0.14 The kinetic molecular theory  (Page 7/8)

This view must be incomplete, though. Each individual particle might create a force proportional to 2v , but there are many particles hitting the wall, generating force, and our pressure gauge can’t possibly measure each tiny impact. So we need to take a different view. The total force generated by all of these tiny impacts will be determined by how many of these impacts there are. If the particles hit the wall more often, then the force will be higher. What determines how frequently the particles hit the wall? One factor should be how dense the particles are in the container. If there are a great many particles in a small volume, then many of the particles will be near the wall and collide with it. So, the frequency of the collisions of the particles with the walls of the container should be proportional to N/V , where N is the number of particles. A second factor would be how large the area of our pressure gauge is, A . A larger surface would be proportionally more collisions. A third factor would be how fast each particle is moving. Faster particles will create more frequent collisions with the wall. Each of these factors individually makes sense.

It is important to note that we have calculated the force of each tiny impact completely independently of the force of impact of any other particles. In fact, from our postulates, we have assumed that the individual particles have no effect on each other since they are so far apart from each other. This is why we can think of the force created by the gas as coming from a huge number of collisions, each one independent of all the others.

Putting these factors together, the frequency of collisions should be proportional to (N/V)Av . If we multiply this by the force of each collision, the total force impacted will be proportional to (2mv)(N/V)Av . Finally, the pressure is the force per area, so we wind up with the result that pressure P must be proportional to 2mv 2 N/V , or:

\text{P}=\frac{\text{kNm}{\text{v}}^{\text{2}}}{\text{V}}

( k is just some proportionality constant which we will need to find. We dropped the 2 since it is just a proportionality constant too.)

This result is very promising. It says that P is proportional to the number of particles N , which we could also write as the number of moles, n . That agrees with the Ideal Gas Law. It also says that P is inversely proportional to V . That also agrees with the Ideal Gas Law.

But there are two ways in which this equation looks different from the Ideal Gas Law. The first is that temperature is missing. This is because there was nothing in our postulates about temperature because we had no experiments which told us about how temperature affects molecular motion. The second is the appearance of the term mv ² . From Physics, this is a very familiar expression, since the kinetic energy of a particle of mass m moving with speed v is ½ mv ² . Notice that the pressure is proportional to the kinetic energy of the particles.

It is hard to solve the first concern. Temperature as we measured it in the previous Concept Development Study is an arbitrary measure of hot and cold. We simply observed that this measure turned out to the proportional to the pressure of an ideal gas. However, if we compare our equation to the Ideal Gas Law, we can make progress. The Ideal Gas Law tells us that pressure is proportional to n/V times the temperature T . Our equation above tells us that pressure is proportional to N/V times the kinetic energy of the particles, ½ mv ² . This tells us that the temperature T is proportional to the kinetic energy of each particle, ½ mv ² .