chapter 20 nuclear chemistry and radioactivity 20.3 rate of radioactive decay
TRANSCRIPT
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