LOREN HEPLER's MAGIC JUMPING BEANS

Gary L. Bertrand    University of Missouri-Rolla

In the book Order and Chaos , Angrist and Hepler discuss Chemical Equilibrium and Kinetics by referring to a system of Magic Jumping Beans, as is shown here. The beans jump about in random order with random direction and velocity.

 

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The drawings here show only a few beans in a relatively small box, jumping about rather slowly.  Try to envision a much larger box, with many more beans jumping about much more rapidly, so that the distribution of beans on the bottom of the box does not appear to be changing appreciably with time.

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If we separate the box with a low partition exactly in the middle and place all of the beans on one side, they will start jumping about

and some will eventually land on the other side of the partition. At first the movement is from the left to the right, but as the number on the right increases there is an increase in the probability for a jump from right to left.

 

Eventually we can expect that about half will be on the left and the rest on the right. The probability of a jump from right to left will then be roughly the same as the probability of a jump from left to right. The number of beans on each side will vary somewhat with time, but if the population builds up on one side or the other, the probability of a jump away from that side will increase.  This self-correction keeps the average population (over a long period of time) the same on both sides.

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If we had many more beans in a much larger box, we would see that only a small percentage of the jumps are as high as the barrier, and many of the higher jumps fall back on the same side without crossing the barrier.  Only a small fraction of the jumps will actually result in a transfer from one side to the other.

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An equal number the beans in the two compartments is the state of maximum probability. We associate this effect of probability with the thermodynamic property entropy. There is not really a drive for the beans to achieve this state - it simply occurs because of the natural (if magic beans can be natural) movements of the beans.  The movements of the beans are associated with their energy, even with magic beans.

 We can expect that this state of nearly equal distribution will occur irrespective of the initial state of the system: whether all start out on the right, all on the left, or any other arrangement.

 Now we consider repeating this exercise, but this time we move the partition so that the area on the right side is larger than the area on the left side.

Again we start out with all of the beans on the left, again the initial movement will be from left to right, and again the beans will occasionally go from right to left as the number on the right increases.  The situation has changed, however, because now there is greater probability of a bean landing in the larger area on the right. The state of maximum probability and maximum entropy now has more beans on the right.

We can consider this state of maximum probability and maximum entropy as a dynamic equilibrium - the numbers on each side stay fairly constant, though the individual beans are continually changing places.

Once the beans are equally distributed, it is extremely unlikely that they will all collect on one side or the other, without some outside influence. However, if there were only two beans, we would statistically expect them to both be on one side or the other half the time. The effect of the size of the sample is an extremely important consideration in statistics. As the size of the sample increases, the percentage deviation from the state of maximum probability decreases. The size (here six) is large in comparison to two, but is infinitesimal in comparsion to Avogadro's number (6.02 x 10^23 = 602,000,000,000,000,000,000,000).  Therefore, we can expect that there would be very little in the way of observable fluctuations when a system of even a billion molecules (still a very small quantity in the REAL world of molecules) reaches this state of dynamic equilibrium. We cannot see the individual molecules at this sub-microscopic level, but at the macroscopic level we can measure the relative amounts (by their mass, their concentration, or their partial pressure) that are in one condition or the other.  This state of equilibrium appears to be unchanging in time, but we know that the individual molecules may be continually undergoing tremendous changes.

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We have made a very important transition here, from a constantly changing dynamic situation (at the molecular level) to an unchanging (at the macroscopic level), but still dynamic, situation.  We associate the dynamic (changing) aspects of chemical reactions with Chemical Kinetics, and the equilibrium (unchanging) aspects of these reactions with Chemical Thermodynamics.

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Let's return to the box with the partition in the middle and consider the effects of changing the height of the barrier relative to the height of a typical jump.  From the animations above, it appears that lowering the barrier will have very little effect.  However, starting with all of the beans on the left, the barrier could be raised to a point such that it is impossible for the beans to jump over it, essentially "locking-in" the number of beans on that side, even though the beans continue to jump around within the box.

 The beans are performing exactly as they did before, but there is now a barrier which prevents them from achieving the state of maximum entropy - an activation barrier.  If we set the barrier a bit lower, at a point such that the beans have a one-in-a-million chance of jumping over it, we will eventually see some make it to the other side.

  Eventually, perhaps after many millions of jumps, the state of maximum entropy will be achieved.  At this point, there will be equal probability of left-to-right and right-to-left jumps.  The same state of maximum probability and maximum entropy will be attained irrespective of the height of the barrier, provided there is a finite possibility that a jump on either side might clear the barrier.  The final outcome, the equilibrium state, is determined by the thermodynamic properties energy and entropy.  The time required to achieve this state depends on the thermodynamic properties, but the time also depends on the activation barrier, a kinetic property.  Of course, the conditions that we impose on a reacting system such as temperature, pressure, and the amounts of the components are also very important factors.

Now let us raise one side of the box with respect to the other.

A bean on the right side will now have to jump higher to clear the barrier.  The sizes of the boxes are the same, so the entropic factors are the same.  Beans on the left will act like those in the box with the lower barrier above, while those on the right will act like those in the box with the higher barrier above.  The difference in height creates an energetic difference between the two sides, since higher objects in the field of gravity have higher potential energy relative to lower objects.

 

This difference in energy is expected to lead to a situation with more beans on the right side than on the left, assuming that the beans on each side jump with the same average height.  The beans on the left side have higher energy than those on the right, and there is a decrease in energy as beans are transferred from left to right.  The transfer of beans from left to right is then an exothermic process.

The equilibrium distribution of beans now favors the right side over the left.  We can expect that this distribution will be achieved irrespective of how the beans are distributed at the start of the process.  If we start with all of the beans on the right, some will be transferred to the left even though this is energetically unfavorable.  This increase in energy as the "reaction" occurs makes this an endothermic process(actually, the words exothermic and endothermic refer to the enthalpy change of a reaction rather than the energy change).

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  Remember, this whole discussion is aimed at helping you make sense of some complicated relationships that we encounter in studying Chemical Reactions - these beans are a device for simplifying these concepts.  At least, when you have read all of this no one will be able to say that, "You don't know beans about Chemistry."

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Now we will separate this discussion into two parts: Thermodynamics and Kinetics.  We discuss Thermodynamics when the beans have reached a state where the average number of beans (averaged over a period of time) in each compartment is staying constant - an equilibrium condition.  We will be concerned with the numbers of beans on each side when this equilibrium condition is reached, and how these numbers are related to the sizes of the compartments and their differences in height.  We will relate these numbers to equilibrium constants, and how these equilibrium constants are related to things like energy and entropy.

When equilibrium is attained, the individual beans continue to move back and forth between the compartments.  The equilibrium situation arises because the number of jumps from left to right over a period of time is exactly matched by the number of jumps from right to left, so that there is no net change.  In Kinetics, we talk about the rates at which things happen.  We may speak of the number of beans crossing the barrier from left to right that occur in a specific amount of time as the rate of jumping from left-to-right.  We may also speak of the rate of jumping from right-to-left as the number of beans that cross the barrier from right to left in that specified amount of time.  These are theoretical terms that are used for building mathematical models to describe what is happening.  If we had huge compartments with millions of beans bouncing back and forth at incredible speeds, it would be impossible to count the beans as they fly over the barrier - and even harder to keep track of which way they were going.  However, we could possibly be weighing the individual compartments as changes take place, or find some other way to keep track of how many beans are on each side, and how the numbers change with time.  This rate of transfer from one side to the other, or the change in the number of beans on one side in a specific amount of time, is a more practical quantity to measure.  The rates of change on the left and right are the same quantities, positive on one side and negative on the other.

It is very easy to get confused about rates, since people and books are not always clear as to which rates they are talking about - and I'll probably be guilty of that somewhere in this article.  If we start out with all of the beans on the left with a barrier that they can jump over, the rate of jumping from right-to-left will be zero because there are no beans on the right.  The rate of jumping from left-to-right will be relatively fast, and this will be the same as the rate of transfer from left-to-right.  However, as the number of beans on the right side builds up, the rate of jumping from right-to-left will increase and the rate of jumping from left-to-right will decrease, until eventually they will become the same.  At this point the rate of transfer from left-to-right will have decreased to zero, and this is the equilibrium state.

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