G25.2651: Statistical Mechanics

Problem set #4

Due: March 10, 2003

1.

Consider a system of N particles interacting via a pair potential. The Hamiltonian for such a system is

In the low density limit, we may assume that each particle interacts with at most one other particle.

a.

Show that the canonical partition function can be written as

where .

b.

Show that g(r) is proportional to in this approximation.

c.

Show that the second virial coefficient in this approximation becomes

where .

2.

One of the most common pair potentials used to model simple liquids and gases is the Lennard-Jones potential:

where is an effective particle radius and measures the strength of the potential.

a.

For a system of N particles interacting via a Lennard-Jones potential, derive an expression for the total force on a given particle. How would the evaluation of your expression depend on the choice of boundary conditions? What would you need to do for periodic boundary conditions?

b.

Find value of corresponding to the minimum of this potential and the value of the potential at this minimum.

c.

Based on the answer to part (b), rank the following systems in terms of how structured they are likely to be locally, i.e., how sharp is the first peak of g(r), at a temperature of T=300K:

i.

Liquid argon ( 3.405Å =0.01eV) at a density of 0.025Å .

ii.

Liquid helium ( =2.556Å = 0.00088eV) at a density of 0.03Å .

iii.

Liquid xenon ( =4.332Å = 0.02eV) at a density of 0.006Å .

3.

The radial distribution function g(r) can be measured in neutron and X-ray scattering experiments. In such experiments, the observed intensity of scattered neutrons or X-rays at a given angle is proportional to the structure factor given by

where is the vector difference between the wave vectors of the incident and scattered neutrons or X-rays and N is the number of particles in the system. (Note that the term j=k is not excluded from this sum!) Assuming a pair potential, show that depends only on the maginitude and is given in terms of g(r) by

where is the number density

4.

Consider a multi-component system with n different chemical species and particles of species i. Suppose the particles interact only through pair potentials with being the pair potential governing the interaction between a particle of species i and a particle of species j. Let be the mole fraction of species i and let be the radial distribution function between species i and species j. Find expressions for the total energy and pressure for this system.

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