1549380323-Statistical Mechanics Theory and Molecular Simulation

278 Grand canonical ensemble

∆N

〈N〉

=

1

〈N〉

√

〈N〉kTκT

v

=

√

kTκT

〈N〉v

∼

1

√

〈N〉

. (6.6.18)

Thus, as〈N〉 −→0 in the thermodynamic limit, the particle fluctuations vanish and
the grand canonical ensemble is seen to be equivalent to the other ensembles in this
limit.

6.7 Problems

6.1. Using a Legendre transform, determine if it is possible to define an ensemble

in whichμ,P, andTare the control variables. Can you rationalize your result

based on Euler’s theorem?

6.2. a. Derive the thermodynamic relations for an ensemble in whichμ,V, and

Sare the control variables.

b. Determine the partition function for this ensemble.

6.3. For the ideal gas in Problem 4.6 of Chapter 4, imagine dividing the cylinder

into rings of radiusr, thickness ∆r, and height ∆z. Within each ring, assume

thatrandzare constant.

a. Within each ring, explain why it is possible to work within the grand

canonical ensemble.

b. Show that the grand canonical partition function within each ringsatisfies

Z(μ,Vring,r,z,T) =Z(0)(μeff(r,z),Vring,T),

whereZ(0)is the grand canonical partition function forω= 0 andg= 0,

Vringis the volume of each ring, andμeff(r,z) is an effective local chemical

potential that varies from ring to ring. Derive an expression forμeff(r,z).

c. Is this result true even if there are interactions among the particles? Why

or why not?

6.4. Consider an equilibrium chemical reaction involvingKmolecular species de-

notedX 1 ,...,XK, where some of the species are reactants and some are prod-

ucts. Denote the chemical equation governing the reaction as

∑K

i=1

νiXi= 0,