Gary L. Bertrand
University of Missouri-Rolla
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This is an animated simulation of a
limited number (50) of "magic beans" randomly jumping around in boxes.
The shapes and sizes of the boxes may be changed, the "beans" may be started
on either side, and one side of the box may be made "sticky" so that the
"beans" cannot jump around in it. The program randomly selects a
position within the boxes. If that position is occupied by a "bean"
the program selects a random direction and initial velocity for the jump.
If the jump will be sufficient to move the bean, the flight of the bean
is shown. The position where the bean will land is completely determined
by its initial position, the direction and initial velocity of the jump.
The "equilibrium" condition for this system is not well-defined because
of the small number of "beans", so the distribution continues to fluctuate
substantially around the "equilibrium" value.
The program starts in a default configuration,
showing both side and top views of the "box". There is a button to hide
or show the top view (hiding the top view speeds up the Windows operation
considerably). Click on "start"
and the "beans" start jumping around. Some will cross the barrier
to the other side. Counters display the numbers of beans in the left
and right compartments. Eventually, some will start to jump back
to the original side.
There is a "pause/resume"
button to stop and resume the action. The action may be set to "faster"
or
"slower" mode for Macintoshes (actually, all
of the action is painfully slow under Windows). There is a "100
jumps" button to speed up equilibrium observations
by hiding the action for 100 jumps, and giving the average of the ratio
of the number of beans on the right to the number on the left over those
100 jumps.
The "setup"
button
shows a diagram so that you can re-configure the box.
Click the mouse on the red triangle and drag it up to raise the box on
the left:
The height of the box on the left may be adjusted with the blue
triangle, and the height of the barrier may be adjusted with the violet
triangle. The location of the barrier may also be adjusted with the
violet
triangle. The relative areas of the two compartments are shown.
The 50 beans may be started on the left or right side by dragging the "balls"
button to the desired side.
Finally, the bottom of the box opposite to where the "beans" start off
may be made "sticky" so that the rates of
the left-to-right or right-to-left
jumps may be studied individually. Click on "Return"
to go back to the animation.
The "sticky" facility has a feature to simplify
kinetic studies. When operated in this mode (whether "faster" or
"slower"), information is retained regarding the time at which the count
changes, so that this "data" may be processed
to determine a "rate constant" for this first-order
process. After observing the transfer, click on "setup"
and a table of the data will be presented. Click on "Calculate",
and a regression is performed to obtain the slope
of a graph of ln(N) vs time. This slope
may be considered as a rate constant IF
the areas of the boxes are always the same.
Otherwise, see the discussion at the bottom of the Kinetics
page. The ratio of the forward rate constant to the reverse rate
constant can be related to the equilibrium constant.