Crystals encourage both teachers and students to ask questions. We should encourage the suggestion that some things must be arranged in a regular array inside crystals, things too small to see called ‘atoms'. Otherwise it is difficult to see why crystals make such regular shapes, and how they ‘know to make the same shape every time’.

(Of course professional crystallographers appreciate that the same material makes a great variety of shapes, as judged by the layman, although they all share the same fundamental pattern. To students, at their first careful look at crystals, the idea of some standard shape is likely to seem quite clear.)

Younger students will certainly have heard the word ‘atom’ but only some will know what it means. Though the word is used easily, most will not have reached the stage of wondering about things that are too small to see. So, the idea needs introducing gently.

When students have been given the idea of small particles or bits of which everything is composed, they are likely soon to take the idea of atoms for granted. So the question ‘How big are atoms?’ can be asked as something to wonder about while leaving it unanswered until more evidence has been gathered.

When introducing crystals to young students, molecules and ions are probably best left aside from the discussion.

As a crude picture that will lead to a rough estimate, assume that each molecule in a liquid occupies a cubical box of side, d , the diameter of the molecule. (Of course real molecules are not hard lumps like billiard balls and certainly not spherical.)

At first glance, this may seem to place the molecules too close together for liquid behaviour. However, the volume of space occupied, d³ , is almost twice the volume of a sphere with diameter d , so the assumed cubical spacing would have liquid behaviour.

In the closely packed array we imagine for a liquid, the spacing for molecules, neighbour to neighbour, is therefore d , one molecule diameter.

'How much greater is the spacing in a gas such as air?'

If the spacing in the gas is D , then a volume of D³ is needed for each gaseous molecule.

The ratio d³ / D³ = volume occupied by a liquid gas/volume occupied by the same mass of the gas.

So, the average separation of the gas molecules is the (volume of gas/volume of liquid) ^1/3 x the molecular diameter.

The change in volume from the liquid to the gaseous phase for:

• air (nitrogen) is about 1: 750
• carbon dioxide about 1: 850
• petrol at 90°C about 1: 800
• water at 100°C about 1: 1650 (water is unusually high)

If we assume that the volume change is about 1: 1000, then we can conclude that in a gas, the molecules are about 10 molecular widths apart, on average. Nine diameters apart would give a volume change of 729 and 12 would give 1728.

This shows the powerful effect of a cube root in estimating molecular separations.

Students will probably have met the simple molecular picture of solids, liquids and gases. Matter is, in every case, made up of atoms or molecules, but with different amounts of arrangement (‘order’) and different types of motion.

The story of the development of the kinetic theory is too good a story to split up and tell as individual experiments. It demonstrates the interaction between observation and experimentation. Thinking about the causes of observed behaviour involves model-building and theorizing.

Show your students models, preferably in several forms, but murmur gentle warnings that a model is not the real thing. This kind of reservation is discouraging when young people first meet it. But they can learn to enjoy devising models and thinking in terms of them with much greater freedom and skill and imagination once they realize the scope of scientific models as scientists use them.

Show your students real balls as models of gases. Imagining their paths and collisions will help in thinking about gases, to suggest a line of investigation, or to illustrate a technical term such as ‘mean free path’. You might contrast such molecular models with mock-up models, such as a tiny wooden model of a fission reactor or a huge wax model of a flower. The latter models are used to aid people in visualising; the former are used for constructive thinking.

Discussion of physical models can lead on to an algebraic model developed from the kinetic theory for the pressure of a gas. The algebraic model can be used to make predictions. The equation of the pressure of a gas can be used, for example, to estimate the molecular speed and mean free path of the molecules. It could also lead to an estimate of Avogadro’s number from some simple measurements. Teaching-models such as these cannot prove a theory right. But they help students to visualize and understand the theory.

All theoretical physics uses models as essential parts of the framework of knowledge, but with great care to remember where the words, phrases, descriptions, are only parts of models. All models have their limitations as well as strengths, and it is vital to understand both. Without imaginative thinking in terms of models, scientific knowledge would be merely a pile of facts, codified here and there in laws, little more than a handbook of data.

Even if you cannot put these ideas over to students, think about the nature of science and the part played by models in it when you use demonstration models as part of your teaching.

Statistical studies, combined with the assumption that every molecular collision is elastic, lead to the conclusion that the molecules of all gases at the same temperature store the same amount of energy kinetically. This ignores one very important physics principle, the equi-partition of energy theorem.

The full form of this theorem states that each degree of freedom will on average have the same energy. The linear motion of molecules entails three degrees of freedom – motions in the x, y and z directions. A gas whose molecules are simple atoms, such as helium or neon, has only these three kinds of motion. A molecule made from two or more atoms, however, can move in additional ways: it will have rotations and vibrations.

At the beginning of the twentieth century, it seemed clear that equi-partition should apply to the energies of rotation and vibration. However, certain experiments, such as those measuring thermal capacities over a wide range of temperatures, threw increasing doubt on this. The full form of equi-partition theorem failed, and quantum theory explained why.

Equi-partition among linear motions still applies. All gas molecules, at a given temperature, have the same average kinetic energy.

Useful results

Average energy for molecules of gas A = average energy of molecules for molecules of gas B, at the same temperature. m_A < v_A²> = m_B < v_B²>

So you can compare molecular speeds if you know the relative molecular masses.

Another equation derived from the kinetic theory of gases: pV = 1/3 Nm < c² > where p is the gas pressure, V is gas volume, N the number of molecules, m their mass and < c² > their mean square speed. This equation tells you that equal volumes of gases at the same pressure and temperature must contain the same number of molecules (Avogadro’s rule).

The atoms or molecules in a gas are widely separated from one another and interact only when they collide. Each one changes its direction of travel only in collisions. This theory of a gas relates macroscopic quantities that we can measure (such as pressure, volume and temperature) to the motion of its molecules. Developed in the nineteenth century, it is called kinetic theory because the molecules are always in motion.

Suppose the pressure of the gas on the walls is produced by the impact of bombarding molecules. Attach a pressure gauge to the box and read the pressure.

Now suppose that you put more and more molecules into the box until there are twice as many as before. The picture below shows a microscopic demon popping in additional molecules one by one, through a trap door.

When there are twice as many molecules as before, what pressure would you expect? With twice as many molecules to bombard the walls, you might expect double the pressure.

Of course with more molecules, collisions would also happen more often in the middle of the box. But those internal collisions would not affect the bombardment of the walls, for the following reason. Suppose there are two molecules moving in opposite ways, heading for opposite ends of the box, where each will add its contribution to the pressure. If they do not collide, but just pass each other, each will arrive at its target end. If they do collide head-on, they will rebound, elastically. Then each of them takes on the other one’s job and does what the other would have done without a collision.

Therefore, you might expect double the number of molecules to produce double the pressure. Yet all the pressure gauge can notice is a doubling of the local population of molecules, in other words, a doubling of density.

There is another way that you could double the density. Push the end wall of the box in, as a piston, so that the air occupies half the volume. Result: the pressure gauge would show the pressure doubling. So collision theory leads to a prediction: Halving the volume of a gas will double its pressure. Does experiment confirm this is true?

Robert Boyle did an experiment to investigate the relationship between pressure and volume more than three centuries ago. He was not testing a theory, but simply taking measurements. In 1661 he announced his result ‘concerning the spring of air’ to the Royal Society. It is now called ‘Boyle’s law’. Provided the temperature is kept constant, the product of gas pressure and volume is constant.

Real gases

Real gases do not behave in this ideal way. Molecules are not point-like particles, and there are likely to be so many of them in the box that space for movement is reduced. Molecules move a shorter distance to and fro. Bombardments happen a little more often, and the pressure is a little larger than for point-like particles.

Also molecules attract each other when they are fairly close, as surface tension shows. This effect decreases gas pressure when molecules are crowded close together and moving slowly, at low temperatures. Some of the slowest molecules are weeded out and never reach the walls. There is a less dense layer near the walls and that exerts smaller pressure.

In real gases, both of these effects are found at high pressure and small volumes, but they can be distinguished because they change in different ways with temperature. At low temperatures the effects of molecule size remain much the same, but the effects of attraction make themselves felt so strongly that gases can liquefy.

Robert Brown is correctly referred to as having observed the jittering motion of small particles. But he did not observe the motion of actual pollen grains. How many text books and other resources continue to hand on this mistake?

A 2001 paper published in Nature alleged that the first recorded observation of what we now call Brownian motion was made in 1785 by Jan Ingenhousz using charcoal dust [Ref: Nature, 7 June 2001 p 641]. This appears not to be the case. See the Microscopy website...

Microscopy

The webpage may also be of interest for its two short videos showing Brownian motion, one in whole milk and the other in a smoke cell.

Having used particles derived from living matter, Brown had to try several other inanimate substances to convince himself that the motion he observed was not something to do with a life force, but a property of all microscopic matter. This systematic investigation is what won for Brown the accolade of having the jittering motion named after him, work that Ingenhousz didn't need to do.

Today's research into nano-technology now routinely fabricates nano-particles. Controlling them suspended in liquids is quite a task. One method is to use a direct current controlled by a feedback system to cancel out the Brownian motion. The position of the 20 nm polystyrene spheres is monitored by a fluorescence microscope and the voltage across the solution altered accordingly. So far nano-particles have been confined to within 1 micron. Alternatively, the path of the particle can be manipulated by suitable changes of the applied voltage. [Ref: Nature , 10 March 2005 p 156]

Even before the recent advent of nano-technology, Einstein's 1905 paper on Brownian motion is his most cited paper (more than for special relativity or his work on photons). It is used by scientists working on such varied topics as aerosol particles (pollution), the properties of milk, paints, granular media (powders) and semiconductors. [Ref Nature, 20 January 2005 p 216]

Thanks to David Walker for pointing out an error on this page, now corrected. Editor

Solid carbon dioxide is known as dry ice. It sublimes at –78°C becoming an extremely cold gas. It is often used in theatres or nightclubs to produce clouds (looking a bit like smoke). Because it is denser than the air, it stays low. It cools the air and causes water vapour in the air to condense into tiny droplets – hence the clouds.

It is also useful in the physics (and chemistry) laboratory.

The Institute of Physics has kindly produced this video to explain how dry ice is formed.

Safety

Dry ice can be dangerous if it is not handled properly. Wear eye protection and gauntlet-style leather gloves when making or handling solid carbon dioxide.

Uses

Dry ice has many uses. As well as simply watching it sublime, you could also use it for cloud chambers, dry ice pucks, and cooling thermistors and metal wire resistors in resistance experiments. It can also be used in experiments related to the gas laws.

Obtaining dry ice

There are two main methods of getting dry ice.

1. Using a cylinder of CO₂

It is possible to make the solid snow by expansion before the lesson begins and to store it in a wide-necked Thermos flask.

Remember that the first production of solid carbon dioxide from the cylinder may not produce very much, because the cylinder and its attachments have to cool down.

What type of cylinder, where do I get CO₂ , and what will it cost?

A CO₂ gas cylinder should be fitted with a dip tube (this is also called a ‘siphon type’ cylinder). This enables you to extract from the cylinder bottom so that you get CO₂ in its liquid form, not the vapour.

NOTE: A plain black finish to the cylinder indicates that it will supply vapour from above the liquid. A cylinder with two white stripes, diametrically opposite, indicates it has a siphon tube and is suitable for making dry ice. A cylinder from British Oxygen will cost about £80 per year for cylinder hire and about £40 each time you need to get it filled up. (The refill charge can be reduced by having your chemistry department cylinders filled up at the same time.)

Don't be tempted to get a small cylinder, it will run out too quickly.

If the school has its own CO₂ cylinder there will be no hire charge, but you will need to have it checked from time to time (along with fire extinguisher checks). Your local fire station or their suppliers may prove a good source for refills.

CLEAPSS leaflet PS45 Refilling CO₂ cylinders provides a list of suppliers of CO₂.

A dry ice attachment for the cylinder

Dry ice disks can be made using an attachment that fits directly on to a carbon dioxide cylinder with a siphon tube. Section 13.3.1 of the CLEAPSS Laboratory Handbook explains the use of this attachment (sometimes called Snowpacks or Jetfreezers). This form is most useful for continuous cloud chambers and low-friction pucks.

You can buy a Snowpack dry ice maker from Scientific and Chemical. The product number is GFT070010.

2. Buying blocks or pellets

Blocks of solid carbon dioxide or granulated versions of it can be obtained fairly easily with a search on the Internet. Local stage supply shops or Universities may be able to help. It usually comes in expanded foam packing; you can keep it in this packing in a deep freeze for a few days.

The dry ice pellets come in quite large batches. However, they have a number of uses in science lessons so it is worth trying to co-ordinate the activities of different teachers to make best use of your bulk purchase.

The experimental graph of pressure against temperature is likely to be straight enough to justify asking whether the plotted points (the true results of the experiment) fit closely to an ideal simple law. This is what we would like to find because it makes our description of nature easy and simple. If some students have graphs that do not seem to suggest a straight line, then a picture gallery of everybody's graph can be organized. That will enable the class to extract a general conclusion, as in a professional research team's work.

Then suggest each student should extend the line backwards to look for absolute zero. They can either draw new axes so the temperature scale can be extended backwards to about 300ºC below the ice point or pages of graph paper stuck to the original graph until the line cuts the temperature axis. It is also possible to calculate the position of absolute zero algebraically, using the slope of the line.

Get students to think on these lines:

"What happens to the motion of the molecules when you cool the air? Think of cooling the air more and more... could you cool it until its molecules had no motion at all? Suppose there was such a temperature: somewhere far down on the scale of the thermometer at which molecules would have no motion at all. What would the pressure be like at that temperature? If we trust our picture of gas models we expect the pressure to fall to nothing."

Students should emerge with a clear idea that, judged by a mercury thermometer, gas pressure runs down an almost straight line as temperature falls. The straight line reaches zero pressure at a temperature somewhere between -250ºC and -300ºC (-273°C), called absolute zero. Tell them this process of extending the graph backwards beyond the reading is called extrapolation. It is a risky process because we do not know if the gas will continue to behave in the same way.

You could ask students to imagine an ideal gas and discover what the temperature would be at which that gas would collapse with no useful motion.

Say:

"The absolute scale of temperature can be defined by shifting the zero from the ice point to this new zero and reckoning all temperatures from there. All we do is add 273 to all Celsius temperatures in order to create the temperature on this new Kelvin scale. The close agreements amongst many gases persuade us to redefine our temperature measurement for gas thermometers and then finally move to the Kelvin scale. The Kelvin scale has several advantages:"

• If you keep the volume of a sample of gas constant, its pressure goes up in direct proportion to the Kelvin temperature. This is automatically true for an ideal gas; fortunately many gases have almost identical behaviour, except at very low temperatures.
• For standard thermometers, you can change from ordinary mercury thermometers, which are convenient, to a gas thermometer. This is a bulb and pressure gauge similar to the class experiment. Instead of using it to investigate a sample of air, turn the argument round and say:

"'Henceforth we choose to measure temperatures on a scale that uses a gas thermometer with an ideal gas in it"

• There is a very fruitful theory of heat engines, thermodynamics, which offers many remarkable predictions, all of which necessarily use the Kelvin scale. Without the idea of that scale, and without practical gas thermometers for measurements, the predictions of thermodynamics would be useless, just hot air.

"You will find that real gases give an almost straight line graph when their pressure is plotted against Kelvin temperature. The expansion of mercury happens, by a lucky chance, to give a fairly straight line when plotted against Kelvin temperature measured by a gas thermometer. That lucky chance makes it comfortable to use mercury thermometers for measuring ordinary temperatures in the laboratory. For higher temperatures, Bunsen flames, mercury thermometers are useless and in very cold weather the mercury freezes. The Kelvin scale extends from zero as high as you like, millions of degrees in nuclear fusion."

A risky guess

When students continue their graph (or calculate) on down to find absolute zero, they are making a risky guess that the behaviour of a gas would stay the same. Explain that continuing beyond all measurements like that is called extrapolation.

You could say that:

Extrapolation is a risky business, trusting or pretending that what you have observed continues on and on.

Did the Sun rise in the East this morning? Did it rise in the East yesterday? Did it rise in the East many a morning before that? Are you willing to extrapolate these observations into the future and say that you are sure the Sun will rise in the East tomorrow? Are you quite sure?...

Suppose you were not a human being but were a butterfly who emerged from a chrysalis on a warm day in early summer and flew about from flower to flower, day after day. What would you observe about the weather? A fine day, another fine day, another fine day, ... If you were that insect, you might extrapolate and predict that every day in the future would be fine. You would not foresee the wintry day which would end your happy flights.

Extrapolation and interpolation

Extrapolation would be very risky if we trusted it for the molecules of a real gas like air or carbon dioxide. If we cool any gas enough, it fails to remain a gas, for example carbon dioxide becomes cold solid crystals of dry ice. Air that has been cooled down and pushed together so that its molecules (moving relatively slowly at that low temperature) hang together in a liquid.

You could say:

Yet we can safely make the extrapolation in imagination and find a useful absolute zero as a starting point for the grand Kelvin scale of temperature.

Interpolation means reading something off a graph between two measured points on it. (Or you can calculate an intermediate value between two values given in some table.) Interpolation is useful in science and engineering; and, if carefully done, it is safe. But in developing new science and technology, extrapolation is more important.

Although extrapolation is risky, it is the way in which some of the great discoveries have been made. Scientists guess what might happen if they continued our knowledge into an unknown region, then they try to test their guess by experiments. And sometimes, those experiments lead them in quite a new direction of knowledge.

Extrapolation is rash but sometimes very fruitful; interpolation is safe but dull.

Theory, modelling, guessing and experimenting are all intertwined. Each step progressing from one idea to the next. However, this is a very cleaned up view of the progress of science. Science is much messier than this and many ideas lead to dead ends and wrong predictions.

Knowing:

• The diameter of an air molecule, 4 x 10^-10m,
• The space occupied by a molecule in liquid, (d³= {4 x 10^-10}³= 64 x 10^-30m³),
• The change of volume from a liquid to gas

You can calculate how many molecules there are in a room, (4m x 3m x 3m = 24 m³) giving about 5 x 10²⁶ molecules.

This is in fact an estimate of the Avogadro number for a kilo-mole. A kilogram mole of any gas contains 6 x 10²⁶molecules. It occupies 22.4 m³at 0 °C, or about 24 m³at room temperature, and atmospheric pressure.

Mass of an air molecule

Number of molecules in a room 24 m³, N = 5 x 10²⁶

Mass of air molecules in a room 24 m³, M = Vp = 24 x 1.2 kg = 28.2 kg

Therefore, Mass of an air molecule = 28.2 / 5 x 10^-26 = 5.6 x 10^-26kg

When students know more about the structure of air (mainly nitrogen and oxygen) then the mass of their atoms can be estimated (they are fairly close in mass).

All this comes from imagining a theoretical picture, guided by the things we know about nature, such as Newton's Laws of Motion, and then making estimates and measurements.

Imagine molecules in a gas; dots spaced far apart. Add arrows to show the random motion, not all speeds (arrows) the same but speeds around the average. You could say:

'Here is a snapshot of air molecules in this room with the camera focused at one distance. To find how one molecule would move through this vast array of moving neighbours is too difficult a business. Instead, pretend that we freeze all the molecules except one and watch that one molecule go hurtling through the crowd.'

Redraw the picture showing each molecule as one round blob without any indication of velocity. Draw the path of the chosen molecule, as it moves to collide with another, as a cylinder swept out between the two molecules. The diameter of the cylinder is equal to the diameter of the molecule and its length is equal to the mean free path. Bend the path at the collision and another cylinder is swept out as shown this diagram:

The mean free path is many times longer than the separation between molecules and so the cylinder should pass many other molecules on the way to a collision.

Now move off to a separate preparatory discussion looking at such a collision in detail. Draw a large round molecule bouncing against another molecule.

'How far apart are the molecules, centre to centre, at the collision? One diameter.'

'I am now going to show you a trick for finding out how far a molecule goes before hitting another. This trick has been invented by scientists and is not what really happens but gives good results. When two molecules collide they must be 2 radii, or 1 diameter apart. Instead of drawing the collision like that, I could pretend that the molecule flying along to make the collision is much bigger, and any other molecule that it hits is much smaller. We get the same result as long as we have the centres of the two molecules 1 diameter apart at the collision. I am now going to push this to the limit and make the flying molecule have double the radius, equal to 1 diameter, and the molecule it hits have no radius at all.'

'Now we start this story all over again. Here is the artificial molecule flying along with radius equal to one molecular diameter. It sweeps out a cylinder of 1 molecular diameter in radius and collides with the artificial point sized molecule where it bends its path.'

'Think about the path swept out by this flying molecule which is possessively patrolling its "share" of the volume of the box. This volume is equal to d²x 10^-7m.'

For justification of mean free path being 10^-7m, see Guidance note...

Further discussion of mean free path

'We need to know the volume of space that belongs to one molecule of air in this room. The volume change from liquid air to air is about 1:750. If for liquid air each molecule of diameter, d, occupies a cubical box of side d, then the volume occupied is d ³on the average.'

750d³= d²x 10^-7

d = 4 x 10 ^-10m

'We have found the diameter of a typical molecule of air. An atom is probably about half that size. This is certainly a rough estimate because our measurements were difficult and we made all kinds of risky moves carrying out our calculations. Yet this is a very good estimate for many working purposes. It is the right order of magnitude.

All we are really measuring here is an order-of-magnitude distance of approach at which inter-molecular forces grow large enough to have a noticeable effect. Air of course is a mixture of different gases, mainly nitrogen (about 78%) and oxygen (about 21%).

Careful measurements for particular molecules give different diameters according to the experiment chosen and the method of interpretation used. After all, the diameter of a molecule is not as definite a thing as the diameter of a steel ball. Both nitrogen and oxygen are diatomic molecules. Not only are diatomic molecules oblong but they behave as if squashy, so more violent collisions are likely to reveal a smaller effective diameter. Nitrogen molecules are very slightly larger than oxygen molecules; in their gaseous state both have effective diameters of about 3 x 10^-10m.