Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
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44 CHAPTER 3. VARIABLE NUMBER OF PARTICLES<br />
and the free energy is a minimum if the chemical potentials are equal, µ1 = µ2.<br />
This, of course, is similar to the conditions we found before for temper<strong>at</strong>ure and<br />
pressure. Although we could have defined the chemical potential as an arbitrary<br />
function of the partial deriv<strong>at</strong>ive, the definition above is the only choice which<br />
is consistent with the experimentally derived chemical potential. The definition<br />
in terms of a deriv<strong>at</strong>ive only holds for large values of N, in general one should<br />
write<br />
µ(T, V, N) = F (T, V, N) − F (T, V, N − 1) (3.3)<br />
which tells us th<strong>at</strong> the chemical potential equals the free energy needed to subtract<br />
one particle to the system. This is another reminder th<strong>at</strong> of a difference<br />
between thermodynamics and st<strong>at</strong>istical mechanics. In thermodynamics, the<br />
number of moles N of a m<strong>at</strong>erial is a continuous variable. In st<strong>at</strong>istical mechanics,<br />
the number of particles N is a discrete variable, which can be approxim<strong>at</strong>ed<br />
as continuous in the thermodynamic limit (showing up again!). Finally, if different<br />
types of particles are present, on specifies a chemical potential for each<br />
type according to<br />
<br />
∂F<br />
µi(T, V, N1, N2, ...) =<br />
∂Ni<br />
Chemical potential, wh<strong>at</strong> does it mean?<br />
T,V,Nj=i<br />
(3.4)<br />
Wh<strong>at</strong> is this chemical potential? The following example will help you to<br />
understand this important quantity. If the systems 1 and 2 are not in equilibrium,<br />
the Helmholtz free energy is reduced when particles flow from the system<br />
with a high value of the chemical potential to the system with the lower value,<br />
since ∆F = (µ2 − µ1)∆N. We can also add some other potential energy to<br />
the systems. For example, the systems could be <strong>at</strong> different heights are have<br />
a different electric potential. This potential energy is Φi. If we bring ∆N<br />
particles from system 1 to system 2, we have to supply an amount of energy<br />
∆E = (Φ2 − Φ1)∆N to make this transfer. Since this is a direct part of the<br />
energy, the total change in the Helmholtz free energy is given by<br />
∆F = (ˆµ2 + Φ2 − ˆµ1 − Φ1)∆N (3.5)<br />
where ˆµ is the chemical potential evalu<strong>at</strong>ed without the additional potential<br />
energy. Therefore, if we choose Φ such th<strong>at</strong><br />
ˆµ2 + Φ2 = ˆµ1 + Φ1<br />
(3.6)<br />
the two systems will be in equilibrium. Of course, the presence of an external<br />
potential Φ leads to a term ΦN in the internal energy U and hence in F. As a<br />
result