University of Malta 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 (iii)
 

(4) Ensembles

(4-1) Introduction (The concept of an ensemble, the canonical ensembles, the canonical partition function, other types of ensembles)

(4-2) The relationship between the canonical partition function and the molecular partition function.

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(4) Ensembles 
 
 

(4-1) Introduction

An ensemble, or more specifically a canonical ensemble is set up as follows:

(a) Take a closed system of a specified composition (N), volume (V) and temperature (i.e. constant NVT)

(b) Create a set of Ñ imaginary replications of this closed system

This collection (ensemble) of replications must satisfy the following conditions:

(a) All the identical Ñ imaginary replications are in thermal contact with each other, so that (i) they all have the same temperature T, and (ii) they can exchange energy in between them.

(b) The total energy of the whole set of ensembles is constant, and equal to . 
 


Fig. A representation of the canonical ensemble, in this case for Ñ = 20. The individual replications of the actual system all have the same composition and volume. They are all in mutual thermal contact, and so all have the same temperature. Energy may be transferred between them as heat, and so they do not all have the same energy. The total energy () of all 20 replications is a constant because the ensemble is isolated overall. 
 

Note that:

(a) Ñ (which bears no connection with N, the number of molecules in a member of the ensemble) can be as large as one wishes. In particular we may wish to set Ñ as approaching infinity, i.e. the so called thermodynamic limit.

(b) The energy of a single member in this ensemble (i.e. one replica) is denoted by Ei (The subscript i is there to indicate that within the same ensemble, members may be in a different state of energy Ei from each other.) 

(c) The number of members (replicas) in a state with energy Ei is denoted by ñi. This means that we now have a scenario where we may speak of a configuration with an associated weight , where:

(i) The total energy of the ensemble is given by:
(ii) The weight of a configuration of the ensemble is given by:
(d) The canonical distribution corresponding to the configuration with a maximum weight is given by:
where Q (a function of T) is the canonical partition function given by:
(e) Note that the total energy of the ensemble is given by:
whilst the average energy of the members of the ensembles is given by:
(f) Although the canonical ensemble (or NVT since members have common N, V, T,) is the more common ensemble, there are also other types of ensembles. For example, the more common are, 
(i) The conical ensemble, or NVT, 
(ii) The microcanonical, or NVE
(iii) The grand-canonical ensemble, or mVT
but we also have NPT, NPH, etc.

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(4-2) The relationship between the canonical partition function for independent molecules and the molecular partition function.

The relationship between the between the canonical partition function and the molecular partition function is as follows:
 
(1) For distinguishable independent molecules, Q = qN

(2) For indistinguishable independent molecules, Q = qN / N!

Note: Although the canonical partition function 'Q' is similar to the molecular partition function 'q' in the sense that it can give all the thermodynamic information about the system, it is superior in the sense that the canonical partition function does not assume the molecules to be independent. This means that we can use Q to tackle a wider range of problems, including real gases and condensed phases, where clearly, the molecules are interdependent. It also means that not all Q's can be related to their respective q's - only the ones which refer to independent molecules.

Before we go to justify these relationships between Q and q, let us clarify the meaning of the terms 'distinguishable' and 'indistinguishable'. If all molecules were to be identical and free to more through space (e.g. in the liquid/gas phase), then the molecules would be indistinguishable. However, if for example, all the molecules in the system were to have a label, say for example, we are describing a system in the solid state, the molecules would be distinguishable as their location is distinguishable. Also, for example, the molecules in the system could all be chemically different! 

Let us now verify these relationships between Q and q for independent molecules. 

Let us first consider a system made up from distinguishable molecules. Let us choose a random member of the ensemble, and let it have an energy Ei, that is, belong to state 'i'. The canonical partition function of this ensemble is given by:

However, the energy Ei of this member of the ensemble can be written as:
which means that the canonical partition function may be written as:
where ei(1) represents the energy of molecule '1' when the system is in state 'i', ei(2) represents the energy of molecule '2' when the system is in state 'i', and so on. (Note that individual molecules are themselves in one of their j molecular states). We can factorise out each particle in turn from the summation over the system states and then gather together all the terms that refer to a given molecule, to give:
To illustrate this, we shall consider a small system with only two molecule (i.e. N = 2) which can assume j-molecular states, where j =3 (we shall denote these as a, b, c). In this case, the member of the ensemble may exist in i states and have energies:
Thus if we where to write out the canonical partition function for this system and expand it, we would get (since molecules are distinguishable):


From the above summation, we may observe that if the molecules where to be indistinguishable, then, for example, the terms when (i) molecule 1 would be in state a, and molecule 2 in state b, and (ii) molecule 1 would be in state b and molecule 2 in state a, would become equivalent. This means that the number of possible different states for a system of indistinguishable molecules is less than then when the molecules are distinguishable. This means that qN is an over-estimation of the canonical distribution for indistinguishable molecules with a factor of N!, that is for system with N fully indistinguishable particles, Q = qN / N!.
 
 

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