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 lnp_i =- be_i – lnq

The entropy is,

or

Partition Functions

In general, g_j is the degeneracy, e_j is the energy:

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

Examples:

A.) Two level system.

B.) 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 Uniform Energy Ladder

Assuming no degeneracy,

or

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 m_i 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.

Indistinguishable Distinguishable

Particles Particles