The Quantum World, Energy and Entropy

 

 

  

 

Quantum Levels

 

The Boltzmann Distribution

 

Statistical Entropy is S = k lnW where W is the weight (number of ways particles can be distributed in a system).

 The weight is:

Conservation of energy requires: 

Conservation of mass requires: 

 

Using S = k lnW we can reason that the system will tend towards the distribution among the ni that maximizes S.

 

This condition is satisfied by d( lnW ) = 0.

 Additionally conservation of energy and mass can be restated as:

 

Using the method of LaGrange undetermined multipliers we have:

 

We can evaluate (lnW/ni) = (lnN!/ni) – Sj (ln nj!/ni) using Stirling's approximation:

lnx! » xlnx – x

Simplification of (lnW/ni)

 

First step is to note that lnW = lnN! - Sj ln nj! = NlnN – N - Sj nj ln nj - Sj nj

 

Since N = Sj nj these two cancel to give lnW = NlnN - Sj nj ln nj

 

These latter derivatives result from the fact that (ni/ni) = 1 and (nj/ni)=0.

Therefore we have:

 

and we have:

 The most probable distribution is:

 

Now we only need to find the undetermined multipliers a and b.

 

 

 

This determines a and defines the Boltzmann distribution.

 

 

The Boltzmann distribution is:

It is not immediately obvious why this is so significant. This is because we have not yet defined b. Looking ahead we will show that b=1/kT and that this distribution represents a thermally equilibrated most probable distribution over all energy levels.

 

The fraction of molecules in state i is pi:

The molecular partition function q gives an indication of the average number of states that are thermally accessible to a molecule at the temperature of the system.

The Boltzmann distribution can be written:

 

We can now proceed to write the total energy:

Substituting in ni from above:

Now, note a little trick:

 

The energy can be expressed as:

Now, to identify b we can study the energy levels and partition function of translation.

Note: The quantum chemical and classical treatments of translation are the same.

 

where px is the momentum in the x direction etc.

 

The partition function, q is:

 

 

This is a Gaussian integral (in three dimensions). In one dimension this becomes:

In the present case s = b/2m and therefore making the substitution x = px etc.

We have:

To calculate the energy we need the derivative dq/db:

 

 The energy then is:

 

This can be compared to the kinetic theory of gases result:

from which we deduce that:

 

 

Statistical Thermodynamics

 

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