The Kinetic Theory of Gases

Temperature as a measure of average kinetic energy of the particles

ObjectivesState the assumptions of the kinetic model of

an ideal gas.

State that temperature is a measure of the average random kinetic energy of the

molecules of an ideal gas.

Explain the macroscopic behavior of an ideal gas in terms of a molecular model.

Write & understand the equation of state.

The Kinetic Theory of Gases

• The properties of gases can be understood in terms of a simple but effective mechanical model. The gas consists of a very large number of molecules moving randomly about with a range of speeds and colliding with each other and the container walls.

The Kinetic Theory of GasesWe can make a model of this by making certain assumptions and seeing what these lead to. The basic assumptions of the kinetic theory of gases are:1. A gas consists of a very large number of molecules.2. Molecules moves with a range of speeds.3. The volume of the molecules is negligible compared with the

volume of the gas itself.4. The collisions of the molecules with each other and the

container walls are elastic (p and KE are conserved).5. Molecules exert no forces on each other or the container except

when in contact.6. The duration of the collisions is very small compared with the

time between collisions.7. The molecules obey Newton’s Laws of Mechanics.

The Kinetic Theory of Gases

Using these assumptions together with the laws of mechanics and the equation of state allows the

derivation of one of the most important formulae in physics, namely the Boltzmann equation:

.

The Kinetic Theory of Gases

Here, the speed v is defined by . So v is the square root of the average of the squares of the speeds of the molecules of the gas (the average of the velocity

vectors of all the molecules is zero since they move randomly in all directions). We call v the root mean

square speed or rms speed. (It must be realized that v is not the average speed of

the molecules – but it is numerically close to the average, so we are usually excused for calling v the

average molecular speed, even if it is not technically correct.)

The Kinetic Theory of Gases

Appearing in this equation is a new constant of physics, the Boltzmann constant k, the value of which is k=1.38x10-23 J K-1 (it is the ratio of the gas constant R

to the Avogadro constant). The meaning of this important relation is that the average kinetic energy

of molecules is proportional to absolute temperature, that is, the absolute temperature is a measure of the

average kinetic energy of the molecules of a substance.

ExampleFour molecules have speeds of 300 m s-1, 350 m

s-1, 380 m s-1, and 500 m s-1. Find the average speed and the root mean square speed.

ExampleIf the root mean square molecular speed is

doubled, what is the new temperature?

Molecular explanation of pressure

• The pressure of a gas originates from the collisions of the molecules with the walls of its container. At every collision, each molecule has its momentum changed, and so a force acts from the wall onto the molecule. By Newton’s Third Law, the molecule exerts an equal and opposite force on the wall. The total force due to all the colliding molecules, divided by the area over which the force acts gives the pressure of the gas.

Molecular explanation of pressure

• Pressure is the normal force per unit area. The pressure in a gas results from the collisions of the gas molecules with the walls of its container (and not from collisions between molecules).

Molecular explanation of pressure

From the molecular point of view, we may identify two factors that affect the pressure of the gas. The first is the average molecular

speed (the higher the speed, the larger the change in momentum of the molecules and so the higher the force). The second factor is the frequency of collisions. The more frequent the collisions, the higher the pressure. Thus in providing molecular explanations for

pressure it is sufficient to remember that roughly. As an application of this, consider a gas that has been heated under constant volume. The molecules are moving faster on average

(increased temperature) and the frequency with which the collisions take place also increases (the time between collisions is reduced since molecules are moving faster). For both reasons, the

pressure increases.

ExampleA gas is compressed slowly by a piston. Explain why the temperature of the gas will stay the same.

Example

A gas is compressed rapidly by a piston. Explain why the temperature of the gas will increase.

Example

A gas expands isothermally. Explain from a molecular point of view why the pressure decreases.

The Equation of State

– Pressure– Volume– R=8.31 J K-1 mol-1– n = number of moles– T = temperature in

Kelvin

• A gas that obeys this law at all temperatures and pressures is said to be an ideal gas.

• Real gases obey this law only for a range of temperatures, pressures, and volumes.

Example

A gas is heated at constant pressure. Explain why the volume must increase as well.

Example

How many moles of gas are there in a gas of temperature 300 K, volume 0.02 m3 and pressure 2 x 105 Pa?