1549380323-Statistical Mechanics Theory and Molecular Simulation

Problems 279

whereνiare the stoichiometric coefficients in the reaction. Using this nota-

tion, the coefficients of the products are, by definition, negative.As the re-

action proceeds, there will be a changeδNiin the numberNiof each species

such that the law of mass balance is

δN 1

ν 1

=

δN 2

ν 2

=···

δNK

νK

.

In order to find a condition describing the chemical equilibrium, we canmake

use of the Helmholtz free energyA(N 1 ,N 2 ,...,NK,V,T). At equilibrium, the

changesδNishould not change the free energy to first order. That is,δA= 0.

a. Show that this assumption leads to the equilibrium condition

∑K

i=1

μiνi= 0.

b. Now consider the reaction

2H 2 (g) + O 2 (g)⇀↽2H 2 O(g).

Letρ 0 be the initial density of H 2 molecules andρ 0 /2 be the initial density

of O 2 molecules, and let the initial amount of H 2 O be zero. Calculate the

equilibrium densities of the three components as a function of temperature

andρ 0.

∗6.5. Prove the following fluctuation theorems for the grand canonical ensemble:

a.

〈NH(x)〉−〈N〉〈H(x)〉=

(

∂E

∂N

)

V,T

(∆N)^2.

b.

∆F^2 =kT^2 CV+

[(

∂E

∂N

)

V,T

−μ

] 2

(∆N)^2 ,

whereCVis the constant-volume heat capacity,F=E−Nμ=TS−PV,

and

∆F=

√

〈F^2 〉−〈F〉^2.

6.6. In a multicomponent system withKcomponents, show that the fluctuations

in the particle numbers of each component are related by

∆Ni∆Nj=kT

(

∂〈Ni〉

∂μj

)

V,T,μi

=kT

(

∂〈Nj〉

∂μi

)

V,T,μj

,

where ∆Ni=

√

〈Ni^2 〉−〈Ni〉^2 , with a similar definition for ∆Nj.