Preview of Period 10: Nuclear Reactions 10.1 Rates of Radioactive Decay How can the half-life of a radioactive source be used to find the age of the source? How can capacitor discharge be used to model radioactive decay? 10.2 Mass as a Form of Energy How can the binding energy of a nucleus be calculated? 10.3 Nuclear Binding Energy How can the binding energy of a nucleus be estimated from a binding energy graph?

10-1

Half -Life ♦ The half-life of a radioactive source is the time required for half of the unstable nuclei to decay. ♦ After one half-life, the material will be only half as radioactive. ♦ The number of the original nuclei remaining will be only half what it was originally. Number of Half Lives 0

Fraction of Original 1

=

1

1 2

=

2

1 4

=

3

1 8

=

Number of Half Lives

Fraction of Original

1

6

20 1

1 64

7

1 128

=

8

1 256

=

9

1 512

=

21 1 22 1 2

4

1 16

=

5

1 32

=

3

1 24 1

10

=

1 = 1024

1 26 1 27 1 28 1 29 1 2 10

25

10-2

Exponential Growth and Decay N = B x 2t

exponential growth: exponential decay:

N =

B x 2

−t

=

B x

1 2t

N = the amount of the quantity at a given time t = the number of time periods elapsed B = the initial amount of the quantity Example A sample of radioactive material has a half life of 15 minutes. If there are 5.0 grams of the material at the beginning of an experiment, how much will be left after 1 hour has passed? After 1 hour, four 15-minute half lives have passed.

N = B x

1 2t

= 5.0 grams

1 2

4

=

5.0 g

1 16

=

0.31 g

10-3

Carbon-14 Dating 14 Carbon-14 ( 6 C ) can be used to date archeological sites

− Carbon-14 decays be emitting a β particle and an antineutrino

14 6C

→

14 7N

+

0 −1 e

+ ν

Carbon-14 is produced when cosmic rays convert stable nitrogen-14 in the air into carbon-14. A

β + particle and a neutrino are emitted. 14 7N

→

14 6C

+

0 +1 e

+ ν

10-4

Carbon-14 Dating ♦ Both stable Carbon-12 and unstable Carbon14 isotopes are present in the atmosphere. ♦ Living organisms absorb both isotopes of carbon. ♦ After an organism dies, it no longer absorbs any new Carbon-14, and the Carbon-14 within it decays. ♦ We can accurately estimate the time of an organism's death, if we know 1)

the ratio Carbon-12 to Carbon-14 in the atmosphere at the time the organism died

2)

the present ratio of Carbon-12 to Carbon14 in the fossil, and

3)

the half-life of Carbon-14 (5,568 years)

10-5

Modeling Radioactive Decay with a Capacitor Graph of Exponential Decay

1800

Charge Q (in microcoulombs)

1600 1400 1200 1000 800 600 400 200 0 0

3

6

9

12

15

Time (in seconds)

What is the Half-Life of the data represented by the graph?

10-6

Finding the Half-Life of a Graph with Background 1)

Pick a data point on your graph and read the Y-axis value (the voltage in our case).

2)

Subtract the background voltage.

3)

Divide the result in half.

4)

5)

Add back in the background voltage. This gives ½ the original voltage, corrected for the background. Find this voltage on your graph.

6)

Read down to the X-axis from this point to find a time in seconds.

7)

The difference in seconds between this time and the time of your original point is the half-life – the time it took for ½ of the capacitor’s charge to be released.

10-7

Nuclear Binding Energy Calculation: E = Mc2 Binding energy = [(mass of unbound protons + neutrons) – (mass of nucleus)] c2 Binding energy = [Z Mp + (A – Z) Mn – Mnuc] c2 Mp = mass of a free proton = 1.6726 x 10–27 kg Mn = mass of a free neutron = 1.6749 x 10–27 kg Mnuc = mass of the assembled nucleus in kg Z

= number of protons in the nucleus

A-Z = number of neutrons in the nucleus c2

= (speed of light)2 = (3 x 108 m/s)2

Binding energy per nucleon = binding energy number of nucleons converting units between MeV and joules: or

1 Mev = 106 eV = 1.6 x 10–13 joules 1 joule = 6.25 x 1012 MeV

10-8

Nuclear Binding Energy Graph

10-9

Period 10 Summary 10.1 The half-life of an unstable element is the time it takes on average for one half of the nuclei in the sample to decay. Radio-carbon dating uses the half life of 14

Carbon-14 ( 6 C ) to determine the age of objects. Carbon-14 has a half life of 5,568 years. 10.2 Energy can be released when nuclei fuse to form a nucleus that is more tightly bound. In fusion reactions, light isotopes release energy by combining (or fusing) into heavier ones. Nuclear fusion is the energy source that fuels stars. 10.3 Nuclear binding energy: the energy required to hold nucleons together into atoms. The most stable nuclei have the greatest binding energy per nucleon. Iron (Fe) has high binding energy, while Uranium (U) has less binding energy. The energy released when nuclei fuse or fission is calculated from E = Mc2 Binding energy = [(mass of unbound protons + neutrons) – (mass of nucleus)] c2

Period 10 Review Questions R.1 Carbon-14 dating cannot be used for objects older than about 70,000 years. Why should this be true? (Hint: the half-life of C-14 is 5568 years.) R.2 The half life of Ba-137 is about 2.6 minutes. If you took a count rate from the Ba-137 30 minutes (about 12 half lives) after it was extracted, could you estimate this elapsed time well from such counting rate data? Why or why not? R.3 In class you used capacitor decay to model radioactive decay and graphed the count rate (voltage) per unit time. What was the shape of your graph? Some graphs leveled off at a count rate greater than zero. Why was this the case? R.4 The amount of matter converted into energy in a chemical reaction is much smaller than the matter converted into energy in a nuclear reaction. Why is this? R.5 You know the mass of a nucleus and the number of protons and neutrons that make up the nucleus. How would you find the binding energy of the nucleus?