Physical Chemistry , 1st ed.

In this chapter on kinetic theory, we will consider the origin of the pressure

of gases. We will find that the speeds of gas particles can have many values but

the distribution of their speeds can be calculated. So can an average speed—in

several different ways. We will also consider how many times gas particles col-

lide with each other, how far they travel between collisions, and how far they

travel from an arbitrary starting point. One of the more curious things from

kinetic theory is the prediction that gas particles are moving very fast indeed,

but because of all their collisions their net displacements change rather slowly.

19.2 Postulates and Pressure

The kinetic theory of gases is based on several postulates, or statements that

are presumed but not proven. In this respect, it is similar to Dalton’s theory of

the atomic structure of matter, which is also based on certain presumed state-

ments. The kinetic theory of gases is based on the following statements:

1. Gases are composed of tiny particles of mass.


2. These tiny particles are in constant motion when in the gas phase.


3. These tiny particles do not interact with each other, nor with the walls of
   the container. That is, there are no forces of attraction or repulsion be-
   tween any two particles or a particle and the wall. (We will clarify this
   statement shortly.)


4. These tiny particles do collide with each other and the walls of the con-
   tainer. However, when a collision occurs, the total energy before the col-
   lision equals the total energy after the collision. One way of expressing an
   ideal collision like this is that the total energy is conserved (it does not
   change) and that collisions are elastic.
   From these statements, the mathematics and predictions of the kinetic the-
   ory of gases can be determined.
   First, let us address a property commonly measured for a gaseous phase: its
   pressure.This is one of the basic observable properties of a gas. Where does the
   pressure of a gas come from?
   If gas particles are constantly moving (which is the second postulate above),
   then each gas particle has some kinetic energy. Classically, kinetic energy has
   the formula
   kinetic energy ^12 mv^2 (19.1)
   where mis the mass of the moving body and vis its velocity. Velocity is a vec-
   tor (although we are not indicating it as such here), and in three dimensions
   we can separate velocity into its three components vx,vy, andvz. If we do this,
   equation 19.1 becomes
   kinetic energy ^12 m(vx^2 vy^2 vz^2 ) (19.2)
   In a collection ofNgas particles in some volume V, each gas particle has its
   own particular kinetic energy (because we haven’t constrained the kinetic en-
   ergy at all; so far, the kinetic energy of any gas particle could be anything).
   Therefore, there are Nequation 19.2’s that when added together give the total
   kinetic energy of the gas.
   As the postulates above mention, the gas particles are constantly moving
   and, in the course of some of the motions, are colliding with the wall of the
   container that holds the gas. Figure 19.1 shows a diagram of a single gas par-
   ticle colliding with the container wall. Before the collision, the particle has

652 CHAPTER 19 The Kinetic Theory of Gases

Gas particle

(vi)

Gas particle

(vf)

Wall

Figure 19.1 A gas particle colliding with the
wall of a container. Initially, the particle has ve-
locity vi. After the collision, the particle has ve-
locity vf. Even if the magnitude of the velocity has
not changed, the gas particle has accelerated be-
cause the velocity’s direction has changed. Kinetic
theory uses this model to understand the pres-
sure of a gas.