Gases

Kinetic-Molecular Theory

The ideal gas equation

PV = nRT

describes how gases behave.

• A gas expands when heated at constant pressure
• The pressure increases when a gas is compressed at constant temperature

But, why do gases behave this way?

What happens to gas particles when conditions such as pressure and temperature change?

The Kinetic-Molecular Theory ("the theory of moving molecules"; Rudolf Clausius, 1857)

1. Gases consist of large numbers of molecules (or atoms, in the case of the noble gases) that are in continuous, random motion
2. The volume of all the molecules of the gas is negligible compared to the total volume in which the gas is contained
3. Attractive and repulsive forces between gas molecules is negligible
4. The average kinetic energy of the molecules does not change with time (as long as the temperature of the gas remains constant). Energy can be transferred between molecules during collisions (but the collisions are perfectly elastic)
5. The average kinetic energy of the molecules is proportional to absolute temperature. At any given temperature, the molecules of all gases have the same average kinetic energy. In other words, if I have two gas samples, both at the same temperature, then the average kinetic energy for the collection of gas molecules in one sample is equal to the average kinetic energy for the collection of gas molecules in the other sample.

• The pressure of a gas is causes by collisions of the molecules with the walls of the container.
• The magnitude of the pressure is related to how hard and how often the molecules strike the wall
• The "hardness" of the impact of the molecules with the wall will be related to the velocity of the molecules times the mass of the molecules

• The absolute temperature is a measure of the average kinetic energy of its molecules
• If two different gases are at the same temperature, their molecules have the same average kinetic energy
• If the temperature of a gas is doubled, the average kinetic energy of its molecules is doubled

• Although the molecules in a sample of gas have an average kinetic energy (and therefore an average speed) the individual molecules move at various speeds
• Some are moving fast, others relatively slowly

• At higher temperatures at greater fraction of the molecules are moving at higher speeds

• The average kinetic energy, e, is related to the root mean square (rms) speed u

Example:

Suppose we have four molecules in our gas sample. Their speeds are 3.0, 4.5, 5.2 and 8.3 m/s.

• The average speed is:

• The root mean square speed is:

• Because the mass of the molecules does not increase, the rms speed of the molecules must increase with increasing temperature

• Constant temperature means that the average kinetic energy of the gas molecules remains constant
• This means that the rms speed of the molecules, u, remains unchanged
• If the rms speed remains unchanged, but the volume increases, this means that there will be fewer collisions with the container walls over a a given time

• Therefore, the pressure will decrease (Boyle's law)

• An increase in temperature means an increase in the average kinetic energy of the gas molecules, thus an increase in u
• There will be more collisions per unit time, furthermore, the momentum of each collision increases (molecules strike the wall harder)
• Therefore, there will be an increase in pressure
• If we allow the volume to change to maintain constant pressure, the volume will increase with increasing temperature (Charles's law)

1996 Michael Blaber