EXPONENTIAL DECAY 

by 

F. Lee Slick

This lesson was created as a part of the SMART website and is hosted by the Illinois Institute of Technology


Time to play with dice (this is not for crap shooting). You will be given a number of dice. Your job is to roll the dice and, as they fall to the table, remove from the pile all dice that have the number 3 (or whatever number your instructor tells you to use) showing. Keep track of the roll number and the number of dice left after roll. Roll the dice down to less than 6 dice, put the dice back together, re-roll the dice and keep track again (do 10 trails). Total the number of dice left after each roll and then plot a graph of the total number left after each throw (N) vs. the throw number.

You should remember that this should be a smooth curve and does not need to go through every point, just close. This will give you a decay curve for dice. Determine the half-life of the dice by looking at your graph. How many throws (to the nearest quarter of throw) does it take for half of the dice to be gone? (i.e. If you started with 1000 dice (N0) how many throws would it take for 500 of them to leave?) How many throws does it take for three quarters of the dice to be gone? (i.e. if you started with 1000 dice how many throws would it take for 750 of them to leave?) 

Attempt to obtain 5 half-lives. Take the first half-life divided by one, the second half-life divided by two, the third half-life divided by three, etc. and average these numbers. This number is the half-life (T) of the dice and, since 0.693 / T (Why?) is the decay constant (the probability of decay per "throw number"), you should be able to calculate the probability of decay for a particular die and compare this to the actual value.

Now for some more fun stuff. Find the ratio N/N0 for each of your trials (Note: N/N0 is the total number you started with and N is the number left after each throw. Also, the first trial is throw zero and N/N0 = 1 since N = N0.) Take out your handy dandy calculator and find the Natural Logarithm; log (we mean ln x -- and not  log x -- which is near the ex key on your calculator) of this ratio for each of your trials. Plot this number (the natural log of N/N0) vs. the throw number on a sheet of graph paper.

Take a sheet of 2-cycle semi-log paper (the reason it is called semi-log paper is because it is made from a part of tree logs) and plot the ratio N/N0 (found in the last part) vs. the throw number. Now, by doing a careful analysis, take the slope of this line. Remember that y = m x+b (m is the slope) or, if you forgot, up-ness / across-ness; or rise / run; or dy/dx.

Wasn't that fun? I thought so. Here's the next part for fun and games, take 100 key board keys and repeat the exercise in the first paragraph. However there are three possibilities. ('break' up, 'break' down, and sideways) You will be assigned one possibility, roll 10 trails and plot data as before. Draw as many conclusions as you can from this data. Check with other groups to compare your data. Are the results nearly the same? If not why not?

With cubes as simulated molecules, you will attempt to verify some of the principles of radioactive decay. Starting with a given number of plastic cubes that have a hole through them, throw the cubes onto a suitable surface and remove every cube that shows a hole on the top. It may be necessary to carefully move some of the cubes so they are not stacked on top of each other. Replace each of these 'decayed molecules' with a plastic die. Do this again, replacing each 'hole' cube with a die but also removing each die that lands with a 6 on top. Your data table will probably have at least five columns:

Throw Number    # Holes Removed    # Holes Remaining    # Dice Removed    # Dice Remaining

Continue to throw and replace/remove them as fast as possible until less than 10 dice are left. Repeat this entire process several times. Plot two graphs on the same axis: total 'hole' cubes vs. throw and total dice vs. throw. (Note: total dice vs. throw number should start at zero, increase, reach some maximum and then go back down. WHY?) You should remember that these curves should be smooth curves. How does this experiment compare with one measuring normal radioactivity with Geiger counters and other electronic equipment?

References



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