Unit 9: Part 3

Contents

• 3. Chance and diffusion
  • The purpose of the discussion about chance and diffusion
  • Diffusion and mixing as one-way processes
  • A collection of many two-way processes can look one-way
  • Large numbers and small numbers
  • Chance is enough to work out what happens on average
  • Counting numbers of ways
  • Processes that happen inexorably, but quite by chance
  • Summary

3. Chance and diffusion

'There are few laws more precise than those 
of perfect molecular chaos.' 

Professor George Porter. From 'The laws of disorder' (1965) BBC Publications.

'The probable is what usually happens.' 

Aristotle.

The purpose of the discussion about chance and diffusion

This Part is, to some extent, a digression from our main theme, which is an understanding of the flow of heat and thermal equilibrium. Its purpose is to prepare the way for that understanding, which will come out of questions about the random sharing of energy amongst atoms. These matters, discussed in Parts Four and Five, are quite subtle, though they are far-reaching in their consequences.

In Part Three, therefore, we look at how similar ideas can be applied to a simpler, if less important problem: the mixing of one substance with another. The results of Part Three are not, however, all trivial. Most chemical reactions involve the mixing of substances, and this fact has direct consequences for the way the progress of a reaction depends upon the proportions in which each substance is present. Some of these consequences are drawn out in Part Six, which deals with applications of the ideas developed in the rest of the Unit.

In Part Three, we also introduce one other idea we shall use later. This is the notion of using dice-throwing games to imitate some features of the behaviour of systems which are subject to chance. This tactic makes it possible to find out how a system behaves by a mathematical experiment, so reducing the need for mathematical calculations.

Diffusion and mixing as one-way processes

As in Part One, there is something to be learned about the one-way nature of some processes by watching films of events, run backwards. While studying Part One, you may have seen the filmed episodes which we shall now discuss in some detail.

Most mornings, the writer, while helping to get breakfast ready, pours hot milk into a jug which has been heated by having had hot water in it. On occasional bad days, he forgets to pour away the hot water before putting the milk in the jug. No doubt the feeling of tragic helplessness this always produces in him is partly the consequence of his half-awake early-morning state of mind, but it also owes something to the impossibility of undoing what has been done, in any simple way. The mixing of milk and water happens all on its own, without any assistance or encouragement, but unmixing them is a different matter altogether.

A film of the mixing of ink and water, shown in reverse, reinforces the point by the very absurdity of what one seems to see. The question is why such spontaneous unmixing looks absurd. Why is it one of those things which never happen? Although it is not at all hard to say more or less why spontaneous unmixing of this kind does not happen, or is at least most unlikely, it turns out that a good, clear account of this rather trivial matter can be used to help in the discussion of much harder matters, like the flow of heat. So it is worth pursuing a little further.

A collection of many two-way processes can look one-way

A swinging pendulum with a massive bob, hung from a stout beam, is about as near as one can easily get in a school laboratory to a process which looks just as good if it is shown as a backwards-running film, as it does if the film goes forwards. Apart from a little damping, the pendulum's energy is not spread around amongst the molecules of nearby material, and its motion is nearly two-way.

A row of such pendula can be hung in a line. If they are not connected together, and the beam from which they hang is rigid, no one pendulum affects any other, and each one continues its undisturbed two-way motion if it is set swinging. Now suppose one saw such a row made of pendula of different lengths, swinging in no special relationship to one another. Then suppose that as one watched, all the pendula gradually began to come together, and all at one moment rose exactly together to their greatest height on one side. If one wasn't dreaming, the only way it could have been achieved deliberately, would be for someone to have worked it all out in advance, and then set each pendulum swinging with just the right motion so that, in the end, as all the swings changed their relationships to one another, the pendula swung together for a moment.

To achieve all this would have been a considerable feat of ingenuity. As you may have seen, another way to make it seem to happen is to start the pendula off together, film their motion, and then show the film backwards. To an excellent degree of accuracy, no one pendulum motion seen on the reversed film is impossible: the motion of a pendulum which is not damped is just as possible either way round. But the collection of many motions, everyone of which is two-way by itself, looks decidedly one-way. The effect solely depends on large numbers.

The effect is not one to puzzle over unduly, though it is worth seeing, if only for its visual charm. If one wants to, one can find some quite deep puzzles in it, which have concerned a number of physicists and mathematicians. For example, wouldn't the pendula come together if one waited long enough? Can a large number of truly reversible processes really add up to something irreversible, or is this just an effect of what one expects to see, not of what one ought to expect? But none of these are puzzles you need follow up now, unless you want to.

Large numbers and small numbers

The particular aspect of mixing which we shall follow up is that aspect which does not involve energy in any way, but simply the muddling together or sorting out of particles among one another, or the spreading out of particles into unoccupied spaces. A simple system which illustrates this aspect is a box containing two sorts of coloured balls. The balls differ only in having one colour or the other, and in no other way. If the box is shaken after the balls have been sorted so that those of each colour are together, everyone knows what to expect. The balls become mixed, and the process seen in reverse looks very strange; no less strange than the act of a conjurer who shuffles a pack of cards and ends up with them in perfect sequence.

What is interesting and illuminating is to see the same shaking process when there are only a few balls in the box, say two of each colour. The constant interchange amongst the balls now looks just as reasonable in reverse as it does in the real life direction. The mixing of many balls is a one-way process, but the mixing of just a few is not. The number of objects involved is an important factor in deciding whether a process is one-way or not.

Q1 When a box containing many marbles of two colours is shaken, is the reason why they do not sort themselves out that they cannot do so?

Q2 Shaking the box containing many marbles rearranges the marbles among each other, going from one way of arranging them to another, and to another. Would you say that there is just one way, or that there are a few ways, or that there are many ways of rearranging them, such that you would say that the marbles were well mixed?

Q3 There is at least one and perhaps there are many possible ways of arranging the marbles so that anyone would agree that they were pretty well sorted out into the two colours. Would you say there were more than, fewer than, or the same number of ways of doing this as of achieving arrangements which one would naturally call muddled or well mixed?

If just four marbles are shaken so as to stay in a two-by-two array, but so as to interchange places, all the possible ways of arranging them can be set down, as in figure 15 (ignoring interchanges of similarly coloured marbles).

Figure 15: Patterns of arrangement of four marbles, having two colours.

One could say, of the two ways shown on the left in figure 15, and of only these two, that the marbles are vertically sorted. If you shook the marbles and then inspected them, they would have to be arranged in one of the ways shown. Suppose that the marbles are smooth and round, and that nothing favours one way of arranging them over another.

Q4 If they start off in one of the vertically sorted ways, and are shaken, could they ever, by chance, end up again in one of the vertically sorted ways?

Q5 If you shook them, and inspected them a great many times, about how often might you expect to see them vertically sorted?

Q6 Why, in question 5, would you not expect to see them vertically sorted quite as often as in some other pattern?

Chance is enough to work out what happens on average

In discussing the interchanges of four marbles (questions 4 to 6) we almost casually introduced the idea that one might reasonably suppose that nothing would favour any one of the ways of arranging the marbles (shown in figure 15) over any other way. We now propose to take this idea more seriously: indeed it is the key to understanding the work of the whole Unit.

Optional experiment: 9.2 Random motion of marbles in a tray

Much can be learned from watching marbles rolling about at random in a shallow tray. If for any reason you have not done so before, you would be well advised to spend some time now looking at the motion of the marbles, at their collisions with each other and with the walls, at how far a marble goes between collisions, and at how this last point depends on how many marbles there are.

For the work of the Unit, at this point, we are concerned with one other aspect. Imagine the tray to be divided into two halves, or draw a line across its middle, and look to see what proportion of the marbles is, on average, to be found in any one half. Look also to see how far the number in one half seems likely to deviate from the average number.

    12     two-dimensional kinetic model kit 
 optional extras: 
   133     camera 
   171     photographic accessories kit 
  1054     film, monobath combined developer-fixer 
           slide projector 

You may, however, feel that you have rolled marbles in trays often enough before. If so, you may prefer to look at the photographs of pucks moving on an almost frictionless surface, reproduced in figures 101 a and b, Appendix D.

Q7 In figure 101 a, a series of snapshots of ten pucks moving about on a horizontal surface is seen. Are all the pucks ever together in the righthand half?

Q8 What is the average number of pucks on the righthand side? Is it what you expect? (There is a quick way to find the average. Guess the answer, then score 0, + 1, + 2, -1, etc. for each frame, according to whether the number in one half is different from your guess. The average is found from the total of these scores, many of which cancel as you go along, divided by the number of frames, and added to your guessed answer. The better your guess, the easier it is.)

Q9 In figure 101 b, there is a similar set of photographs, but with only four pucks. Are all the pucks ever together in the righthand half?

Q10 As it happens, all four pucks are never together in the lefthand half. Has chance been unkind, do you think, or was that to be expected?

Game: 9.3 Simulation of the behaviour of marbles in a tray

The photographs in figures 101 a and 101 b, Appendix D , were in fact taken with moving pucks, photographed at intervals as they moved. They look as if they might have been produced with less trouble, by putting the pucks down at random for each picture, so as not to bother with making them move.

Q11 Suggest a way of putting the pucks down at random, which you think ought to divide them between the two halves in the same kind of way as they are divided in the pictures. (To imitate the pictures one would have to place the pucks randomly within each half as well, but you can ignore this aspect.)

Try the idea of letting the random fall of a die or the spin of a coin decide into which half of a box an object will go, using a sheet of paper ruled into two, and some counters. You might try it with ten counters, seeing if the average number of counters in one half over many trials is as you expect. If you use ten counters, you may also form some idea of why all ten rarely go into one half together.

Naturally, this game does not say anything about where one should put the counters down within each half. But it is quite good at getting the numbers in each half right. This is typical of random simulation methods: they often predict average behaviour well, without representing or predicting the detailed behaviour of a system at all.

     1053     counter 10 
     1054     graph paper 
     1055     die or coin 

Often, the average behaviour is all one needs to know. If the air in a car tyre is to maintain a steady pressure on the walls of the tyre, all that is needed is a reasonably steady hail of molecules on the rubber. This can be achieved by a totally random movement of many molecules, and the details of every molecule's path, speed, and collisions are of no importance.

The first game, 9.3, suggests what the average division of pucks or molecules between two halves of a space might be. The next game, 9.4, suggests how one might simulate some aspects of how a system reaches that more or less steady average behaviour if it starts a long way from the average behaviour. The main point of introducing it is that a later game, 9.8, which is crucial to the whole Unit, uses a similar idea, but it will also help in working out a detailed analysis of how the spreading of particles in a mixing or diffusing process depends on the numbers of ways in which the particles can be arranged.

Game: 9.4 Simulation of spreading into an empty space

In the game 9.3, chance seemed to be enough to give the average behaviour of particles divided between two parts of a box. Suppose now that we agree to start with all the counters already in one half of the box, as in figure 19.

Q 12 Is the distribution which has all counters in one half near to the average behaviour if the counters can go in either half?

The question is, how might one now use a die to move the counters randomly between the halves of the box? If there are six counters, as in figure 19, and each is numbered, a six-sided die can be thrown and the counter whose number comes up can be moved to the half it is not now in.

Figure 19

Q 13 What must happen on the first throw?

Q 14 On the second throw, out of the six possible moves, how many will move a second counter to the initially unoccupied half?

Q 15 On the second throw, how many of the possible moves will take the first counter moved over (counter number five in figure 19) back again to the full half?

Q 16 If everyone in a large class makes two moves, what proportion might end up with all the counters back in the full half?

It is worth going on to make several more moves. Although the die does not know or care when all the counters are in one half, you should find that that distribution arises rather rarely. You should be able to say why it arises infrequently.

      1053    counter 6 
      1054    graph paper 
      1055    die 

Counting numbers of ways

In several discussions in this Part, it has been helpful to think about the number of ways in which a state of affairs can arise. This idea is a crucial one. We shall now use it to calculate one aspect of the behaviour of particles spreading into an unoccupied space. This calculation is the basis of a chemist's thinking about the mixing of reacting molecules. It also serves as an example of a method which we shall put to more important uses in Part Five, in discussing why heat goes from hot to cold, and what hot and cold mean at the level of molecules sharing energy amongst themselves.

One way six counters can be divided between two halves of a box is to have them all in one half. But there are many other ways of arranging them. Suppose there are altogether W ways of arranging the counters between the two halves (including the one way with all in the left and the other one with all in the right).

Q17 Out of many observations, in what fraction of them might you reasonably expect to see all the counters in, say, the lefthand half, in terms of W?

The total number of ways W is easy enough to calculate. Figure 20 shows how to do it. The first counter can be put down in two ways, as in figure 20 a. The second counter can also be put down in two ways, each of which can go with either of the two Ways for the first counter, as in figure 20 b. The third counter can again be put down in two ways, as in figure 20 c.

Figure 20: Counting numbers of ways for particles in two halves of a box.

Each of the two ways for the third counter can go with any one of the previous four ways, giving now 4 × 2 = 8 ways (not 4+2). This last point is of the greatest importance: numbers of ways multiply when they are independent of one another.

If there are four counters, there will be 2 × 2 × 2 × = 2⁴ = 16 ways in all. If there are six, there will be 2⁶ = 64 ways. If there are N counters, there will be a total number of ways given by W= 2^N .

The calculation makes a definite prediction. If N pucks move about at random on a horizontal surface, or if N gas molecules move about in a container, on one observation out of 2^N, on average, all N particles may be expected to be in one particular half.

We tried it out using a computer, which was asked to play the game in 9.3 in its head, using four imaginary counters. Instead of spinning a coin, the computer calculated random numbers to decide the fate of each counter. Using N = 4, we expected 1/2⁴ = 1/16 out of a great many trials to end up with all four in one particular half. In 1 minute 11 seconds, the computer made 16 000 trials and 1/16 therefore is just 1000. The outcome is shown in table 3.

Number of counters in the 
'lefthand half'              0      1      2      3      4

Number of occasions          1086   3823   6188   3900   1003

    1053     counter 4 
    1054     graph paper 
    1055     die or coin 

Processes that happen inexorably, but quite by chance

In the game 9.4, with six counters, the counters were more likely to spread into both halves of the box, starting all in one half, than to stay or go back into one half. The larger the number of particles, the smaller is the fraction 1/2^N , which indicates the proportion of occasions on which all the particles may concentrate by chance in one half.

Think, for example, of an ordinary gas jar full of air. There will be perhaps 10²² molecules in the jar. (One mole of molecules, N = 6 × 10²³ , occupies about 24 dm³ at room temperature. A gas jar might contain 1/50 of a mole.) On less than one occasion in 2¹⁰²² , all the molecules might by chance be in the bottom half of the jar. The number 2¹⁰²² is unimaginably large. Its logarithm to base 10 is given by

lg 2¹⁰²² = 10²² lg 2 ≈ 0.3 × 10²² = 3 × 10²¹

The number whose logarithm is 3 is 1000, that is, 1 followed by three zeroes.

The number 2¹⁰²² is 1 followed by 3 × 10²¹ zeroes. If it were written out, with each zero only 1 mm across, it would stretch a distance 3 × 10²¹ mm, or 3 × 10¹⁸ m. In a year, a flash of light travels nearly 10¹⁶ m (a year is just over 3 ≈ 10⁷ s, and the speed of light is 3 ≈ 10⁸ m s^-1 ). So the number 2¹⁰²² would stretch out for 300 light-years, nearly as far as the Pole Star, and more than thirty times as far as the star Sirius.

It follows that the chance of seeing all the molecules in one half is so small as to be negligible, even if one only allows as little as a microsecond for the time needed for molecules to rearrange themselves. (This is not generous, since a molecule takes over a tenth of a millisecond to cross a jar.) See questions 18 to 20. The chance is there, but it is smaller than the chance that all the houses in a country will burn down by accident on the same day, or that all the people in a country will just happen to catch measles. Insurance companies do well enough despite the chance of such disasters, while physicists, chemists, and engineers can rely on chance working the way they expect, so huge are the numbers of molecules involved. It follows that if all the molecules are in one half, but are allowed to pass into the other half, they will do so with all the appearance of inevitability. Diffusion is a one-way process because it is so very, very likely, that in effect it always happens.

Nor should you suppose that very large numbers of particles have to be involved for the chance of reverse diffusion to be negligible.

Q18 Suppose there are only 100 molecules in a jar. Write down the logarithm of the number 2¹⁰⁰ as a power of ten.

Q19 Now suppose you look at the jar every microsecond. For how many seconds might you have to wait on average before, by chance, all the 100 molecules were in one half?

Q20 People think the Universe is about 10¹⁰ years old, which is about 3 ≈ 10¹⁷ s. How many Universe-lifetimes does the answer to question 19 represent?

Everyone is disturbed by these large numbers when they are first encountered. Their very vastness, which so defeats the imagination, is what lies behind the paradox in the previous heading, Processes that happen inexorably, but quite by chance.

Summary

Chance is blind: it tries out everything impartially. Things which can happen in many ways therefore happen often. Molecules which can spread into a larger space will do so. This need not be because they are pushed into the empty space, but can simply be the result of chance. There are more ways of arranging the molecules when they are spread than when they are not, and chance will try out all the ways open to the molecules. When there are many molecules, there are many more ways of being arranged when they can spread, and so they do spread. They are so busy trying out all these ways that only rarely will chance take them back to the original unspread condition. The larger the number of molecules, the less likely is it that chance will soon produce a spontaneous reversal of diffusion.

If chance tries out all ways of arranging molecules impartially, an event which arises in twice as many ways as another will be observed twice as often.

Numbers of ways multiply if the choices are separate ones that do not influence each other.

6 1 0 5 6 6 1 8 7 6 8 2 1 1 1 8 6 1 9 
0 9 0 3 6 7 8 5 7 3 2 0 6 3 9 9 5 7 5