Vacancies in solids 117theformation energyofa vacancyis,andthat the vacancies are sufficiently dilute
that they are weakly interacting. The increase inUdue to the vacancies is thenn.
Theincreasein entropy canbecalculatedin a manner similar to theisotopicdisorder
ofsection 1 0 .1.1. Thesolid withnvacancieshas(N+n)lattice sites of whichN
arefullandnare empty. Neglecting small surface effects, the number of possible
arrangementsis(N+n)!/(N!n!).Theincreasein entropydue to thedisorder ofthe
vacanciesisthuskkkBln[(N+n)!/N!n!].In practice thevibrationalentropyofthesolid
will also increase slightly for secondary reasons, but we shall ignore this.
Following(10.7), thefree energy increase whennvacancies are presentis
F(n)=U(n)−TS(n)
=n−kkkBTln[(N+n)!/N!n!]
=n−kkkBT[(N+n)ln(N+n)−NlnN−nlnn]EquilibriumisgivenbytheminimumF,andthere are no constraints onn.Hence the
requiredn(strictlyn∗)satisfiesdF/dn=0,i.e.
0 =dF(n)/dn
=−kkkBTln[(N+n∗)/n∗]Bearinginmindthatn∗ N(since meltinghas not occurred), we mayreplace
(N+n∗)byN,togive the final result for the vacancyconcentration
n∗/N≈exp(−/kkkBT) (10.11)Itisinterestingthat the answerisagainjustasimpleBoltzmannfactor. For most
solids the value of formation energyis about 1 eV, for example its value for copper
is 1.4 eV. Equation (10.11) would give a vacancy concentration in copper of only
about 1in1 024 at room temperature (one vacancyper mole!), although in practice
thermodynamic equilibrium would not be reached. Vacancy movement, i.e. diffusion,
also requires an activation energy anddisorderisfrozenin at room temperature,
another exampleofa metastable state. However, as thecopperisheatedthe number of
vacancies in equilibrium rises, and the time to reach equilibrium falls. By the melting
point (108 3 ◦C), the number givenby (10.11)is1in1 05. For some substances, the
numberis evengreaterbefore meltingoccurs, andtheincreasein entropyofthesolid
can be observed directly as an additional heat capacity.