chapter 20 nuclear chemistry and radioactivity 20.3 rate of radioactive decay

CHAPTER 20

Nuclear Chemistry

and Radioactivity

20.3 Rate of Radioactive Decay

2 20.3 Rate of Radioactive Decay

What is carbon dating?

How can we tell how old fossils are?

3 20.3 Rate of Radioactive Decay

What is carbon dating?

How can we tell how old fossils are?

We introduce the time variable

Reaction rates

4 20.3 Rate of Radioactive Decay

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!

Reaction rates

5 20.3 Rate of Radioactive Decay

Some reactions take place very quickly; they have a short half-life, t1/2.

half-life: the time it takes for half of the atoms in a sample to decay.

Decay

6 20.3 Rate of Radioactive Decay

Half-life

7 20.3 Rate of Radioactive Decay

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 20.3 Rate of Radioactive Decay

Carbon dating

Carbon dating revolves around carbon-14, a radioactive isotope.

1 147 1

14 10 6n N C H

Carbon-14 is generated in the upper atmosphere through a bombardment reaction:

Neutrons generated by cosmic rays

becomes 14CO2 in the atmosphere

9 20.3 Rate of Radioactive Decay

Carbon-14 goes through the same cycle as carbon-12

Carbon dating

14

10 20.3 Rate of Radioactive Decay

In living organisms:

121 24 166 : ~ 1:10C ratioC

Carbon dating

This ratio stays constant while the organism is alive

11 20.3 Rate of Radioactive Decay

In living organisms:

14 12 126 6: ~ 1:10C C ratio

Carbon dating

Over time, carbon-14 decays by emission:

14 14 06 7 1C N e 1/2 5,730t years

This ratio stays constant while the organism is alive

12 20.3 Rate of Radioactive Decay

Over time, carbon-14 decays by emission:

14 14 06 7 1C N e 1/2 5,730t years

Carbon dating

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 20.3 Rate of Radioactive Decay

Carbon dating

Ratio not to scale

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.

14 20.3 Rate of Radioactive Decay

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 20.3 Rate of Radioactive Decay

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 20.3 Rate of Radioactive Decay

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 = 8.90 billion / 8 = 11.25 billion

17 20.3 Rate of Radioactive Decay

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 = 8.90 billion / 8 = 11.25 billion

Answer: After three half-lives the amount of carbon-14 atoms is reduced by a factor of 8, from 90 billion to 11.25 billion.

18 20.3 Rate of Radioactive Decay

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 20.3 Rate of Radioactive Decay

Rate of decay

The number of nuclei in the sample (N) is constant

1/2 0.693 /t k

A short half-life implies a large rate constant, k.

20 20.3 Rate of Radioactive Decay

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 20.3 Rate of Radioactive Decay

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 = 0.693 / t1/2

22 20.3 Rate of Radioactive Decay

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 = 0.693 / t1/2

Solve: For C-14, t1/2 = 5,730 years, and

.

120.693 1 1 1 1

5,730 365 24 60 60

3.84 10year day h mink

years days h min s s

23 20.3 Rate of Radioactive Decay

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 = 0.693 / t1/2

Solve: For C-14, t1/2 = 5,730 years, and

For Ra-220, t1/2 = 1 min, and

120.693 1 1 1 1 3.84 10

5,730 365 24 60 60

year day h mink

years days h min s s

21.155 10.693 1

1 0

0

6

mink

m s sin

24 20.3 Rate of Radioactive Decay

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 = 0.693 / 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.

120.693 1 1 1 1 3.84 10

5,730 365 24 60 60

year day h mink

years days h min s s

20.693 1 1.155 10

1 60

mink

min s s

25 20.3 Rate of Radioactive Decay

Decay rate law

The rate of decay of a radioactive sample

is also called the activity of the sample

26 20.3 Rate of Radioactive Decay

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 20.3 Rate of Radioactive Decay

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 01/2

0.693exp

tN N

t

28 20.3 Rate of Radioactive Decay

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:

01/2

0.693exp

tN N

t

0.693 4

2.8610 2.71 3.798

years

yearsN mg mg

29 20.3 Rate of Radioactive Decay

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.

01/2

0.693exp

tN N

t

0.693 4

2.8610 2.718 3.79

years

yearsN mg mg

30 20.3 Rate of Radioactive Decay

Radioactive dating

the composition of the atmosphere over time

the age of rocks that are billions of years old

the age of a once-living organism

Information can be extracted from the ratio of specific isotopes

Carbon-14 and carbon-12

Oxygen-18 and oxygen-16

Uranium-238 and plutonium-239

31 20.3 Rate of Radioactive Decay

Radioactive dating

00

lnkt

NN

N N e so tk

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

32 20.3 Rate of Radioactive Decay

An ancient Greek scroll written on an animal skin was discovered by archeologists in 2008. 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 20.3 Rate of Radioactive Decay

An ancient Greek scroll written on an animal skin was discovered by archeologists in 2008. 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 20.3 Rate of Radioactive Decay

An ancient Greek scroll written on an animal skin was discovered by archeologists in 2008. 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:1/2

40.693 0.693

5,730

1.209 10

yearsk

t years

35 20.3 Rate of Radioactive Decay

An ancient Greek scroll written on an animal skin was discovered by archeologists in 2008. 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:

4

1/2

0.693 0.693 1.209 10

5,730k

t years years

04

111ln ln150

1.202,491

9 10

NN

tk

ye

yea

s

r

ar

s

36 20.3 Rate of Radioactive Decay

An ancient Greek scroll written on an animal skin was discovered by archeologists in 2008. 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.

4

1/2

0.693 0.693 1.209 10

5,730k

t years years

04

111ln ln150 2,491

1.209 10

NN

t yearsk

years

37 20.3 Rate of Radioactive Decay

0

lnNN

tk

1/2 0.693 /t k

Mathematics of radioactive decay