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**Nuclear Chemistry and Radioactivity**

CHAPTER 20 Nuclear Chemistry and Radioactivity 20.3 Rate of Radioactive Decay

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

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**Reaction rates What is carbon dating?**

How can we tell how old fossils are? We introduce the time variable

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**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!

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**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.

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Half-life

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**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.

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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

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

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**Carbon dating In living organisms:**

This ratio stays constant while the organism is alive

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**Carbon dating In living organisms:**

This ratio stays constant while the organism is alive Over time, carbon-14 decays by b emission:

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**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.

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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

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**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?

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**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

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**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

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**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.

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**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

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

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

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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.

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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

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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 .

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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

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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.

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

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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?

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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

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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:

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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.

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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

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

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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.)

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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.

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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:

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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:

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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.

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**Mathematics of radioactive decay**

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