Physical Chemistry Third Edition

1124 27 Equilibrium Statistical Mechanics. III. Ensembles

Exercise 27.1

Choose several different functions ofEsuch as sin(E), cos(E),E,E^2 , and so on, and show that

they do not satisfy Eq. (27.1-5). Show that the function of Eq. (27.1-6) does satisfy this relation.

We can derive Eq. (27.1-6). Letxandyrepresent the two subsystem energies, so

that

p(x+y)p(z)p(x)p(y) (27.1-7)

wherex+yz. By the chain rule (see Appendix B),

dp

dx



dp

dz

∂z

∂x



dp

dx

p(y) (27.1-8a)

dp

dy



dp

dz

∂z

∂y



dp

dy

p(x) (27.1-8b)

Sincezx+y,∂z/∂y∂z/∂x1, and we can write

dp

dx

p(y)

dp

dy

p(x) (27.1-9)

We divide byp(x)p(y):

1

p(x)

dp

dx



1

p(y)

dp

dy

(27.1-10)

We have separated the variablesxandy. A function ofxthat equals a function ofyfor

all values of the independent variablesxandymust be a constant function ofx, since

ycan be held fixed whilexvaries. We let the constant functions equal−β, so that

1

p(x)

dp

dx

−βconstant (27.1-11)

We multiply bydxand do an indefinite integration

1

p(x)

dp

dx

dx

1

p

dpd[ln(p)]−βdx (27.1-12)

ln(p)−βx+ln(A) (27.1-13)

where ln(A) is a constant of integration. We take antilogarithms:

pAe−βx (27.1-14)

which is the same as Eq. (27.1-6), withxreplaced byE.

The Canonical Partition Function

We now need to eliminate the parametersAandβfrom the distribution shown in

Eq. (27.1-14). The total number of systems in the ensemble is fixed, so that

∑

k

nkn (27.1-15)