Kinetic Theory of Gases

A container is filled with a quantity of gas.  The assumptions of the Kinetic Theory of gases are the following:

1. All the molecules of a pure gas have the same mass and are in random motion.  The mass is small so gravity is ignored.
2. The gas molecules are separated by large distances.
3. The molecules exert no forces on each other except when they collide.
4. All collisions of gas molecules with each other or the walls of the container are perfectly elastic.

A molecule of gas can be analyzed using the laws of mechanics.  If a molecule of gas collides with the a wall of the container...

Impulse = Δ Momentum

Force_avg · Δ Time = Δ p

F_avg · Δ t = m · v_f - m · v_o

It is assumed that the direction of the molecule is perpendicular to the wall of the container.

If the gas molecule - wall collision is elastic then v_f  = - v_o The final velocity is equal in magnitude but opposite in direction.

F_avg · Δ t = [ m · v ] - [ m · (-v) ]

F_avg · Δ t = 2 m · v

If the container is a cube with sides of length L then the elapsed time Δ t  between collisions with a particular wall will be 2 L / v

F_avg · 2 L / v = 2 m · v

F_avg  =  m · v² / L

Since N particles move randomly in three dimensions, one third of them [ N / 3 ] on average strike the one particular wall of our container.  Therefore the total force is

F  = [ N / 3 ] · m · v² / L

James Clerk Maxwell (1831-1879) figured out a distribution of molecular speeds that a container full of molecules will have.  The molecules will have an average speed, but what we need is an average value of the squared speeds.  The square root of the average value of the for the squared speeds is called the root mean-square speed, v_rms.

F  = [ N / 3 ] · m · (v_rms)² / L

Pressure is force per unit area, so the pressure P acting on a wall of area L² is

P = F / L²

P = [ N / 3 ] · m · (v_rms)² / L   /   L²

P = [ N / 3 ] · m · (v_rms)² / L³

P = 1/3 · N · m · (v_rms)² / L³

The volume of the box V = L³.

P = 1/3 · N · m · (v_rms)² / V

P V = 1/3 · N · m · (v_rms)²

Since Kinetic Energy = 1/2 m v² then the Average Molecular Kinetic Energy of a molecule of gas, KE_avg = 1/2 m (v_rms)²

P V = 1/3 · N · 2 · KE_avg

P V = 2/3 · N · KE_avg

The ideal gas law P V  =  N k_B T looks very similar.

2/3 · KE_avg = k_B · T

The result is an equation that allows us to calculate the Average Molecular Kinetic Energy of a molecule of gas.

KE_avg = 3/2 k_B T

This equation indicates that Temperature and Average Molecular Kinetic Energy are directly proportional.

KE_avg =  1/2 m (v_rms)² = 3/2 k_B T

As you would expect, the average molecular kinetic energy is zero at a temperature of absolute zero.

(v_rms)² = 3 k_B T / m

A monatomic ideal gas is composed of single atoms.  Because the atoms are not connected by chemical bonds, there is a lack of  intermolecular forces meaning that the atoms have no potential energy. Therefore the total Internal Energy of a ideal monatomic gas would be the sum of the Kinetic Energy of each individual atom.

.Internal Energy = Number of Molecules · KE_avg

U = N · KE_avg

U = N · 3/2 · k_B  · T

But it is a little impractical to measure the amount of gas in terms of number of molecules of gas.  Typically we measure amounts of gas in terms of moles.  Remember that 6.022 x 10²³ molecules of gas = 1 mole of gas. The equation can be rewritten as

U = n · 3/2 · R  · T

n is the number of moles of gas & R = 8.31  J / (mol K)