Chapter 9: Review

The chapter problem, solved

The hospital scanning unit uses a tracer with 43 protons and 56 neutrons, a half-life of 6.0 hours and a starting count rate of 800 counts per second.

1. The particles in the tracer atom. The atomic number is the number of protons.

Z = 43

The mass number counts protons and neutrons together.

A = 43 + 56 = 99

A neutral atom has the same number of electrons as protons, so it has 43 electrons.

2. The count rate after 12 hours. A sample is prepared at 08:00 and used at 20:00, which is 12 hours later.

\frac{12}{6.0} = 2 \ \text{half-lives}

After two half-lives the count rate is

\frac{800}{2^2} = 200 \ \text{counts per second}

3. The count rate after 24 hours. The stored sample is now four half-lives old.

\frac{24}{6.0} = 4 \ \text{half-lives}

\frac{800}{2^4} = 50 \ \text{counts per second}

4. The background correction. A detector near the stored sample reads 70 counts per second. The background count rate is 20 counts per second, so the count rate from the tracer alone is

70 - 20 = 50 \ \text{counts per second}

This agrees with the value predicted in part 3.

5. The net decline. After four half-lives the fraction remaining is

\left(\frac{1}{2}\right)^4 = \frac{1}{16}

The net decline is

1 - \frac{1}{16} = \frac{15}{16}

so the net decline is a ratio of $15:16$ of the original emission.

6. Why the tracer is a gamma emitter with a short half-life. Gamma radiation passes out through the body, so a detector outside can measure it. An alpha or beta emitter would be stopped before the radiation reached the detector. A half-life of 6.0 hours is long enough to carry out the scan, but short enough that the activity falls to a low level soon afterwards, so the patient receives only a small dose.

7. The hospital's electricity. The reactor uses nuclear fission. A uranium-235 nucleus absorbs a neutron and splits into two smaller nuclei, releasing two or three neutrons, gamma rays and energy. The chain reaction is controlled so that, on average, only one neutron from each fission causes the next, so the energy is released at a steady rate.

8. Is the scan worth its risk? The scan can find a problem that other tests would miss, so the patient can be treated sooner. The tracer gives a dose of ionising radiation, which carries a small risk of damaging cells. The dose is kept small by choosing a short half-life and a small amount of the isotope. This final judgement is a reasoned answer. A computer cannot check it, so compare your reasoning with the evidence above.

Mixed practice — choose the method first

For each question, say which equation or idea you will use before you calculate. Keep your units with your numbers.

In the chapter problem the tracer isotope has 43 protons and 56 neutrons. State its atomic number, its mass number and the number of electrons in a neutral atom.
The tracer has a half-life of 6.0 hours and a starting count rate of 800 counts per second. Say which method you will use, then calculate the count rate after 12 hours.
The tracer is now 24 hours old. Its half-life is 6.0 hours and its starting count rate was 800 counts per second. Calculate the count rate now.
H After four half-lives, calculate the net decline in the emission from the tracer as a ratio.
A detector near the stored tracer reads 70 counts per second. The background count rate is 20 counts per second. Calculate the count rate from the tracer alone, and compare it with the value predicted after four half-lives.
Explain why a tracer for exploring an internal organ should emit gamma radiation and have a half-life of a few hours.
Compare alpha, beta and gamma radiation in terms of their ionising power and their range in air.
A nucleus of radon-222, $^{222}_{86}\text{Rn}$ , emits an alpha particle. Write the balanced nuclear equation.
Describe how nuclear fission in a reactor releases energy, and how a chain reaction is controlled.
Describe nuclear fusion and explain where it happens naturally.
A 2.0 kg block of iron is heated from 20 °C to 50 °C. The specific heat capacity of iron is 450 J/kg°C. Say which equation you will use, then calculate the energy transferred to the block.
A car of mass 1200 kg travels at 8.0 m/s. Calculate the energy in its kinetic store.
A wave has a frequency of 50 Hz and a wavelength of 0.40 m. Calculate its speed.
A lamp is connected to a 230 V supply and takes a current of 0.20 A. Calculate the power of the lamp.
A motor is supplied with 500 J of energy and transfers 350 J usefully. Calculate its efficiency as a decimal.
A hospital needs a tracer. Say which type of radiation the tracer should emit and what its half-life should be, and explain each choice.

Answers and concise solutions

1. Atomic number, mass number and electrons

Answer: Atomic number $Z = 43$ . Mass number $A = 43 + 56 = 99$ . A neutral atom has 43 electrons.

2. Count rate after 12 hours

Answer: Count the half-lives: $\dfrac{12}{6.0} = 2$ . Halve the count rate twice: $\dfrac{800}{2^2} = 200$ counts per second.

3. Count rate after 24 hours

Answer: $\dfrac{24}{6.0} = 4$ half-lives. $\dfrac{800}{2^4} = 50$ counts per second.

4. Net decline after four half-lives

Answer: The fraction remaining is $\left(\dfrac{1}{2}\right)^4 = \dfrac{1}{16}$ . The net decline is $1 - \dfrac{1}{16} = \dfrac{15}{16}$ , a ratio of $15:16$ of the original emission.

5. Corrected count rate

Answer: Correct the reading by subtracting the background: $70 - 20 = 50$ counts per second. This matches the predicted value of 50 counts per second after four half-lives.

6. Why the tracer is a gamma emitter with a short half-life

Gamma radiation passes through the body instead of being stopped by it.
A detector outside the body can measure the gamma radiation, so the organ can be imaged. An alpha or beta emitter would be stopped before the radiation reached the detector.
A half-life of a few hours means the activity falls to a low level soon after the scan, so the patient receives a small dose.

7. Ionising power and range

Alpha is the most strongly ionising, beta is less ionising, and gamma is the least ionising.
Alpha has the shortest range in air, a few centimetres. Beta travels about a metre, and gamma has a very long range.

8. The balanced alpha decay equation

Answer: An alpha particle is $^{4}_{2}\alpha$ . The mass number falls by 4 and the atomic number falls by 2, giving $^{222}_{86}\text{Rn} \rightarrow {}^{218}_{84}\text{Po} + {}^{4}_{2}\alpha$ .

9. Fission and the controlled chain reaction

A large unstable nucleus absorbs a neutron and splits into two smaller nuclei, releasing two or three neutrons and gamma rays. Energy is released.
The released neutrons can cause more fissions, so the reaction can spread through the fuel.
In a reactor the reaction is controlled so that on average one neutron from each fission causes the next one, releasing energy at a steady rate.

10. Fusion and where it happens

Fusion joins two light nuclei to form a heavier nucleus.
Some of the mass is converted into the energy of radiation.
Fusion happens naturally in the core of a star such as the Sun.

11. Energy transferred to the iron block

Answer: Use $\Delta E = mc\Delta\theta = 2.0 \times 450 \times 30 = 27000\,\text{J}$ , so $2.7 \times 10^4\,\text{J}$ to 2 significant figures.

12. Energy in the kinetic store

Answer: $E_k = \tfrac{1}{2}mv^2 = 0.5 \times 1200 \times 8.0^2 = 38400\,\text{J}$ , so $3.8 \times 10^4\,\text{J}$ to 2 significant figures.

13. Speed of the wave

Answer: $v = f\lambda = 50 \times 0.40 = 20\,\text{m/s}$ .

14. Power of the lamp

Answer: $P = VI = 230 \times 0.20 = 46\,\text{W}$ .

15. Efficiency of the motor

Answer: $\text{efficiency} = \dfrac{350}{500} = 0.70$ .

16. Choosing a tracer, and why

The tracer should emit gamma radiation.
It should have a half-life of a few hours.
Gamma radiation passes out through the body, so a detector outside can measure it. A half-life of a few hours is long enough for the scan but short enough that the patient receives only a small dose.

What you can now do

describe the structure of an atom and the particles inside it;
state the atomic number, the mass number and the number of electrons of an atom or an ion;
explain what isotopes are, and describe how the model of the atom changed;
describe alpha, beta and gamma radiation, and write balanced decay equations;
explain what half-life means, and use it to find a count rate or an activity;
compare contamination with irradiation, and describe the hazard of each;
describe background radiation, and explain why the dose a person receives varies;
describe and evaluate the medical uses of nuclear radiation;
describe nuclear fission and nuclear fusion, and explain where each happens.

Your relevance reflection

Return to the prompt from the chapter opener: where have you met radioactivity, or a use of it, outside this book? Write one more example now that you have finished the chapter. Has the chapter changed what you notice about a hospital scan, a news story or a smoke alarm?