stat_thermo

CH237 - Chemical Thermodynamics and Kinetics

Dr. Joseph N. Grima, Department of Chemistry
University of Malta, Msida, MSD 06, MALTA
http://staff.um.edu.mt/jgri1/teaching/ch237

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Statistical Thermodynamics (ii)

(5) Calculation of the various properties from the partition function

(5-1) The internal energy
(5-2) The statistical entropy (5-2-1) Derivation of S = k ln W
(5-2-2) Derivation of the statistical entropy in terms of the partition function
(5-2-3) Residual entropies (5-3) The Helmholtz energy
(5-4) The pressure
(5-5) The enthalpy
(5-6) The Gibbs energy
(5-7) The heat capacities

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(5) Calculation of the various properties from the partition function

We have said in the discussion above that the partition functions q (valid for independent molecules) and Q (valid for dependent and independent molecules) carry all the thermodynamic information of the system. This is a very strong statement, since the partition functions where derived by looking at the system from the molecular level (statistically) whist the thermodynamic properties refer to the properties of the bulk. Let us illustrate how this relationship between the bulk properties and the molecular level properties q and Q are related with some examples.

The derivation of properties in terms of q is more 'basic' and hence, we shall produce full derivations for U and S in terms of the molecular partition function for distinguishable independent particles, q. We shall then proceed by another derivation, a general one in terms of Q, the canonical partition function.

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(5-1) The internal energy

The total energy of the system relative to the zero point energy* is given by:

(*) Recall that we has arbitrarily chose to set the lowest energy level (ground state) at 0J.

For distinguishable independent molecules, we have seen before that since the most probable configuration (i.e. the one with a highest W) is so strongly dominating, then we may re-express n_i in this energy equation in terms of the Boltzmann distribution, i.e. since (assuming q = q^tot):

we have:

But mathematics tells us that:

i.e. assuming that the partition function is only a variable of b (see below), we have

From this energy expression we may calculate the value of the internal energy by:

where U(0) is the zero point energy, that is the energy correction that we need to make since in our derivation of q we had assumed that the energy of the lowest states is equal to 0, i.e. e₀ = 0. This energy correction equates to the energy at T=0K, since at this point, all the molecules are in their lowest energy states.

We should also not that the partition may also depend on other variables apart from T (e.g. volume). In particular, we can replace the full derivative of the partition function by a partial derivative where all the other variables apart from T (or rather b) are kept constant, i.e.:

or in terms of ln(q) by: (since dq/q = ln q )

NOTE: A similar relationship may be derived for dependent molecules in terms of the canonical partition function, i.e.:

But we know that:

whist for most dominant canonical configuration (i.e. the most important one):

i.e.:

which means that (see derivation using q):

NOTE: An expression of U in terms of the molecular partition function can be obtained from the expression of U in terms of the canonical partition function by recalling that:

(1) For independent distinguishable particles, Q = q^N , i.e.:

i.e.:

(2) For indistinguishable particles, Q = q^N / N! , i.e.:

i.e. as in the case of independent distinguishable particles,

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(5-2) The statistical entropy

In this section we shall first justify one of the most important entropy relationships in thermodynamics, that is,

and then show how S can be derived from the partition functions.

We shall then also have a look at the residual entropy, that is, the entropy that is presnt at OK due to 'imperfections' in the crystal structure.

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(A) Verification of S = k ln W

From our internal energy equation we had seen that:

i.e.:
Now since (a) the energy levels do not change when the system is heated at constant volume (they would have changed if work would have been done on the systems), but (b) the populations change, then, in the absence of all changes other than heating:
But we know that under the same conditions:
i.e.:
The most dominant/important configuration (and the only one we need considering) is the one which has maximum W, or maximum ln(W)). This configuration may be obtained by solving:
dW=0 or d[ln(W)] = 0
where:
with the additional constraints that:
and
These three equations may be combined using the Lagrange method of undetermined multipliers to get (using arbitrary constants -b and a to get:

i.e. if we want to satisfy the condition that for the most dominant/important configuration dW=0 (i.e. d[ln(W)] = 0), for all dn_i we must have:
i.e.:
Thus,
But, since the number zero, the sum over dn_i is zero, i.e.:
and hence:
which integrates to:

QED

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(B) Derivation of S from the partition functions.

As in the case of the internal energy U, we shall commence our discussion by deriving the deriving S in terms of q, the molecular partition function for a system of N independent distinguishable particles.
We have seen that statistical entropy S relates to the weight W of the configuration through:
where
However, we have seen that for large W, we may write:
i.e.:
where from the Boltzmann distribution, we have:
i.e.:
and hence:
i.e.:
Note that the expression of S in terms of the molecular partition function for independent particles is different from the one derived here. This is discussed below (derived from Q).
As in the case of the internal energy, let us now have a look on how we may derive S in terms of the canonical partition function, Q.
The total weight of a configuration of the ensemble is the product of the average weight W of each member of the ensemble, i.e.:
Thus we have:
Thus as in the case of the derivation of S in terms of the molecular partition function, we now have:
Let us now obtain the expressions for S in terms of the molecular partition function from the expression of S in terms of the canonical partition function.:
(1) For independent distinguishable particles, Q = q^N , i.e.: lnQ = N lnq

(2) For indistinguishable particles, Q = q^N / N! , i.e.:
i.e.

which is not the same as the one derived for independent distinguishable particles.

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(C) Residual entropy

As we have said in the discussion of entropy in classical thermodynamics, we sometimes make the assumption that the entropy of a system at 0K is zero. However, this is only true for perfect crystals, and in practice, such perfection cannot be achieved. This 'entropy at 0K', S(0) is referred to as the residual entropy.
The origin and magnitude of the residual entropy can be explained by consider crystal composed of AB molecules, where A and B are similar atoms (such as CO, with very small electric dipole moment). There may be so little energy difference between … AB ABABAB … , … AB BA BA AB …, and other random arrangements that the molecules adopt in the solid. The entropy arising from this residual disorder can be calculated by the Boltzmann formula S = k ln W. Suppose that two orientations are equally probable, and that the sample consists of N-molecules. Because the same energy can be achieved in 2^N different ways (each molecule can take either of two orientations, AB or BA ), the total number of ways of the same energy is W = 2^N. Thus,
More generally, for solids composed of molecule that can adopt s orientations at T=0, the molar residual entropy is given by: