VELOCITY DISTRIBUTION

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

DISTRIBUTION OF PARTICLES IN ENERGY LEVELS

Given N particles with total energy E,

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

EXPERIENCE:

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.

Distribution of Maximum Probability

Different distribution of 10 energy units between 6 particles.

Distribution of 10 energy units between 6 particles: 5 on 1 and 1 on 5 Ways to distribute 6 particles in this configuration: 6
Distribution of 10 energy units between 6 particles: 2 on 1 and 4 on 2 Ways to distribute 6 particles in this configuration: 15
Distribution of 10 energy units between 6 particles: 4 on 1, 1 on 2 and 1 on 4 Ways to distribute 6 particles in this configuration: 30
Distribution of 10 energy units between 6 particles: 3 on 1, 2 on 2 and 1 on 3 Ways to distribute 6 particles in this configuration: 60

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

Distribution of Maximum Probability

Distribution of 40 units of energy between 15 particles with high probability.

Distribution of 40 units of energy between 15 particles with high probability    Ways to distribute 15 particles in this configuration: 37.837.796

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.

Boltzmann factor

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 "Controls", put 200 or 300 particles and press "Run". 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:

Difference of pressures between bottom and top layers

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

Differential equation

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

Equation for n

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.

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 can be obtained by summing up all the molecules with the same modulus of velocity, independently of the direction. Then

Equation F(v)dv

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

Volume of a spherical shell

Equation F(v)dv

(*) STATISTICAL PHYSICS. Berkeley Physics Course, Vol. 5, F. Reif.
(**) STATISTICAL PHYSICS. Berkeley Physics Course, Vol. 5, F. Reif. 6.2 Maxwell's Velocity Distribution.