Physical Chemistry of Foods

This clearly asks for an explanation, and an attempt will be made
below. To that end it is useful first to say something about the kinetic energy
of molecules.

4.3.1 The Maxwell–Boltzmann Distribution

All atoms, molecules, or particles constantly move (velocityv), except at
absolute zero temperature. They thus have a translational kinetic energy,
according to basic mechanics given byð 1 = 2 Þmv^2 , wheremis mass. (In
addition, they have rotational and vibrational kinetic energy, but these do
not concern us at this moment.) It can be shown that for all particles,
whether small or large, the average kinetic energy hUi depends on
temperature only, according to

hUi¼



1

2



mhv^2 i¼



3

2



kBT ð 4 : 8 Þ

where Boltzmann’s constantkBis a fundamental constant of nature, its
magnitude being 1: 38? 10
^23 J?K
^1. Incidentally, Eq. (4.8) also can be seen
as the definition of temperature; notice that temperature thus can only refer
to a fairly large ensemble of molecules or atoms, since it is defined in terms
of an average.
As mentioned, there is a spread in kinetic energy, and the energy
distributionof the molecules or particles is at any temperature given by

dlnN¼

dN

N

¼ 2



U^0

p

 0 : 5

expð
U^0 ÞdU^0 ð 4 : 9 Þ

which is known as the Maxwell–Boltzmann distribution; it can also be given
as a velocity distribution.Nstands for the number of molecules etc., and
U^0 ¼U=kBT. The distribution is depicted in Figure 4.3a. It can, of course,
be recalculated in terms of Ufor any T, and then it is seen that the
distribution becomes wider and flatter for a higher temperature: Figure 4.3b.
The essential point is that a certain proportion of the molecules has a
kinetic energy above a specified level (say, 10
^20 J), which proportion is
given by the area under the curve in Figure 4.3b above the specified level
relative to the total area. For a given temperature, this proportion is the
same for any kind of molecule (or atom or particle). For a higher
temperature the proportion is higher. If a higher energy level is specified, the
proportion is smaller.

Notes For small species, the translational kinetic energies are very

significant, but for large ones they may be (very) small compared to