Exponential Decay Our last Exponential model was for Growth. For radioactive decay, we also use an exponential model. However, the rate is now negative to represent decay. Example 1a: If there were 20 grams of Iodine131 8 days ago and now there is only 10 grams, write a decay model to represent this. Q = ae^{bt} Q = end amount a = initial amount b = rate t = time (days) 10 = 20e^{b(8)} Isolate e .5 = e^{8b} Take ln of both sides ln .5 = 8b Solve for b
ln.5 = b =  0.086643397
Note: b is negative because we have
decay.
Example 1b: Predict how much Iodine131 will be left in 3 weeks from day 0. if b < 0 , there is exponential decay. Halflife: the amount of time it takes for radioactive material to reduce to half its original amount. What is the halflife of Iodine131? 8 days (See EX 1a) Example 1c: How much Iodine131 is present after 16 days? 24 days? Since 8 days is the halflife of Iodine131, then in 16 days ½ (10 grams) = 5 grams In 24 days ½ (5 grams) = 2.5 grams. The halflife of a radioactive substance does not depend on its initial amount. Page 625, Figure 9.16. EX 2: Halflife of cobalt60 is 5.3 years. If you store 12.4 grams of cobalt60 today, what will you have one year from now? Since b is not given, use halflife information to derive model. a/2 = ae^{b(5.3)} Isolate e ½ = e^{5.3b} Take ln of both sides ln ½ = 5.3b Solve for b
ln ½ =
b = .130782487
Store in memory Q = 12.4e^{.130782487(1)} = 10.9 grams remaining after one year. Compare relative decay rate to actual rate using EX 2.
D Q
= 10.9 – 12.4 = 1.5
grams/1 yr. Relative Decay Rate: 1.5 gr./1 yr/ 12.4 gr =  .12259 /yr =  12.26%/year 12.26 % vs. –13.08% Radiocarbon Dating: Based on the fact that two types of carbon occur in nature; carbon12 is stable, carbon14 is radioactive. When an organism dies, the carbon14 begins to decay while the carbon12 remains the same. Using this ratio helps to determine the age of a fossil or artifact. Example 3: Determine the model for carbon14. a/2 = ae^{b(5730)} ½ = e^{5730b} ln ½ = 5730b b = .000120968 Q = ae^{.000120968t} Example 4: Find the percentage of carbon14 present in the Shroud of Turin that was determined to be dated in the 14^{th} century. Let x = % of initial amount ax = ae^{.000120968(600)} x = .92999999 = 93% What percentage must be remaining for the Shroud to be considered authentic? 2001 – 33 = 1968 years ax = ae^{.000120968(1968)} x = .788 = 78.8% Back to Exponenials and Logarithms Main Page Back to the Survey of Math Ideas Home Page Back to the Math Department Home Page email Questions and Suggestions
