Page 75 - Essentials of physical chemistry

3 The Kinetic Molecular

Theory of Gases

INTRODUCTION
We remind ourselves we are trying to present the essential aspects of physical chemistry and we
consider this one of the most essential topics. In our treatment of the van der Waals gas, we have
already mentioned the ideas of the collisions of small atoms, which have a lot of space between
them as in Dalton’s law. Here, we go into further detail regarding the behavior of gas molecules
using the ideas of Ludwig Boltzmann (1844–1906), who was one of the intellectual giants of the late
nineteenth century and whose ‘‘Boltzmann principle’’ of energy distribution is one of the pillars of
modern science. The breakthrough here was due mainly to Boltzmann’s PhD thesis on the theory of
gases. Here, we will first review the freshman chemistry derivation of part of kinetic molecular
theory of gases (KMTG) and then introduce Boltzmann’s amazing energy principle.

KINETIC ASSUMPTIONS OF THE THEORY OF GASES
1. A gas is made up of a large number of particles (molecules or atoms) that are small in
comparison with both the distance between them and the size of the container.
2. The molecules=atoms are in continuous random motion.
3. Collisions between the molecules=atoms themselves and between the molecules=atoms and
the walls of the container are perfectly elastic.

Let us consider the idea that gas pressure is caused by impacts of atoms=molecules with the wall of a
container (or the diaphragm of a pressure gauge). We know a gas will fill any shaped container, but to
make the derivation simpler, we assume a cubical container of dimension L L L where each side is
of length L (Figure 3.1). Thus, each inner face of the container has area A ¼ L L. Looking ahead to
the idea that pressure is force=area, we put just one atom in an empty cubical box and analyze the force
on one face of the box. Since force is a change in momentum, let us consider the left face of the box in
the y–z plane and assume the atom is moving only in the negative x-direction. This simplifies the
elastic bounce back into the positive x-coordinate, although in general, the assumed random direction
of a molecule would produce a series of random paths throughout the 3D volume of the box. Our
thinking is also constrained by a convention in thermodynamics that ‘‘change ¼ after before,’’ so
we arbitrarily choose the molecule initially traveling in the negative x-direction so that (~ v after ~ v before )
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is positive in the derivation. Similarly, the box is cubical to make V ¼ L , but the container could be of
any shape. Since we assumed the atom is in continuous motion and all collisions are elastic, it will
bounce off the wall and go in the reverse direction until it reaches the other wall and bounces back
again and so on. Since a change in momentum is a force, the collision with the wall causes the
(force=area) pressure. Since the collisions are perfectly elastic, the atom will bounce back and forth
rapidly. ‘‘Perfectly elastic’’ means that no momentum is lost in the collision, which is an approxima-
tion since a hot gas will cool and lose energy, but it is a good approximation over a short period of time.
Thus, for convenience, we show the particle moving in the negative x-direction and then reversing:

dp
¼ (mv x ) after ( mv x ) before ¼ 2mv x :
dt
A
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