## Episode 603: Kinetic model of an ideal gas

Lesson for 16-19

- Activity time 100 minutes
- Level Advanced

This episode relates the gas laws to the behaviour of the particles of a gas.

Lesson Summary

- Discussion and demonstration: explaining pressure in terms of particles (15 minutes)
- Discussion: deriving an equation for the pressure of a gas (30 minutes)
- Discussion: the link between energy and pressure (20 minutes)
- Worked example and student questions: Calculating molecular speeds (20 minutes)
- Discussion: internal energy of a gas (15 minutes)

#### Discussion and demonstrations: Explaining pressure in terms of particles

In the previous episode, we looked at the macroscopic behaviour of a gas, in terms of its temperature, pressure and volume. Now we can go on to relate this behaviour to the underlying microscopic behaviour of the particles of which the gas is made.

Remind your students of the description of pressure arising from bombardment by particles of the walls of the container. Ask how the pressure will change if:

- the particles move faster (i.e. higher
*T*)? - the particles have greater mass
*m*? - there are more particles?

(All of these will result in greater pressure. You may wish to point out at this stage that increasing v has two effects: a greater force on impact, and more frequent impacts.)

You can usefully demonstrate the heating effect when the air in a bicycle pump is compressed.

Episode 603-1: Warming up a gas by speeding up its particles (Word, 46 KB)

#### Discussion: Deriving an equation for the pressure of a gas

pV=13Nm{*c*^{ 2}}

The discussion above will prepare your students for the derivation and use of the equation for the pressure of an ideal gas, where N is the total number of molecules in the volume *V*, and the bar indicates an average. Check whether your specification requires the derivation. It is covered in most of the major texts. Even if the derivation is not required, you will have to explain the terms used, and show that the equation is plausible. Point out that the quantities on the left are macroscopic, while those on the right are microscopic.

Stress the underlying assumptions of the model. The gas:

- has zero volume at zero temperature so the volume of the actual molecules is negligible
- has zero pressure at zero temperature so heating is the only way to get the molecules moving
- have atoms or molecules which behave as elastic spheres with no long-range intermolecular forces

There are a number of points to look out for in the derivation. Students will need convincing about the idea of averaging the *square* of a velocity; the average velocity is zero, because particles with opposite velocities cancel out. The change in momentum is 2*m**v* when an atom rebounds. The square of the velocity arises because *v* increases both the momentum change and the frequency of impact. We can average a series of impulses into a smooth net force to give an impulse Ft. All of this will rely on having covered momentum and impulse thoroughly.

If necessary try a demonstration to help convince the students. Balls strike a force sensor at increasing frequencies and with increasing velocity. The force sensor records the pressure which results.

Episode 603-2: One collision: many collisions (Word, 97 KB)

#### Discussion: The link between energy and pressure

You can now compare the expression:

pV=13Nm{*c*^{ 2}}

With the ideal gas law to show the equivalence between temperature and the average energy per molecule (Given the elastic sphere assumption, this energy is only stored kinetically). This idea follows directly from a statistical analysis of elastic collisions but the maths involved is too advanced for the post-16 level. However the idea may be demonstrated by computer models that have programmed into them only the laws of dynamics for collisions, and nothing about thermodynamics. Such programs demonstrate the plausibility of the idea and are a powerful visual stimulus.

Episode 603-3: Kinetic theory applets (Word, 24 KB)

If your specification requires it, make that link formal with:

12*m**v*^{ 2}=32*k**T*

Where *k* is Boltzmann’s Constant, and
*k* = *R**N*_{A}.

#### Worked example and student questions: Calculating molecular speeds

This equation allows us to calculate mean molecular speeds, which should be done as a worked example. Calculate the rms speed for oxygen molecules at room temperature; ask the class to repeat the calculation for nitrogen, or for other molecules in air. They will find that the lighter the molecule, the faster its particles move.

Compare these speeds with the speed of sound in air (~330 m s^{-1}), and with the Earth’s escape velocity (11 km s^{-1}). You could also draw a comparison with familiar diffusion rates (e.g. speed with which a stink bomb is detected), leading to the idea of collisions and mean free path, which need not be entered into too deeply. The following question would be excellent to work through, leaving students to do some parts, and helping them through trickier concepts such as ratios.

Episode 603-4: Speed of sound and speed of molecules (Word, 58 KB)

#### Discussion: Internal energy of a gas

- Energy flowing into a liquid initially raising the temperature, then boiling it, then heating it as a gas. All stages increase internal energy
- Joule’s paddle wheel experiment where instead of heating increasing the internal energy, mechanical working is converted directly to internal energy. The temperature is raised and the equivalence of heating and doing mechanical work as two means of transferring energy is demonstrated. This is leading towards the first law of thermodynamics
- Cooling something down by evaporation. Compare when it is on a surface that then supplies further energy to keep the temperature constant (people sweat to cool down) and evaporation when there is no such reheating, so that the liquid cools (puddles evaporate away totally). A drop of isopropyl alcohol on the back of the hand demonstrates the cooling effect clearly – it feels cold as it evaporates

You are now in a position to talk about *internal energy* as the total energy stored kinetically and chemically in the molecules. Energy that is stored chemically in the bonds must be considered as you need to transfer energy to melt a solid or boil a liquid; this does not go into the energy stored kinetically by the molecules as the temperature stays constant. Another key idea here is that the energy calculated from the temperature using
32*k**T* = 12 *m**v*^{ 2}
is an average and, therefore, there must be a distribution: some particles will have greater than this and some less. This leads to explanations of everything from evaporation to chemical reaction rates.

All of the above are quite subtle, sophisticated ideas that require the students to absorb definitions of closely related concepts. As such it cannot be rushed, and discussion should be spaced out with examples.