical potential of this mixture? There the chemical potential would definitely depend on the question: Where do you want to insert which sort of particles?

• I have found out that there will propably be a different chemical potential for each species, so one for water and one for oil. but then there is still the question. where do we need to insert the particle? – Xin Wang Jul 19 '13 at 19:12
• This is a good question, but perhaps it should be asked in the chemistry forum. – Mark Rovetta Jul 20 '13 at 4:42
• (it's not a forum, it's a Q&A site) Chemical potential is a thermodynamic concept, which means this likely qualifies as physics and is thus on topic here. But an emulsion is probably not the best example. Lipschitz, would your question apply as well if it were, say, a mixture of two gases? (Air, approximated as 79% nitrogen and 21% oxygen, for example?) If you're asking something about an emulsion specifically, then this wouldn't be on topic here, but I don't think that's the case. – David Z Jul 20 '13 at 7:56
• Here is how I think intuitively about chemical potential. So far this is the only way I know to "get" it at any kind of gut level. Chemical potential is to chemical equilibrium what temperature is to thermal equilibrium. Temperature is a "knob" for adjusting average energy; chemical potential is a "knob" for adjusting particle number (or density). Temperature gradients cause heat flow; chemical potential gradients cause particle flow. – Michael Brown Jul 20 '13 at 8:00
• On topic on physics or on chemistry. – dmckee Jul 21 '13 at 2:53

Following Wikipedia, if you have a chemical system containing $n$ constituent species, adding $dN_i$ particles of the $i$-th species, at $T$ and $p$ constant, make a change of the Gibbs free energy :

$$dG = \sum_{i=1}^n \mu_i ~dN_i$$

(If you prefer $T$ and $V$ constant, you will use the Helmholtz free energy $F$ instead of $G$)

So, fundamentally, the chemical potential is a (potential) energy by specy particle.

At chemical equilibrium, the free energies $F$ or $G$ have to be minimum, which means that, at equilibrium :

$$\sum_{i=1}^n \mu_i ~dN_i = 0$$

So, chemical potential is important in all the equilibrium chemistry

Another interesting use, if you consider only one specy, is the grand-canonical situation (http://en.wikipedia.org/wiki/Grand_canonical_ensemble) where a system is in chemical (and thermal) equilibrium with a reservoir. In this case, the temperature and the chemical potential are constant, and, for a microstate, the probability to have a energy E, and a number N of particles, is given by :

$$p(E,N) \sim e^{-\beta(E - \mu N)}$$