9.1 Distribution of particles in energy levels. The Boltzmann factor.



9.2 Molecular speed distribution.

The first part of this Section is devoted to the introduction of the Boltzmann factor. It will be first introduced in a qualitative way, to be used in an experience in which a certain amount of energy is distributed among a given number of particles. In the second part, the velocity distribution of the particles of an ideal gas will be analyzed and compared with that expected from a theoretical basis according to the Boltzmann factor.

9.1 Distribution of Particles in Energy Levels. The Boltzmann factor.

Given N particles with total energy E,
- How is the energy distributed provided that there is a random exchange of energy between the particles?
- Shall the energy distribute in such a way that all particles have the same energy?
As discussed in the previous Section, the energy will be distributed according to the most probable configuration.

The problem to be posed is therefore a combinatorial one:

Given N particles with total energy E, how many configurations are possible with n1 particles having energy e1, n2 particles having energy e2, ... with n1+n2+.. =N and n1*e1+n2*e2..=E?
To investigate this we shall carry out the following

By clicking the button a window appears where particles may be placed at different energy levels, from level 1 to level 20. Let us begin with few particles, say six particles and ten energy units. We first place particles at level 1 (the arrows pointing upwards and downwards increase and decrease the number of particles in each level,respectively). The energy counter shows how much energy has been already spent. When five particles have been placed at level 1, there is only one particle left and five energy units, so that the only possibility is placing that particle at level 5.
What is the number of different ways of choosing five particles from six? Clicking "Compute" gives the answer: 6. 

Try now other configurations compatible with the constraints of having ten energy units to be distributed among six particles, and find out that one with the largest number of possibilities. This will correspond to the "Maximum Probability Configuration". Click here to show this configuration. 

Try now to find out the most probable configuration corresponding to fifteen particles with forty energy units. Click here to get some help. 

Q9.1. The "Maximun Probability Configuration" is obtained:
- When the largest number of particles have the same energy
. - Giving the highest energy to a particle and the lowest energy to the rest.
- Placing the maximum number of particles in the lowest level and quickly decreasing the occupation of the levels as their energy increase.

The Boltzmann factor.

We have found in a qualitative way that the particles in a system are distributed among the different energy levels following an exponential law, with less particles at higher energies. The question can be posed of finding the probability that a particle has an energy E when it is at equilibrium with a system at temperature T. This probability is found to be inversely proportional to the exponential of the energy.

Particle distribution in the atmosphere.

Let us see how the Boltzmann factor can be obtained in the case of a system of particles in a gravitational field.

By clicking the button a column of an ideal gas is shown, in a uniform gravitational field as it is the case in the earth atmosphere. The atmospheric pressure at the earth's surface is caused by the weight of the column of air placed above a unit area. Press CPanel, put 200 or 300 particles and press Execute. The particles distribute nonuniformly due to gravity.

Let us analyze an infinitesimal layer of thickness dh in this column. The difference of pressures between the bottom and the top bases of the layer will equilibrate the weight of the air within the layer:


where m is the mass of a molecule, n the number of molecules per unit volume and g the acceleration of gravity. Taking into account the state equation of an ideal gas, P=nkT, differentiating and substituting into the previous expression leads to


a differential equation whose solution is a function of the height h which is an exponential.


The Boltzmann factor appears again. In this case it is the potential energy what is found in the exponent. The density of particles decreases exponentially with h.

9.2 Molecular speed distribution. (*)

Let us consider an ideal gas in a container of volume V, in equilibrium at temperature T (**). Pick a molecule of the gas. What is the probability that its speed is between v and v+dv?

Consider the quantities:
- F(v)dv, average number of molecules, per unit volume, with speed between v and v+dv.

- f(v)dv, average number of molecules, per unit volume, with velocity between v and v+dv.

F(v)dv can be obtained by summing up all the molecules with the same modulus of velocity, independently of the direction. Then


f(v) is constant in the integral, and then may be taken out. The resulting integral is just the volume of a spherical shell of radius v and thickness dv in velocity space, so that we finally get


We consider now an experience in which a number of particles are given initially the same kinetic energy, and then are allowed to interact between them. The experience shows the evolution of the distribution of kinetic energy among the particles.
Click here to create two windows. In the upper window there is a cube containing a number of particles placed in a plane, to optimize their mutual interaction and reach an equilibrium state in a reasonable time.
The lower window contains the controls to initialize and run the simulation, rotate the cube, change the number of particles which initially is set to 100, and also give us information about the energy and temperature of the system.
The window under the label "Energy distribution" shows the theoretical distribution of particles as a function of their energy with a continuous line. Since all particles have the same energy at the beginning, there is a single bar. When the particles are put into motion and the energy redistributes among them the bar plot results in a good agreement with the theoretical curve, despite the low number of particles in the system.
The distribution of energy among the particles of an ideal gas in equilibrium
Depends on the initial distribution of energy among the particles.
Increases proportional to the square of the particle speed and decreases exponentially with the square of the speed.
Depends on the volume occupied by the system.
(*) STATISTICAL PHYSICS. Berkeley Physics Course, Vol. 5, F. Reif.
(**)STATISTICAL PHYSICS. Berkeley Physics Course, Vol. 5, F. Reif. 6.2 Maxwell's Velocity Distribution.

Specific Heat
Index 1.Introduction 2.Pressure 3.Temperature 4.Internal Energy
5.Heat 6.Work 7.First Law 8.Entropy 10.Specific Heat