So far our discussion has been about the property and behaviour of a gas on a big (macro) scale i.e. pressure and volume, which involve billions of atoms or molecules.

Kinetic theory takes the view that if you really want to understand what’s going on you need to look at how individual atoms are behaving.

Ideal gases and Internal Energy

Normally we consider the internal energy of a substance to be made up of two components - E_k and E_p.

The E_k is the vibration, rotation or translation of the atoms.

The E_p is due to the interaction of one atom and its neighbours.

The impact of these two components on the temperature and state of the substance they belong to is discussed at the start of the ‘Thermal Properties’ section.

Once again as with our earlier gas work, we need to imagine an Ideal Gas that behaves in a perfect way. We’ve already stated that an Ideal Gas is one that perfectly obeys the Gas Laws. Here’s something else about Ideal Gases.

An Ideal Gas is one in which all its internal energy is in E_k form. i.e. it has no E_p. How can you justify this? Well E_p is as a result of one atom exerting a force on its neighbours. In ideal gases, each atom acts independently. It is unaware of the fact that any other atoms exist (except during collisions - but they only last for an instant!)

Brownian Motion

About 150 years ago a botanist, Robert Brown, observed pollen grains moving in a random way under his microscope. You can see the same effect with smoke today. We call it Brownian Motion.

Brown thought the movement of the grains may be explained by the fact that the pollen was alive! But in fact it was some years before it was explained that the movement was due to collisions between the pollen and millions of smaller (and therefore invisible) gas atoms.

The pollen was small enough to be knocked sideways when more air particles collided with one side of it than with the other. This was evidence of the continuous movement of gas atoms. Gas atoms have E_k.

Kinetic Theory Assumptions

Kinetic Theory then developed to explain this movement. The following assumptions are made to help the theory:-

1) There are a very large number of particles (molecules or atoms) involved. This means we can apply statistics to our solutions.

2) The particles are involved in perfectly elastic collisions with their containers and each other. So no E_k is lost.

3) The length of time involved in a collision is negligible compared to the time between collisions (i.e. we can ignore the moments when the potential energy component of the internal energy is not zero).

4) The volume occupied by the particles is negligible compared to the volume of the container. i.e. there are big gaps between particles so they have little or no effect on each other.

5) the particles are involved in random motion i.e. there is no resultant force on them - due to gravity, for example.

Using Kinetic Theory it is possible to derive the equations that describe and explain things like pressure and temperature in terms of the movement of individual molecules.

where

p = pressure

V = volume

N = No of atoms/molecules

m = mass of one atom/molecule

= mean square speed of the atoms

and

where

= density of the gas

because

You are  required to learn how to derive these from scratch.  

Root mean square speed, rms speed

The mean square speed, in the equation above is found by taking the average of all the “squared speeds” of the molecules. If you take the square root of this you get the root of the average squared speeds or root mean square speed.

This is not the same as the mean speed or average speed of the molecules but it is easily measurable (using ) whereas the average speed isn’t directly measurable.

Boltzmann constant and E_k

Remember:

pV = nRT

where

n = number of moles of gas

and

R = universal molar gas constant.

Boltzmann decided it would be useful if we know what the gas constant was per molecule, not per mole. He came up with a new constant - named the Boltzmann constant, of course - which was:

where

k = the Boltzmann constant

R = the universal molar gas constant

N_A = Avogadro’s number, the number of particles in one mole.

Then it was possible to rewrite

pV = nRT

as

pV = NkT

where

N = no of atoms or molecules involved.

If you combine

and

pV = NkT

to give

or, in words

The average E_k of the particles = a constant x temperature (in kelvin).

This confirms what we’ve mentioned a number of times in this section, that the temperature of a gas depends only on the E_k of the molecules that make it up.