Statistical Entropy

Note the we have calculated E = U – U(0), which is the internal energy referenced to the value, U(0) at absolute zero (T = 0 K).

 

We can now calculate the entropy S = k ln W

 

 

Now recalling the definition of the Boltzmann distribution lnpi =- bei – lnq

The entropy is,

 

or

 

 

Partition Functions

In general, gj is the degeneracy, ej is the energy:

We assume that the energy of the lowest energy level, the ground state is e0 = 0.

 Examples:

  1. Two level system.
  2. Infinite energy ladder.

 

Two Level System

Assume that there is only a single state at each level. The degeneracy g0 = g1 = 1.

Therefore,

Note that as T ® 0, q ® 1 and as T ® ¥, q ® 2.

The ratio of the population in the two is states e-be where e is the energy difference between the two states.

 

Infinite Energy Ladder (Uniform)

Assuming no degeneracy,

 

This can be written q = 1 + x + x2 + x3 + …

where x = e-be

xq = x + x2 + x3 + x4 + … = q – 1, therefore xq = q – 1

Solving for q we find,

which means (after substituting back in for x)

 

The Canonical Partition Function

  • The canonical ensemble represents a large number of replications of the system with constant N, V, and T.
  • Energy fluctuations between the members of the ensemble are allowed.
  • The weight W of a configuration is:

 

where M is the number of replications of the system and mi is the number of those with configurations 0,1,2 etc.

 The configuration with the greatest weight is

where Q is the grand canonical partition function.

Q differs from q, the molecular partition function because it is not based on the assumption that the molecules in the system are independent.

Distinguishable Particles

Indistinguishable Particles