Chemical potential and fixed number of particles

[cryptist]

Can we talk about the chemical potential of a system with fixed number of particles? Is this physically meaningful? Why/why not?

P.s: I know that chemical potential is the partial derivative of free energy with respect to number of particles. But in the formulation of grand canonical ensemble, we write N=Ʃf (for example f would be fermi-dirac distribution function) and can't we fix N in this formula, and solve chemical potential μ by changing energy ε?

 

[DrClaude]

Can we talk about the chemical potential of a system with fixed number of particles? Is this physically meaningful? Why/why not?

Yes, for the same reason you can talk about the temperature of a system with a fixed entropy or the pressure of a system with a fixed volume.

 

[jfizzix]

You can indeed solve for the chemical potential \mu(N,V,T) in the way you suggest.

Consider the ideal gas as an example. Here, the occupancy comes from the classical maxwell-Boltzmann statistics f_{MB}(\epsilon) = e^{-(\frac{\epsilon-\mu}{k_{B}T})}.

The number of particles N can be expressed as

N=\sum_{j}g_{j}f_{MB}(\epsilon_{j})
where g_{j} is the number of states at energy level \epsilon_{j}

The partition function for a single particle Z_{1} is

Z_{1} = \sum_{j}g_{j} e^{-(\frac{\epsilon_{j}}{k_{B}T})}=\eta_{q}V : \eta_{q}=(\frac{m k_{B}T}{2 \pi \hbar^{2}})^{\frac{3}{2}}

Here \eta_{q} is a characteristic quantum concentration (you can see it has dimensions of \frac{N}{V}). It's a large concentration indicating when quantum effects (bose/Fermi statistics) must be taken into account. For ideal gases, the real concentration \eta=\frac{N}{V} is much less than \eta_{q}.

We can relate the total particle number N to the partition function Z_{1}, giving us

N=Z_{1}e^{-\frac{\mu}{k_{B}T}}

We can solve for \mu to find

\mu =k_{B}T \ln (\frac{N}{Z_{1}})

Then, substituting out expression for Z_{1}, we arrive at the final result

\mu =k_{B}T \ln (\frac{\eta}{\eta_{q}}) :\eta=\frac{N}{V}.

It's a total non-sequitur, but it seemed worth explaining since I had already written lecture notes on the subject.