# Chemical potential different for different systems?

1. May 12, 2007

### pivoxa15

1. The problem statement, all variables and given/known data
For a given phenomena of macroscopic particles.

If we model it using the canonical ensemble then we get a certain chemical potential u.

But if we model it using the Grand Canonical system, we get a varying chemcial potential that depends on the average number of particles in the system.

So two different chemical potentials for the two systems. But in the thermodynamic limit of n-> infinity, both u equals each other.

That sounds right dosen't it?

2. May 13, 2007

### cepheid

Staff Emeritus
Yeah, you're basically right.

In any of the systems in the canonical ensemble, the chemical potential is given by

$$\mu(T,V,N)$$

whereas in any of the systems in the grandcanonical ensemble, it's given by

$$\mu(T,V,\langle N \rangle)$$

Now, in the "thermodynamic limit" as you have called it, the number of particles in any of the grandcanonical systems is *MUCH* more likely to be equal to $\langle N \rangle$ than to any other number. So it's equivalent to having a closed system with a fixed number of particles $\langle N \rangle$.

If I have an open system and I want it to be equivalent to a closed system with a particular number of particles $N_0 [/tex], I'd have to somehow pick or set the value of [itex] \mu [/tex] such that [itex] \langle N \rangle = N_0$ for that system.

Last edited: May 13, 2007