Chemistry: Decay Reaction Rates Guided Inquiry
Reaction rate is the rate at which a reaction occurs; the rate at which reactants disappear; and the rate at which
products are created.
Fission, fusion and artificial transmutation reactions happen very quickly, on the order of thousandths to
millionths of a second. Decay reactions can be fast or slow. We will learn about the reaction rates of decay
reactions.
A scientist is studying the alpha decay of uranium-231:
. She measured the mass of
uranium-231 every 2.1 days for 29 days. Her data is shown in Table 1 below:
Table 1 – Decay rate of Uranium-231
Time
Mass of
Reaction
(days)
Uranium, g
Rate, g/day
Time
(days)
Mass of
Uranium, g
Reaction
Rate, g/day
16.8
6.25
1.30
0
100.00
2.1
70.71
14.0
18.9
4.42
0.87
4.2
50.00
9.9
21.0
3.13
0.61
6.3
35.36
7.0
23.1
2.20
0.44
8.4
25.00
4.9
25.2
1.56
0.30
10.5
17.68
3.5
27.3
1.10
0.22
12.6
12.50
2.5
29.4
0.78
0.15
14.7
8.88
1.7
1. Calculate the reaction rate in units of grams per day, and add the results to Table 1. The first several rates are
calculated as examples.
2. Does the uranium decay at a constant rate? In other words, does the same amount of uranium decay every
2.1 days?
3. If the uranium doesn’t decay at a constant rate, how does the reaction rate change over time?
4. Is there a direct or indirect mathematical relationship between the mass of uranium and the reaction rate?
5. How long does it take for the mass of uranium to decrease to half its original mass?
6. How long does it take for the mass to drop in half again (from ½ to ¼ of the original mass)?
7. How long does it take for the mass to drop in half again (from ¼ to 1/8 of the original mass)?
8. How long does it take for the mass to drop in half again (from 1/8 to 1/16 of the original mass)?
9. What pattern do you notice for the decay of uranium-231?
Half-life, t1/2, is the amount of time it takes for one-half of the atoms of a radioactive element to decay.
10. What is the half-life of the uranium-231 alpha decay reaction?
STOP – Show your teacher your answers.
Nuclear decay reactions do not happen at a constant rate. The rate of decay is the greatest at the beginning of
the reaction when the amount of the radioactive element is the greatest. The decay rate decreases as the
amount of remaining radioactive element decreases. However, the amount of time it takes for half of the
radioactive element to decay is a constant, which is called its half-life. Here is the generic decay pattern for all
half-life reactions:
Time
Fraction of Original Amt. Remaining
0
1
1 half-life
½
2 half-lives
1/4
3 half-lives
1/8
…
…
Let’s look at the decay of 100.0 grams of a radioactive element that has a half-life of 15.00 days.
Time, days
Amount, g
0.00
100.0
15.00
50.00
30.00
25.00
45.00
12.50
60.00
6.250
75.00
3.125
11. 40.0 grams of radioactive element has a half-life of 5.00 years. How much remains after 15.0 years?
The above method only allows us to find the amount of radioactive material remaining if the time is exactly a
multiple of the half-life. To find the amount of radioactive material remaining for any given time we need to use
the following formula:
( )
⁄
where Nt is the amount of element remaining at time, t; N0 is the original amount of element; t is the elapsed
time; and t1/2 is the half-life of the element.
Example – 40.0 grams of a radioactive element has a half-life of 5.00 years. How much remains after 2.50 years?
⁄
( )
12. 40.0 grams of a radioactive element has a half-life of 5.00 years. How much remains after 13.5 years?
13. Carbon-14 has a half-life of 5,730 years. An ancient artifact originally contained 0.0235 grams of carbon-14.
How much carbon-14 remains in the artifact after ten thousand (1.00 x 104) years?
14. How old is an artifact if it now has one-half its original amount of carbon-14?
15. How old is an artifact if it now has one-fourth its original amount of carbon-14?
16. How old is an artifact if it now has one-sixteenth its original amount of carbon-14?
STOP – Show your teacher your answers.
In #14 - #16 we could determine the age of the artifact because the amount of carbon-14 remaining was a power
of one-half, i.e. 1/2, 1/4, 1/16. We can’t use that approach if the ratio of the remaining amount to the original
amount isn’t a power of one-half. However, we can rearrange the half-life equation to find out how old the
sample is regardless of how much material is left.
( )
⁄
( )
⁄
( )
⁄
Example – How old is an artifact that has 37.5% of its original carbon-14 remaining?
17. A fossil has 9.55% of its original carbon-14 remaining. How old is it?
18. [Challenge Problem] Cobalt-60 is has many commercial uses including the sterilization of medical instruments.
It has a half-life of 5.271 years. A 275 day old sample contains 21.7 grams of cobalt-60. What was the original
mass of cobalt-60?
STOP – Show your teacher your answers.