# the equilibrium concentration of vacancies

In the derivation of the equilibrium concentration of vacancies by statistical mechanics method, I was stumped by this procedure (marked by "?").

$\textbf{Physical Model}:$

1.Solid viewed as a collection of $N$ atomic sites;

2.Each site may or may not be occupied, and assume now that $N_o$ sites are occupied and $N_v$ sites are vacant;

3.If a site is not occupied then the system has an additional energy, namely the formal energy $E_v$;

$\textbf{Solution}:$

1.multiplicity function

$$\Omega = C_N^{N_o} = \dfrac{N!}{N_o!N_v!}$$

2.entropy:

$$S=k_B\ln \Omega = -N k_B (c\ln c+(1-c)\ln(1-c)) \qquad (c=\dfrac{N_v}{N} \quad ;\quad (1-c) = \dfrac{N_o}{N})$$

3.the internal energy ($\textbf{?}$)

$$U = N c E_v$$

(Why we don't consider the internal energy of the whole system rather than the vacancies ? )

4.The Helmholtz free energy

$$F = U-TS = N(cE_v + k_B T (c \ln c + (1-c) \ln (1-c)))$$

and taking $c \ll 1$

$$\dfrac{F}{N} = c E_v + k_B T c \ln c$$

5.equilibrium concentration (by minimizing the Helmholtz free energy.)

$$c \rightarrow e^{-\dfrac{E_v}{k_B T}}$$

So what's the missing points to understand the marked procedure above?

• Surely you mean $NcE_v$ rather than $NcN_v$ right below the question mark. I'm assuming that that's a typo rather than the source of your confusion. – David Hammen Feb 10 '17 at 12:11
• It's just a typo. – Jack Feb 10 '17 at 12:46