138 Chemical thermodynamics
as usedinChapter 2. Stirringinthedefinition oftheGibbsfree energyG=F+PV=
U−TS+PVwe obtain dG=−SdT+VdP+μdNand hence an equivalent definition
ofμis:
μ=(
∂G
∂N
)
T,P
Thislast resultis particularly useful,since temperature andpressure areintensive
parameters, whereas number aloneinG(T,P,N)isextensive.This means thatif we
double the number ofparticlespresent of a substance at constantTandP,in order
forGtodouble,itis necessary simplythatGandNare proportional.Inother words
itfollowsimmediatelythat
μ=G/N (13.2)andwe see that another wayoflookingat thechemicalpotentialisthatitistheGibbs
free energy per particle.
Thisideaimmediatelyrelates to thediscussion ofphase transitionsinChapter 11.
Consider again a pure (one-component) substance whichcan existin two phases,
labelled 1 and 2. We wish to decide which of the phases is the stable one in equi-
librium at a particular pressure andtemperature. The answeris easy. As always, the
equilibrium conditionisthat whichminimizesfree energy. Suppose that wehavea
totalNparticles, of whichN 1 particles are in phase 1 andNNN 2 in phase 2, so that
N=N 1 +NNN 2 .Using(13.2), we writefor the totalfree energyofthesystem
G=G 1 +G 2 =N 1 μ 1 +NNN 2 μ 2whereμ 1 andμ 2 are thechemicalpotentialsinthe two phases. Itis now clear that
which phase is stable is determined bythe magnitude of the two chemical potentials.
Ifμ 1 >μ 2 , thenGis lowest whenN 1 = 0 and all the substance is in phase 2, the
phase withthelower chemicalpotential.Ontheotherhand,ifμ 1 <μ 2 thenNNN 2 = 0
and the stable phase is phase 1. In other words, particles move from highμto lowμ
and it is seen that chemical potential is indeed a ‘potential for particle number’.
Ifon theotherhandμ 1 =μ 2 ,thenGisthe same whatever theparticle number
ratioN 1 /NNN 2 .This is the case ofphase equilibrium. An indeterminate mixture of the
two phases can be present. It is now worth turning back to Fig. 11.1, which shows a
graphofGfor the two phases. Essentiallythisisjust the same as agraphofG/N,i.e.
μ, for the two phases; and the above commentary describes precisely the equilibrium
condition.
Itis worthremarkinghere that the equalityofchemicalpotentialinanequilibrium
system is a very powerful tool. As an example, it enables one to discuss a whole
range ofproblemsin semiconductor physics. In this case the appropriate potential
isthe so-calledelectrochemicalpotential, a sum ofelectric potentialenergyofan
electron and the chemical potential. (Conduction electrons are charged and have their
potentialenergies changedbyanelectric potential,whatever theirkinetic energies).
Thedirection ofelectron motionisdeterminedbythegradient ofthiselectrochemical