H.2 Euler-Maclaur in Sum Formula
21.2 Ideal Systems: Orbitals and Factorization
21.2.1 Factorization for Independent Sites
In this section, we present several examples of factorization of the grand partition function for cases in which particles can reside on a number Ntot of noninteracting sites that can be occupied by one or more particles. This would be expected if such sites are sufficiently dilute; they are separated by distances that are large relative to the range of forces applicable to each site. In these examples, we shall suppose for simplicity that the chemical potentialμ, and hence the activityλ=exp(βμ)is imposed by a classical ideal monatomic gas.
Example Problem 21.2. Calculate the probability of adsorption of an ideal gas on Ntot independent sites that are either unoccupied, with energy zero, or singly occupied with energyε1.
Solution 21.2. The grand partition function for a single site isZ(1)=1+λe−βε1so the total grand partition function isZ = (Z(1))Ntot. The average number of adsorbed atoms, which in this case happens to equal the number of occupied sites, is
N =λ∂q
∂λ=Ntot λe−βε1
1+λe−βε1 (21.72)
and the average energy is
U= −∂q
∂β =Ntot ε1λe−βε1
1+λe−βε1. (21.73)
Except for the very important factors ofλ, Eq. (21.73) resembles the energy for independent two- state systems. In order for the gas to adsorb at low temperatures, we wantε1<0. The fraction of occupied sites isθ= N/Ntot, so
θ= λe−βε1
1+λe−βε1 =λe−βε1
Z(1) . (21.74)
Of course the fraction of unoccupied sites is 1−θ = 1/Z(1), so these results could have been deduced entirely from the ratios of the corresponding terms inZ(1)toZ(1)itself. From Eq. (19.66), the chemical potential of an ideal gas isμ=kBTln(n/nQ)=kBTln(p/(nQkBT)), wherenis the number density andnQ(T)= (mkBT/2πh¯2)3/2is the quantum concentration.
The absolute activity is therefore
λ= n
nQ(T) = p
nQ(T)kBT, (21.75)
which is the ratio of the actual pressure to a quantum pressure. We can therefore define a temperature-dependent pressure
p0(T):=nQ(T)kBTeβε1 =nQ(T)kBTe−β|ε1|, (21.76) forε1<0, which increases with temperature. Then Eq. (21.74) takes the simple form
θ= p
p0+p. (21.77)
Equation (21.77) has the form of a Langmuir adsorption isotherm and is plotted inFigure 21–1.
See Kittel and Kroemer [6, p. 142] for a plot of data for adsorption of an oxygen molecule by a heme group of myoglobin, which closely follows such an isotherm.
2 4 6 8 10
0.2 0.4 0.6 0.8 1
p/p0
θ
θ
2 4 6 8 10
0.2 0.4 0.6 0.8 1
p, arbitrary units
FIGURE 21–1 Langmuir adsorption isotherms for the fractional adsorption of an ideal gas onNtotindependent sites. The curves on the right correspond to temperatures in the ratios 1:4:8, from left to right.
372 THERMAL PHYSICS
Example Problem 21.3. Calculate the probability of adsorption of an ideal gas onNtotin- dependent sites that are either unoccupied, with energy zero, or singly occupied with partition functionz(T). What is the canonical partition functionZN for a system havingN occupied sites?
Solution 21.3. The grand partition function for a single site is
Z(1)=1+λz(T), (21.78)
so Eq. (21.77) still applies; however, the pressure in Eq. (21.76) is replaced by
p0(T):=nQkBT/z(T). (21.79) The canonical partition function for N adsorbed atoms is the coefficient of λN in Z=(Z(1))Ntotwhich is readily found from the binomial theorem to be
ZN = Ntot!
N!(Ntot−N)![z(T)]N. (21.80) The binomial coefficient accounts for the degeneracy that arises because we do not know which of theNtotsites are occupied, but they are distinguishable by virtue of their position. The reader is invited to verify that the chemical potential for such a system is
μ= −kBT∂lnZN
∂N =kBTln N
Ntot−N 1 z(T)
=kBTln θ
1−θ 1 z(T)
. (21.81)
Equating this μ to that of a classical ideal gas, Eq. (21.75), gives p/p0(T)=θ/(1 − θ) withp0(T)given by Eq. (21.79). Then solving forθgives the consistent result Eq. (21.77).
Example Problem 21.4. Calculate the probability of adsorption of a monatomic ideal gas onNtotindependent sites that are either unoccupied with energy zero or singly occupied with energyε1or doubly occupied with energyε2. Note thatε2is not necessarily equal to 2ε1, so atoms on a doubly-occupied site can interact.
Solution 21.4. The grand partition function for a single site isZ(1)=1+λe−βε1+λ2e−βε2. The probabilities of a site being unoccupied, singly occupied, or doubly occupied are
p0=1/Z(1); p1=λe−βε1/Z(1); p2=λ2e−βε2/Z(1). (21.82) The average number of adsorbed gas atoms is thereforeN =Ntot(p1+2p2), where the factor of 2 enters because of the double occupancy. Alternatively, one can use the total grand partition functionZ= [Z(1)]Ntotto calculate
N =λ∂q
∂λ =Ntot λe−βε1+2λ2e−βε2
1+λe−βε1+λ2e−βε2, (21.83)
where the factor of 2 occurs automatically, or U= −
∂q
∂β
λ=Ntotε1λe−βε1+ε2λ2e−βε2
1+λe−βε1+λ2e−βε2 (21.84) where there is no such factor of 2.
Example Problem 21.5. Calculate the probability of adsorption of either anAatom or aB atom onNtotindependent sites that are either unoccupied with energy zero or singly occupied with energiesεAandεB, respectively.
Solution 21.5. See Eq. (21.169) for an obvious generalization of the GCE to a binary system.
In the present case, we have
Z(1)=1+λAe−βεA+λBe−βεB. (21.85) Thus the fractional occupations are
θA= λAe−βεA
1+λAe−βεA+λBe−βεB; θB= λBe−βεB
1+λAe−βεA+λBe−βεB, (21.86) and the fraction of unoccupied sites is 1−θA−θB. We would have to determineλAandλBfrom the chemical potentials of the environment, say ideal gases ofAandB. We see in this case that theAandBatoms compete for occupancy of the sites. Moreover, a small difference betweenεA andεBcan make an enormous difference between the relative adsorption ofAandBif|βεi| 1.
For the examples in this section,Z = (Z(1))Ntot, so viewed as a series in powers ofλ, the series cuts off after a finite number of terms. For the first two examples, the highest power is(λ)Ntot and for the third example it is(λ)2Ntot. These cutoffs occur because of the restrictions on maximum occupancy of a site. In terms of the general formula Eq. (21.21), they can be imposed formally by assuming that any state of the entire system having greater occupancy than allowed would have an infinite energy, so its Boltzmann factor would be zero. On the other hand, for ideal Fermi and Bose gases, there are an infinite number of orbitals available for occupation, so the expression for Z for those gases contains all powers ofλ, as shown in the next section.