The Maxwell-Boltzmann Distribution Valentim M. B. Nunes ESTT - IPT April 2015

The aim of statistical thermodynamics is the prediction of macroscopic properties (such as internal energy or entropy) from the properties of microscopic systems (atoms, molecules…). Assumes that the systems consist of large number of particles (~ ), then they present average values of the properties of interest.

Assembly – a set of systems (a particle or molecule, for example) Number of complexions,  - number of independent states accessible to an assembly, i.e. the number of ways in which we can distribute the particles by accessible energy levels.  = f (E,V,N) Principle of equal probability - all distributions of energy are equally probable. Some basic definitions :

From the macroscopic point of view the State of equilibrium of an isolated system is characterized by a maximum entropy, S. From the microscopic point of view is characterized by a maximum value of . k B = R/N A ≈  J.K -1 Boltzmann formula for entropy

The configuration is the distribution of the particles by the several energy levels,  i, possible. For N systems (particles) in which there are N 0 particles with energy  0, N 1 with energy  1, N 2 with energy  2, etc., the number of complexions is;

Maximizing  gives the most probable distribution, that is the more probable distribution, N 0, N 1,….., N i of the N particles (discernible) by the energy levels  0,  1, …,  i Is subject to the following restrictions :

Although all distributions are equally likely, there is one which imposes. As N tends to infinity,  max tends to  total, and the sum of the remaining terms to zero.

Maximizing a function subject to constraints, in this case total energy constant and constant number of particles. Method of Undetermined Multipliers of Lagrange

The most probable population of the state energy  i is given by : Maxwell-Boltzmann distribution

The Maxwell-Boltzmann distribution allows to know how N discernible* particles distributes by the several energy levels,  i. If there is degeneracy of the energy levels, then the distribution will come: * If the systems are indistinguishable, that does not affect the Boltzmann distribution, but only the macroscopic properties.

The molecular partition function, z, is a sum extended to all energy states accessible to each system. Plays a key role in statistical thermodynamics. We will see later that  = 1/k B T, then:

When T  0, z  g0 – all the terms of the sum are nulls except for the first one: Only the fundamental state is accessible When T  , z   All states are accessible We can conclude that the partition function gives an indication of the average number of states that are thermally accessible to systems, at a given temperature.

Consider a molecule with equally spaced energy levels and non- degenerate.  0 22 33

Rotational levels  E << k B T Vibrational levels  E  k B T Electronic levels  E >> k B T

The importance of the partition function is that it contains all the information needed to calculate the thermodynamic properties of a system of independent molecules at equilibrium.

As the average energy of the system identifies itself with the internal energy, U, and  = 1/k B T, we obtain the following expression for the internal energy of an assembly for which it is valid the Maxwell-Boltzmann statistics :

Entropy

Helmholtz energy: A= U-TS Pressure: Heat capacity:

Helmholtz energy : In the case of indistinguishable particles (e.g. gases) we have to divide the total number of complexions by a factor N! and the thermodynamic functions suffers some changes*. Entropy : * see tables 15.1 and 15.2 of the reference: Azevedo E.G., Termodinâmica Aplicada, 2ª ed. Escolar Editora, Lisboa