1549380323-Statistical Mechanics Theory and Molecular Simulation

Problems 215

d. Denote the single-molecule partition function asq(n,V,T) =V f(n,T).

Now suppose that the system is composed ofdifferent types of molecules

(see comment following eqn. (4.3.14)). Specifically, suppose the system

containsNAmolecules of typeA,NBmolecules of typeB,NCmolecules

of typeCandNDmolecules of typeD. Suppose further that the molecules

can undergo the following chemical reaction:

aA+bB⇀↽cC+dD,

which is a chemical equilibrium. The Helmholtz free energyAmust now

be a function ofV,T,NA,NB,NC, andND. When chemical equilibrium

is reached, the free energy is a minimum, so that dA= 0. Assume that

the volume and temperature of the system are kept constant. Letλbe a

variable such that dNA=adλ, dNB=bdλ, dNC=−cdλ, and dND=

−ddλ.λis called thereaction extent. Show that, at equilibrium,

aμA+bμB−cμC−dμD= 0, (4.14.27)

whereμAis the chemical potential of speciesA:

μA=−kT

∂lnQ(V,T,NA,NB,NC,ND)

∂NA

with similar definitions forμB,μC, andμD.

e. Finally, show that eqn. (4.14.27) implies

ρcCρdD

ρaAρbB

=

(qC/V)c(qD/V)d

(qA/V)a(qB/V)b

and that both sides are pure functions of temperature. Here,qAis the

one-molecule partition function for a molecule of typeA,qB, the one-

molecule partition function for a molecule of typeB, etc., andρAis the

number density of type A molecules, etc. How is the quantity on the right

related to the usual equilibrium constant

K=

PCcPDd

PAaPBb

for the reaction? Here,PA,PB,... are the partial pressures of speciesA,

species,B,..., respectively.

4.13. Consider a system ofNidentical particles interacting via a pair potential

u(r 1 ,...,rN) =

1

2

∑

i,j,i 6 =j

u(|ri−rj|),

whereu(r) is a general repulsive potential of the form