Historical burdens on physics
103 The Maxwell speed distribution
Subject:
The distribution function of molecular speeds in a gas calculated by Maxwell admits the value zero for v = 0. For increasing speed values it goes
over a maximum value and tends again to zero for v → ∞. The most probable speed vmp, the mean value of the speed v and the root mean square
value vrms are different. The distribution is measured by means of a molecular beam. It can be visualized in a model experiment: Small moving spheres
that exit a model gas, are classified according to their speed.
Deficiencies:
1. The aspect of the curve of Fig. 1, that is usually called Maxwell distribution might surprise. It is to be expected that high speeds are seldom, as the
diagram shows. But shouldn’t the probability increase the smaller the
speed? In many texts this manifest question is not discussed. Actually, this
behavior of the function can be considered an artifact of an inconvenient
representation.
Fig. 1. Distribution of the absolute value of the velocity
The figure shows the distribution of the speed, i.e. the absolute value of the
velocity. Velocity is a vector quantity. The laws of mechanics get complicated and clumsy when formulating them for the absolute values of the mechanical quantities (velocity, momentum, force). If in our case, we do not
ask for the probability of finding a molecule with a speed in a given interval
dv, but for finding a molecule with a velocity vector in the interval dvx dvy
dvz, we get a Gaussian distribution centered at v = 0 (zero vector). Fig. 2
Fig. 2. Distribution of the x component of the velocity

Particles with velocity vector v
mv 2
–

F1(v ) = A·e 2kT
(1)
Particles with speed v
F2 (v ) = B ·v 2 ·e
–
mv 2
2kT
(2)
Particle flow with velocity v in a molecular beam
F3 (v ) = C ·v 3 ·e
–
mv 2
2kT
(3)
Particles with energy E
F4 (E ) = D · E ·e
–
E
kT
(4)
Fig. 3. Various probability distributions. Constants have
been merged into the factors A, B, C and D respectively.
shows the probability distribution for one component of the velocity vector,
see also equation (1) in Fig. 3. The reason why for v → 0 the absolute values of the velocity (the speed) tends toward zero is that we do not compare
equal volumes of the velocity space, but equal intervals dv. The volume
4π v 2 dv in the velocity space that belongs to dv, increases for a given dv
with the square of the speed, see equation (2) in Fig. 3. Hereby great
speeds are “privileged” and small speeds are “penalized”. In Maxwell’s
original work the two representations are clearly distinguished.
2. It is often emphasized that the curve of Fig. 1 is asymmetrical, but usually it is not said what is meant by that: Is it the fact the curve itself does not
have an axis of symmetry, or is it meant that it is not placed symmetrically
to the axis of ordinates. It is also said that because of this asymmetry the
values of vmp, v and vrms are different from one another. Sometimes it is
insisted that one has to clearly distinguish between these values. The problem for the student is, that he or se does not know in which context this distinction is important. Most probably students will never get the opportunity
to confound them.
3. Equations (1) and (2) make statements about the speed distribution of
the molecules of a gas, that is in thermodynamical equilibrium. Often it is
said or suggested that the same distribution hold for the particles in a particle beam, and that the distribution can be measured directly by means of
such a beam. Actually, the speed distribution in a particle beam has a similar shape as that of the particle speed distribution in the case of equilibrium.
However the functions are not the same, equation (3), Fig. 3. Here, the
speed in front of the Boltzmann factor is at the power of three [1]. (For
geometrical reasons there is a factor of v 2, but there is an additional factor
v since a fast molecule contributes stronger to the current density than a
slow one.)
4. Often, the importance of the speed distribution is emphasized without
saying what it is needed for. The distribution of the components of the velocity allows to calculate the pressure. For many other purposes the distri-
bution of the kinetic energy is needed. Also this distribution displays a certain similarity with the speed distribution, equation (4). It answers for instance the following questions [2]: “How many molecules of a gas have
enough energy to initiate an endothermal chemical reaction, or to ionize an
atom or molecule or to excite an atom, or escape from the gravitational field
of the Earth or another planet, or to overcome the electrostatic repulsion
between two atomic nuclei (in order to allow for a nuclear fusion reaction)?”
The only distribution that is not needed is that of the speed, i.e. the absolute
value of the velocity.
Origin:
1. The speed distribution is found in Maxwell’s work [3]. Maxwell’s results
are handed on from generation to generation, since Maxwell was a great
physicist.
2. Maybe an effort to justify the claim that the average speed is a measure
for the temperature.
3. The uncritical interpretation of the model experiment with the small
spheres.
Disposal:
Other distributions are more useful, like that of the components of the momentum vector or of the kinetic energy. The model experiment would better
be omitted.
[1] Döring, W.: Einführung in die theoretische Physik V, Statistische Mechanik, Sammlung Göschen, Band 1017, p. 16.
[2] Vogel, H.: Physik, Gerthsen - Kneser - Vogel, 13. Auflage, Springer-Verlag, Berlin, 1977, p. 169.
[3] Maxwell, J. C.: On the dynamical theory of gases; Philosophical Transactions of the Royal Society of London, Vol. 157 (1867) p. 49-88
Friedrich Herrmann, Karlsruhe Institute of Technology