Chemical potential different for different systems?

[pivoxa15]

Homework Statement

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 doesn't it?

 

[cepheid]

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 [itex]N_0 [/tex], I'd have to somehow <b>pick </b> or <b>set</b> the value of [itex]\mu [/tex] such that $\langle N \rangle = N_0$ for that system.[/itex][/itex]