1549380323-Statistical Mechanics Theory and Molecular Simulation

214 Canonical ensemble

4.10. The canonical ensemble version of the classical virial theoremis credited to

Richard C. Tolman (1918). Prove that the canonical average

〈

xi

∂H

∂xj

〉

=

1

N!h^3 NQ(N,V,T)

∫

dxxi

∂H

∂xj

e−βH(x)=kTδij

holds. What assumptions must be made in the derivation of this result?

4.11. Prove that the structure factorS(q) of a one-component isotropic liquid or

gas is related to the radial distribution functiong(r) via eqn. (4.6.31).

4.12. Consider a system ofN identical noninteracting molecules, each molecule

being comprised ofn atoms with some chemical bonding pattern within

the molecule. The atoms in each molecule are held together by a potential

u(r

(i)

1 ,...,r

(i)

n),i= 1,...,N, which rapidly increases as the distance between

any two pairs of atoms increases, and becomes infinite as the distance be-

tween any two atoms in the molecule becomes infinite. Assume the atoms in

each molecule have massesmk, wherek= 1,...,n.

a. Write down the Hamiltonian and the canonical partition function for this

system and show that the partition function can be reduced to a product

of single-molecule partition functions.

b. Make the following change of coordinates in your single-molecule partition

function:

s 1 =

1

M

∑n

k=1

mkrk

sk=rk−

1

m′k

k∑− 1

l=1

mlrl k= 2,...,n,

where

m′k≡

k∑− 1

l=1

ml,

Mis the total mass of a molecule, and theisuperscript has been dropped

for simplicity. What is the meaning of the coordinates 1? Show that if

u(r 1 ,...,rn) only depends on therelativecoordinates between pairs of

atoms in the molecule, then single molecule partition function is of the

general form

Q(N,V,T) =

(V f(n,T))N

N!

,

wheref(n,T) is a pure function ofnandT.

c. Show, therefore, that the equation of state isalwaysthat of an ideal gas,

independent of the type of molecule in the system.