Question

half - life practice

1. what is radioactivity?
2. what is nuclear radiation?
3. what is half - life?
4. if we start with 400 atoms of a radioactive substance, how many would remain after one half - life? after two half - lives? after three half - lives? after four half - lives?
5. if we start with 48 atoms of a radioactive substance, how many would remain after one half - life? after two half - lives? after three half - lives? after four half - lives?
6. if we start with 16 grams of a radioactive substance, how much will remain after three half lives?
7. if we start with 120 atoms of a radioactive substance, how many will remain after three half lives?

use the following graph to answer questions 9 - 13

1. how long is a half - life for carbon - 14?
2. if only 25% of the carbon - 14 remains, how old is the material containing the carbon - 14?
3. if a sample originally had 120 atoms of carbon - 14, how many atoms will remain after 16,110 years?
4. if a sample known to be about 10,740 years old had 400 carbon - 14 atoms, how many atoms were in the sample when the organism died?
5. if you have a 550 gram sample of carbon - 14, how much remains after 7 half - lives.

Explanation:

Step1: Recall half - life concept

The amount of a radioactive substance remaining after \(n\) half - lives is given by \(N = N_0\times(\frac{1}{2})^n\), where \(N_0\) is the initial amount and \(n\) is the number of half - lives.

Step2: Answer question 4

• After one half - life (\(n = 1\)) with \(N_0=400\), \(N = 400\times\frac{1}{2}=200\).
• After two half - lives (\(n = 2\)), \(N = 400\times(\frac{1}{2})^2=400\times\frac{1}{4} = 100\).
• After three half - lives (\(n = 3\)), \(N = 400\times(\frac{1}{2})^3=400\times\frac{1}{8}=50\).
• After four half - lives (\(n = 4\)), \(N = 400\times(\frac{1}{2})^4=400\times\frac{1}{16}=25\).

Step3: Answer question 5

• After one half - life (\(n = 1\)) with \(N_0 = 48\), \(N=48\times\frac{1}{2}=24\).
• After two half - lives (\(n = 2\)), \(N = 48\times(\frac{1}{2})^2=48\times\frac{1}{4}=12\).
• After three half - lives (\(n = 3\)), \(N = 48\times(\frac{1}{2})^3=48\times\frac{1}{8}=6\).
• After four half - lives (\(n = 4\)), \(N = 48\times(\frac{1}{2})^4=48\times\frac{1}{16}=3\).

Step4: Answer question 6

With \(N_0 = 16\) grams and \(n = 3\) half - lives, \(N = 16\times(\frac{1}{2})^3=16\times\frac{1}{8}=2\) grams.

Step5: Answer question 7

With \(N_0 = 120\) atoms and \(n = 3\) half - lives, \(N = 120\times(\frac{1}{2})^3=120\times\frac{1}{8}=15\) atoms.

Step6: Answer question 8

From the graph, the half - life of carbon - 14 is 5730 years.

Step7: Answer question 9

If 25% of the carbon - 14 remains, since \(25\%=\frac{1}{4}=(\frac{1}{2})^2\), two half - lives have passed. So the age of the material is \(2\times5730 = 11460\) years.

Step8: Answer question 10

Since 16110 years is \(16110\div5730 = 2.8115\approx3\) half - lives (rounded to the nearest whole number) for carbon - 14. With \(N_0 = 120\) atoms, \(N = 120\times(\frac{1}{2})^3=15\) atoms.

Step9: Answer question 11

Since 10740 years is \(10740\div5730 = 1.8743\approx2\) half - lives (rounded to the nearest whole number). Let the initial number of atoms be \(N_0\). We know \(N = 400\) and \(N = N_0\times(\frac{1}{2})^2\). So \(N_0=400\times4 = 1600\) atoms.

Step10: Answer question 12

With \(N_0 = 550\) grams and \(n = 7\) half - lives, \(N = 550\times(\frac{1}{2})^7=550\times\frac{1}{128}\approx4.297\) grams.

Answer:

1. After one half - life: 200 atoms; After two half - lives: 100 atoms; After three half - lives: 50 atoms; After four half - lives: 25 atoms.
2. After one half - life: 24 atoms; After two half - lives: 12 atoms; After three half - lives: 6 atoms; After four half - lives: 3 atoms.
3. 2 grams.
4. 15 atoms.
5. 5730 years.
6. 11460 years.
7. 15 atoms.
8. 1600 atoms.
9. Approximately 4.297 grams.