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Gary L. Bertrand
University of Missouri-Rolla

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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

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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

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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.

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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.

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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.

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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.

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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.