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

Statistical thermodynamic provides the necessary link between the microscopic properties of matter (i.e. properties that can be directly linked to the atoms/molecules in the system) and the bulk properties, including the all the thermodynamic properties (e.g. T, p, H, …) we have encountered before. 

The thermodynamic properties (e.g. T, p, H, …) are concerned with an average behaviour. They are characteristic of large assemblies of molecules, not individual ones. 

This means that at any time, the actual instantaneous value of a thermodynamic property in a particular sector of the system may differ slightly from the 'average value' of the bulk. However, such fluctuations turn out to be negligible compared with the established average since the number of particles contributing to the average is so large (e.g. 1 mole 6.02 x 10²³ particles, that is, 602,000,000,000,000,000,000,000 particles!).

Let us assume that we have a system with:

• a total N molecules, 
• a total energy E.

It is NOT possible to show how E is distributed amongst the individual molecule because of ceaseless re-distribution of energy (i) between different molecules, and (ii) in between different modes.

Instead, we can report on 'population of a state' that is, report that on average, there are n_i molecules in a state of energy e_i . Note that:

• (1) Population of states remain almost constant, even though the actual molecules (if we were to name them individually) that occupy the states may change with collisions. 

 
• (2) Our problem may be redressed as 'the calculation of the population of states for any type of molecule, in any mode of motion, and in any temperature. 

 
• (3) We apply the following restrictions:

(i) the molecules are independent, that is, that the sum of energy is the sum of the energies of the single molecules (i.e. we ignore any contribution to the energy that may arise as a result of intermolecular interactions).

(ii) We are adopting the principle of equal a priori probabilities, that is, assuming that all possibilities for the distribution of energy are equally probable. Thus for example we assume that (a) vibrational states of energy e, and (b) rotational states of the same energy e, are equally likely to be populated. 

• Any individual molecule may exist in states with energy e₀, e ₁, e ₂, etc. We shall let e₀ = 0 (arbitrarily). 
• Other e_i are reported relative to e₀. This means that if we want to compute U, we may need to add a constant to the calculated energy of the system (e.g. if we are considering the vibrational contribution to the internal energy, we must add the total zero-point energy of any oscillators in the sample (see for example Fig. 1 for the case of a harmonic oscillator) energies.

Fig. 1: (i) The parabolic potential energy V = 1/2 kx² a harmonic oscillator, where x is the displacement from equilibrium. The narrowness of the curve depends on the force constant k: the larger the value of k, the narrower the well. (ii) The energy levels fo a harmonic oscillato are evenly spaced with separation where w = (k/m)^1/2. Note that even in its lowest state, an oscillator has an energy greater than zero.

At any instant there will be n₀ molecules with an energy e₀, n₁ molecules with an energy e₁, n₂ molecules with an energy e₂, etc. This instantaneous configuration may be written in the form of:

Fig. 2: Whereas a configuration {5, 0, 0, . . . } can be achieved in only one way (part i), a configuration {3, 2, 0, . . . } can be achieved in the ten different ways (part ii). The tinted blocks represent different molecules.

Thus may be generalised by saying that there are W ways for achieving a configuration {n₀, n₁, n₂,…} of N molecules, where W (the weight of the configuration) is given by:

This may be approximated by the fact that for large x,

(2.2) The dominating configuration

From our discussion above, we have seen that some configurations are more important than other. In particular, there is a particular configuration that is the most dominant (i.e. the one which has maximum W, or maximum ln(W)), and this may be obtained by solving:

dW=0 or d[ln(W)]=0

In addition to this we must also impose the constraints that:

(i) The total energy of the system must remain constant, i.e.:

(2-3) The Boltzmann distribution and the molecular partition function

At maximum W, the populations at a temperature T in the configuration depend on the energy of the states according to the Boltzmann distribution:

is the fraction of moles in state i

is the molecular partition function (summing over states)

k = Boltzmann's constant, a constant that is related to the gas constant R and Avogadro's number, N_A through: 

 

NOTE: Sates and levels

Note that the molecular partition function q is sometimes written to include degeneracy. Several distinct states may correspond to the same energy level, that is, energy levels may be degenerate (see Fig. 4)

If energy level e_j is g_j-fold degenerate, then the molecular partition function may be written as:

Fig. 4:  Several distinct states may correspond to the same energy. That is, each energy level may be degenerate. Three energy levels are shown here, possessing one, three, and five distinct states.

(2-4) More on energy states, levels, etc.

The energy of a molecule may be classified into four sub-headings:

1. Translational energy
2. Rotational energy
3. Vibrational energy
4. Electronic energy. 

A typical energy separation between the ground state and the first electronically excited state of an atom or molecule is about 3 eV, which corresponds to 300 kJ mol^-1. In a sample at 25 °C (298 K) the ratio of the populations of the two states is about e^-121, or about 10^-53. Therefore, essentially every atom or molecule in a sample is in its electronic ground state The population of the upper state rises to about 1 per cent of the population of the ground state only when the temperature reaches l0,000^oC! 

The separation of vibrational energy levels is very much less than that of electronic energy levels (about 0.1 eV, corresponding to l0kJmol^-1), but nevertheless at room temperature only the lowest energy level is significantly populated. Only about 1 in e⁴ (c. 60) molecules is not in its ground state. 

Rotational energy levels are much more closely spaced than vibrational energy levels (typically, about 100 to 1000 times closer), and even at room temperature we can expect many rotational states to be occupied. In the case of translational energy levels, under most normal circumstances, the energy levels are close to each other hence forming a so called 'virtual continuum'. 

Therefore, when considering the contribution of rotational/translational motion to the properties of a sample, we need to take into account the fact that molecules occupy a wide range of different states, with some rotating/translating rapidly and others slowly. Moreover, more states are occupied at high temperatures than at low temperatures. The distribution over rotational/translational states has a very a peculiar shape (see fig. **). This is because each energy level actually corresponds to a number of degenerate states in which the molecule is 'rotating at the same speed but in different orientations' or 'translating with the same speed, but with different values of the velocity components relative to the x,y,z-axes'. This degeracy increases at high speeds. Therefore, although the populations of individual states decrease with increasing energy (Boltzmann distribution), there are many more states of a given energy at high energies and the product of this rising degeneracy and the falling exponential function has a bulge at an intermediate energy.
 
 
 

(a) A representation of the quantization of the energy of different types of motion. Free translational motion in an infinite region is not quantized, and the permitted energy levels form a continuum. Rotation is quantized, and the separation increases as the state of excitation increases. The separation between levels depends on the moment of inertia of the molecule. Vibrational motion is quantized, but note the change in scale between the ladders. The separation of levels depends on the masses of atoms in the molecule and the rigidities of the bonds linking them. Electronic energy levels are quantized, and the separations are typically very large (of the order of 3 eV).

(b) The Boltzmann distribution for three types of motion at a single temperature. There is a change in scale between the three stacks of levels. Only the ground electronic state is populated at room temperature in most systems, and the bulk of the molecules are also in their ground vibrational state. Many rotational states are populated at room temperature as the energy levels are so close. The peculiar shape of the distribution over rotational states arises from the fact that each energy level actually corresponds to a number of degenerate states in which the molecule is rotating at the same speed but in different orientations. Each of these states is populated according to the Boltzmann distribution, and the shape of the distribution reflects the total population of each level.

Fig. 5: The different types of energies

NOTE: For non-interacting gas molecules, different translational energies now correspond to different speeds, so the Boltzmann formula can be used to predict the proportions of molecules having a specific speed at a particular temperature. The expression giving the proportion of molecules that have a particular speed is called the Maxwell distribution.  (see fig. 6). Notice how the tail towards high speeds is longer at high temperatures than at low, which indicates that at high temperatures more molecules in a sample have speeds much higher than average. The speed corresponding to the maximum in the graph is the most probable speed, the speed most likely to be found for a molecule selected at random.