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.
The problem to be posed is therefore a combinatorial one:
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
at level 1 (the arrows pointing upwards and downwards increase and
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
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.
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
column. The difference of pressures between the bottom and the top
of the layer will equilibrate the weight of the air within the layer:
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.
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.
can be obtained by summing up all the molecules with the
same modulus of velocity, independently of the direction. Then
is constant in the integral, and then may be taken out.
The resulting integral is just the volume of a spherical shell of
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
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
|5.Heat||6.Work||7.First Law||8.Entropy||10.Specific Heat|