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.
If there were 20 grams of Iodine-131 8 days ago and now there is only 10 grams, write a decay model to represent this.
Q = aebt Q = end amount a = initial amount b = rate t = time (days)
10 = 20eb(8) Isolate e
.5 = e8b Take ln of both sides
ln .5 = 8b Solve for b
ln.5 = b = - 0.086643397
Note: b is negative because we have
Predict how much Iodine-131 will be left in 3 weeks from day 0.
if b < 0 , there is exponential decay.
Half-life: the amount of time it takes for radioactive material to reduce to half its original amount.
What is the half-life of Iodine-131? 8 days (See EX 1a)
How much Iodine-131 is present after 16 days? 24 days?
Since 8 days is the half-life of Iodine-131, then in 16 days
½ (10 grams) = 5 grams
In 24 days
½ (5 grams) = 2.5 grams.
The half-life of a radioactive substance does not depend on its initial amount.
Page 625, Figure 9.16.
Half-life of cobalt-60 is 5.3 years. If you store 12.4 grams of cobalt-60 today, what will you have one year from now? Since b is not given, use half-life information to derive model.
a/2 = aeb(5.3) Isolate e
½ = e5.3b 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.
= 10.9 – 12.4 = -1.5
Relative Decay Rate:
-1.5 gr./1 yr/ 12.4 gr = - .12259 /yr = - 12.26%/year
-12.26 % vs. –13.08%
Based on the fact that two types of carbon occur in nature; carbon-12 is stable, carbon-14 is radioactive. When an organism dies, the carbon-14 begins to decay while the carbon-12 remains the same. Using this ratio helps to determine the age of a fossil or artifact.
Example 3: Determine the model for carbon-14.
a/2 = aeb(5730)
½ = e5730b
ln ½ = 5730b
b = -.000120968
Q = ae-.000120968t
Find the percentage of carbon-14 present in the Shroud of Turin that was determined to be dated in the 14th century.
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%