1. Nuclear Chemistry and Radioactivity
CHAPTER 20
Nuclear Chemistry and Radioactivity
20.3 Rate of Radioactive Decay

2. What is carbon dating?
How can we tell how old fossils are?

3. Reaction rates What is carbon dating?
How can we tell how old fossils are?
We introduce the time variable

4. Reaction rates What is carbon dating?
How can we tell how old fossils are?
We introduce the time variable
In Chapter 12 we studied reaction rates for chemical reactions
Nuclear reactions also involve rates!

5. Some reactions take place very quickly; they have a short half-life, t1/2.
Decay
half-life: the time it takes for half of the atoms in a sample to decay.

6. Half-life

7. Half-life Every radioactive isotope has a different half-life.
Isotopes with short half-lives do not occur in nature, but must be generated in the laboratory.

8. Carbon dating
Carbon dating revolves around carbon-14, a radioactive isotope.
Carbon-14 is generated in the upper atmosphere through a bombardment reaction:
becomes 14CO2 in the atmosphere
Neutrons generated by cosmic rays

9. Carbon dating
Carbon-14 goes through the same cycle as carbon-12 14

10. Carbon dating In living organisms:
This ratio stays constant while the organism is alive

11. Carbon dating In living organisms:
This ratio stays constant while the organism is alive
Over time, carbon-14 decays by b emission:

12. Carbon dating Over time, carbon-14 decays by b emission:
When the organism dies, it no longer consumes carbon from the environment.
The number of carbon-14 atoms in the dead organism will decrease over time.

13. Carbon dating
An archeologist looks at the ratio of carbon-14 to carbon-12.
Carbon dating works reliably up to about 10 times the half-life, or 57,300 years (beyond that time, there is not enough carbon-14 left to detect accurately).
Carbon dating only works on material that has once been living: tissue, bone, or wood.
Ratio not to scale

14. About 18% of the mass of a live animal is carbon
About 18% of the mass of a live animal is carbon. If 1 g of live bone contains about 90 billion carbon-14 atoms (t1/2 = 5,730 years), how many C-14 atoms remain in 1 g of bone 17,190 years after the animal dies?

15. About 18% of the mass of a live animal is carbon
About 18% of the mass of a live animal is carbon. If 1 g of live bone contains about 90 billion carbon-14 atoms (t1/2 = 5,730 years), how many C-14 atoms remain in 1 g of bone 17,190 years after the animal dies?
Given: The half-life and the number of carbon-14 atoms

16. About 18% of the mass of a live animal is carbon
About 18% of the mass of a live animal is carbon. If 1 g of live bone contains about 90 billion carbon-14 atoms (t1/2 = 5,730 years), how many C-14 atoms remain in 1 g of bone 17,190 years after the animal dies?
Given: The half-life and the number of carbon-14 atoms
Solve: Since 17,190 years is three half-lives, the initial amount must be reduced by a factor of 2 x 2 x 2 = billion / 8 = billion

17. About 18% of the mass of a live animal is carbon
About 18% of the mass of a live animal is carbon. If 1 g of live bone contains about 90 billion carbon-14 atoms (t1/2 = 5,730 years), how many C-14 atoms remain in 1 g of bone 17,190 years after the animal dies?
Given: The half-life and the number of carbon-14 atoms
Solve: Since 17,190 years is three half-lives, the initial amount must be reduced by a factor of 2 x 2 x 2 = billion / 8 = billion
Answer: After three half-lives the amount of carbon-14 atoms is reduced by a factor of 8, from 90 billion to billion.

18. Every radioactive isotope has a different half-life, t1/2
Carbon dating is based on the knowledge that t1/2 for carbon-14 is 5,730 years
Ratio not to scale

19. Rate of decay The number of nuclei in the sample (N) is constant
A short half-life implies a large rate constant, k.

20. Rate of decay
Carbon-14 and radium-220 have half-lives of 5,730 years and 1 minute, respectively. Calculate the rate constants for their decay, in units of 1/s.

21. Rate of decay
Carbon-14 and radium-220 have half-lives of 5,730 years and 1 minute, respectively. Calculate the rate constants for their decay, in units of 1/s.
Asked: The rate constant k
Given: The half-life t1/2 for each radioactive decay process.
Relationships: The equation that relates t1/2 to k: k = / t1/2

22. Rate of decay
Carbon-14 and radium-220 have half-lives of 5,730 years and 1 minute, respectively. Calculate the rate constants for their decay, in units of 1/s.
Asked: The rate constant k
Given: The half-life t1/2 for each radioactive decay process.
Relationships: The equation that relates t1/2 to k: k = / t1/2
Solve: For C-14, t1/2 = 5,730 years, and .

23. Rate of decay
Carbon-14 and radium-220 have half-lives of 5,730 years and 1 minute, respectively. Calculate the rate constants for their decay, in units of 1/s.
Asked: The rate constant k
Given: The half-life t1/2 for each radioactive decay process.
Relationships: The equation that relates t1/2 to k: k = / t1/2
Solve: For C-14, t1/2 = 5,730 years, and
For Ra-220, t1/2 = 1 min, and

24. Rate of decay
Carbon-14 and radium-220 have half-lives of 5,730 years and 1 minute, respectively. Calculate the rate constants for their decay, in units of 1/s.
Asked: The rate constant k
Given: The half-life t1/2 for each radioactive decay process.
Relationships: The equation that relates t1/2 to k: k = / t1/2
Solve: For C-14, t1/2 = 5,730 years, and
For Ra-220, t1/2 = 1 min, and
Discussion: Note that a small t1/2 gives a large k. The rate constant k gives us an indication of the number of decays over a certain period of time.

25. Decay rate law
The rate of decay of a radioactive sample is also called the activity of the sample

26. Decay rate law
Plutonium-236 decays by emitting an alpha particle and has a half-life of 2.86 years. If we start with 10 mg of Pu-236, how much remains after 4 years?

27. Decay rate law
Plutonium-236 decays by emitting an alpha particle and has a half-life of 2.86 years. If we start with 10 mg of Pu-236, how much remains after 4 years?
Asked: N, the amount left after 4 years
Given: The half-life t1/2, the initial amount N0, and the elapsed time t
Relationships: The equation that relates t1/2 and N0 to N is

28. Decay rate law
Plutonium-236 decays by emitting an alpha particle and has a half-life of 2.86 years. If we start with 10 mg of Pu-236, how much remains after 4 years?
Asked: N, the amount left after 4 years
Given: The half-life t1/2, the initial amount N0, and the elapsed time t
Relationships: The equation that relates t1/2 and N0 to N is
Solve:

29. Decay rate law
Plutonium-236 decays by emitting an alpha particle and has a half-life of 2.86 years. If we start with 10 mg of Pu-236, how much remains after 4 years?
Asked: N, the amount left after 4 years
Given: The half-life t1/2, the initial amount N0, and the elapsed time t
Relationships: The equation that relates t1/2 and N0 to N is
Solve:
Discussion: After 4 years, the initial 10 mg is reduced to 3.79 mg, which is 37.9% of the initial amount of Pu-236.

30. Radioactive dating
Information can be extracted from the ratio of specific isotopes
Carbon-14 and carbon-12
the age of a once-living organism
Oxygen-18 and oxygen-16
the composition of the atmosphere over time
Uranium-238 and plutonium-239
the age of rocks that are billions of years old

31. Radioactive dating
The amount of sample remaining, compared to the initial amount of sample, can be used to determine the age of the sample.

32. An ancient Greek scroll written on an animal skin was discovered by archeologists in They isolated 10 g of it and measured the carbon-14 decay rate to be 111 disintegrations/minute. Calculate the age of the scroll. (Assume that living organisms have a carbon-14 decay rate of 15 disintegrations per minute per gram of C.)

33. An ancient Greek scroll written on an animal skin was discovered by archeologists in They isolated 10 g of it and measured the carbon-14 decay rate to be 111 disintegrations/minute. Calculate the age of the scroll. (Assume that living organisms have a carbon-14 decay rate of 15 disintegrations per minute per gram of C.)
Given: The initial decay rate of C-14: N0 = 15 disintegrations/(min·g). The present decay rate of C-14 is 11 disintegrations/(min·g). The half-life of C-14 is 5,730 years.

34. An ancient Greek scroll written on an animal skin was discovered by archeologists in They isolated 10 g of it and measured the carbon-14 decay rate to be 111 disintegrations/minute. Calculate the age of the scroll. (Assume that living organisms have a carbon-14 decay rate of 15 disintegrations per minute per gram of C.)
Given: The initial decay rate of C-14: N0 = 15 disintegrations/(min·g). The present decay rate of C-14 is 11 disintegrations/(min·g). The half-life of C-14 is 5,730 years.
Solve: For the rate constant k:

35. An ancient Greek scroll written on an animal skin was discovered by archeologists in They isolated 10 g of it and measured the carbon-14 decay rate to be 111 disintegrations/minute. Calculate the age of the scroll. (Assume that living organisms have a carbon-14 decay rate of 15 disintegrations per minute per gram of C.)
Given: The initial decay rate of C-14: N0 = 15 disintegrations/(min·g). The present decay rate of C-14 is 11 disintegrations/(min·g). The half-life of C-14 is 5,730 years.
Solve: For the rate constant k: And the time is:

36. An ancient Greek scroll written on an animal skin was discovered by archeologists in They isolated 10 g of it and measured the carbon-14 decay rate to be 111 disintegrations/minute. Calculate the age of the scroll. (Assume that living organisms have a carbon-14 decay rate of 15 disintegrations per minute per gram of C.)
Given: The initial decay rate of C-14: N0 = 15 disintegrations/(min·g). The present decay rate of C-14 is 11 disintegrations/(min·g). The half-life of C-14 is 5,730 years.
Solve: For the rate constant k: And the time is:
Discussion: The animal skin on which the scroll was written was 2,491 years old. It was written in about 483 BC.

37. Mathematics of radioactive decay