Thermal Physics Tutorial 3 - Molecular Kinetic Theory

Molecular Kinetic Theory Model

The model, which we will study in this tutorial, tells us how gases behave.  The topic fits quite well with mechanics because you need to understand about collisions to make sense of the Kinetic Theory model.  It may be worth revising these ideas in Mechanics in Tutorials 11 and 12.

In mechanics we have been looked at big things, like tennis balls, cars, people, to name but a few.  This can be referred to as the macroscopic level.  In this piece of work we will go down to the molecular level, where things are very small.  We call this the microscopic level.

A Scottish botanist, Robert Brown, first suggested that gases were in a state of random motion. He observed that pollen particles seem to jiggle about for no apparent reason. We can see the same by observing smoke particles under a microscope.  The random jerky movement of the particles is called Brownian motion and is due to the continuous and random bombardment by air molecules.

The work of Einstein, Maxwell, and Boltzmann gave us the kinetic theory, as we know it today.

Pressure Exerted by an Ideal Gas

Pressure is exerted by the impact of molecules on the sides of a container. We can use the idea to derive an expression for gas pressure. We must make the following assumptions:

The gas consists of a large number of identical molecules, such that the statistical treatments are meaningful.

• All collisions between molecules and the walls of the container are perfectly elastic.

• Intermolecular forces are negligible, as is gravity.

• Molecules move in straight lines and at constant speed between collisions.

• Collision times are negligible compared with time between collisions.

• Volume of the gas molecules is negligible compared to that of the gas.

• Newton’s laws of motion are applicable.

Health Warning:  The derivation of the kinetic theory equation is rather tedious but it's on the syllabus. 

(Fabio Capello led a dance troupe called Fabio's Fairies.  Their last performance was accompanied by a chorus of boos and B-flat vuvuzelas.)

Here we go:

Stage 1 Momentum Change

Consider a large box, side l, full of molecules:  

Stage 3 Working out the pressure

Since pressure = force ÷ area, we can work out the pressure on face 1.

The area of face 1 = l².

Suppose we have N molecules of gas. The pressure will be:

Now l³ is the volume of the gas, so we can rewrite the equation:

This can also be written as:

The term <c_x ²> ("cx-squared bar") is called the mean square speed of the molecules in the x direction.  The bar is written over the c², as above.  (Notice the different way of writing it in the text.  You may see this in some books.  It is also almost impossible to write c-bar in this web-page editor!) 

Stage 4 The Statistics

However the molecules are moving randomly in the container and very few would be moving exactly parallel to the x-axis.  However we can consider each molecule’s velocity to be the resultant of three components, c_x, c_y, and c_z.

As in two dimensions, the three components can be combined by Pythagoras to give the resultant velocity:

Similarly we can combine the mean square velocities:

Since there are a large number of molecules we can assume that there are equal numbers moving in each of the co-ordinate directions:

So we can write:

So our final equation becomes:

Which we can rewrite as:

Stage 5 Density

However we can go further.  Since Nm is the total mass of the gas and V is the volume, we can express the pressure in terms of the density.

density =   mass   =    Nm

             volume       V

This gives us the equation:

Internal Energy and Kinetic Energy of Molecules and Temperature

The internal energy of a material is the sum of its potential and kinetic energies.  In an ideal gas the molecules are so far apart that intermolecular forces can be ignored.  Therefore there is no potential energy.  So all the energy is kinetic.  The total kinetic energy is shared randomly throughout the molecules in the gas.  Therefore there is a random distribution of the speeds of the molecules, which we can show on a graph.

If no energy is being transferred as heat between an object and its surroundings, we say that it is in thermal equilibrium.  If a gas is in thermal equilibrium and it is not being compressed or expanded, the average kinetic energy will remain constant, so will the temperature.

We have seen that:

 pV = nRT

and:

It does not take a genius to see that:

We also know that kinetic energy,

or:

We can now rewrite our expression as (since 2/3 ×1/2 = 1/3):

We can rearrange this to write:

Since N is the total number of molecules and n is the number of moles, N/n will always be N_A, which is Avogadro's number, 6.02 ×10²³.

Now R/N_A is called Boltzmann's constant and is given the code k. 

We can easily work out the value for k.

So the equation now becomes: