Temperature and Kinetic Theory

so what is the significance of the equation    

The quantity   means very little ( the average value of all the velocities squared ) but the square root of this, is significant

imagine there are 10 molecules in the gas with the velocities below.

average velocity	average speed
0.1	5.1	30.7	5.5

for a very large number of molecules moving randomly the average velocity is actually zero ( as velocity is a vector quantity )

notice that the r.m.s. velocity ( root mean square ) is a pretty good approximation for the average speed.

The air pressure in the room you are in is about 1 x 10⁵ N/m² and its density is about 1.2 kg / m³. Use the equation above to get a value for the r.m.s. velocity of the air molecules in your room. You might find the answer surprisingly high, but then remember Brownian motion.

Not all molecules will have the same speed. The variation is called a Maxwell speed distribution

If we combine the equation above with the equation of state for an ideal gas we get some very interesting conclusions.

for 1 mole of an ideal gas p = RT / V  and      where     = M / V  so  1/3 M = R T

now M = L m  where L is Avagadro's constant and m is the mass of a particle so 1/3 L m = R T

we now define a new quantity k = R / L called Boltzmann's contant  so 1/3 m = k T

multiplying both sides by 3/2 gives       1/2 m = 3/2 k T

The left hand side of this equation is the average kinetic energy of the particles in the gas. This equation shows that the absolute temperature of the gas is directly proportional to the average kinetic energy of its particles. An important result which greatly enhances our understanding of temperature.

In the following pages we are going to approximate the average energy to be just k T