Exercises: Radioactivity and nuclear processes

Unstable nuclei emit three types of radiation: alpha ($\alpha$), beta ($\beta$) and gamma ($\gamma$).

1. a) State the nature of each: is it a helium nucleus, an electron or a photon?
2. b) Order them from smallest to largest mass.
3. c) Indicate which ones have an electric charge and what sign it is.

In a laboratory there are three protective materials: a sheet of paper, an aluminium plate a few millimetres thick, and a thick wall of lead or concrete. Observe the diagram:

1. a) Match each material with the type of radiation (alpha, beta or gamma) it is sufficient to stop.
2. b) Which of the three types of radiation is the most penetrating? And the least?
3. c) Explain why technicians working with gamma sources use lead shields or aprons.

Uranium-238 decays by emitting an alpha particle (a helium-4 nucleus, ${}^{4}_{2}\text{He}$) and transforms into thorium.

1. a) Write the complete equation for the process, ${}^{238}_{92}\text{U} \rightarrow {}^{A}_{Z}\text{Th} + {}^{4}_{2}\text{He}$, calculating the values of $A$ and $Z$.
2. b) State the two conservation laws you have applied: conservation of mass number and conservation of atomic number.

Carbon-14, used in the dating of archaeological remains, decays by beta-minus emission: a neutron in the nucleus converts into a proton and an electron (${}^{0}_{-1}\text{e}$) is emitted.

1. a) Write the equation for the process, ${}^{14}_{6}\text{C} \rightarrow {}^{A}_{Z}\text{N} + {}^{0}_{-1}\text{e}$, calculating $A$ and $Z$.
2. b) Explain why in beta decay the mass number does not change, but the atomic number does.

The half-life of a radioactive isotope is the time taken for the amount of undecayed nuclei to halve. A laboratory source contains $100\ \text{g}$ of a certain isotope. The figure shows its decay curve.

1. a) How many grams remain undecayed after one half-life has passed?
2. b) And after two half-lives? And three?
3. c) Write down, in terms of $n$, the general expression for the fraction remaining after $n$ half-lives.

After many alpha or beta decays, the daughter nucleus is left in an excited state and de-excites by emitting gamma radiation. Observe the energy level diagram:

1. a) Explain what gamma radiation is: how is it similar to visible light and how is it different?
2. b) Do the mass number $A$ or the atomic number $Z$ of the nucleus change when it emits gamma? Justify your answer.

The activity of a radioactive sample is the number of decays that occur in it per second, and is measured in becquerels ($1\ \text{Bq} = 1$ decay per second).

1. a) In a laboratory sample, $12000$ decays are counted in one minute: calculate its activity in becquerels.
2. b) Explain what happens to the activity of a sample as time passes, relating it to the decrease in the number of nuclei.

The two ways to obtain energy from nuclei are fission and fusion.

1. a) Define in your own words what each one consists of.
2. b) Give an example of where each one occurs, naturally or technologically.
3. c) Which of the two processes is the Sun's energy source? Justify your answer.

Thorium-232 (${}^{232}_{90}\text{Th}$) first emits an alpha particle (${}^{4}_{2}\text{He}$) and the resulting nucleus then emits a beta-minus particle (${}^{0}_{-1}\text{e}$). Observe the diagram of the decay chain:

1. a) Write the equation for the first decay and identify the daughter nuclide (it is an isotope of radium, with $Z = 88$).
2. b) Write the equation for the second decay and identify the final nuclide (it is an isotope of actinium, with $Z = 89$).
3. c) Verify that mass number and atomic number are conserved in each step.

Iodine-131, used in nuclear medicine, has a half-life of $8$ days. A hospital receives a sample of $80\ \text{mg}$.

1. a) How many half-lives have elapsed after $32$ days?
2. b) Calculate the mass of iodine-131 remaining undecayed after $32$ days.
3. c) Express this mass as a percentage of the initial mass.

Technetium-99m, the most widely used radiopharmaceutical in diagnostics, has a half-life of $6\ \text{h}$. A patient is administered a dose whose initial activity is $800\ \text{MBq}$ (remember that the prefix mega means $10^{6}$).

1. a) Calculate the activity remaining after $6\ \text{h}$.
2. b) Calculate the activity remaining after a full day, $24\ \text{h}$.
3. c) Reason why such a short half-life is an advantage for the patient and, at the same time, a logistical problem for the hospital.

In bone remains found near the Altamira caves, the current proportion of carbon-14 is measured to be a quarter of what the bones had when the animal died. The half-life of carbon-14 is $5730$ years. The figure shows the decay curve of carbon-14.

1. a) How many half-lives have elapsed since the animal's death?
2. b) Calculate the age of the remains.
3. c) Explain why the carbon-14 in an organism stops being replenished when it dies, and why this allows it to be used as a "clock".

In the wood of a ship attributed to the Vikings, the concentration of carbon-14 is measured to be one eighth of that of modern wood. The half-life of carbon-14 is $5730$ years.

1. a) Express the fraction $\frac{1}{8}$ as a power of $\frac{1}{2}$ and deduce how many half-lives have passed since the tree was cut down.
2. b) Calculate the age of the wood.
3. c) The Vikings sailed about $1000$ years ago: comment on whether the result is consistent with that period and what conclusions could be drawn about the ship.

One of the possible fission reactions of uranium-235 is ${}^{235}_{92}\text{U} + {}^{1}_{0}\text{n} \rightarrow {}^{141}_{56}\text{Ba} + {}^{92}_{36}\text{Kr} + x \cdot {}^{1}_{0}\text{n}$. Observe the diagram of the process:

1. a) Applying conservation of mass number, calculate the value of $x$.
2. b) Verify that atomic number is also conserved.
3. c) Explain why the release of several neutrons in each fission makes a chain reaction possible.

Experimental fusion reactors use the reaction between deuterium (${}^{2}_{1}\text{H}$) and tritium (${}^{3}_{1}\text{H}$): ${}^{2}_{1}\text{H} + {}^{3}_{1}\text{H} \rightarrow {}^{4}_{2}\text{He} + {}^{A}_{Z}\text{X}$. Observe the diagram:

1. a) Identify the particle ${}^{A}_{Z}\text{X}$ that is emitted, calculating its values of $A$ and $Z$.
2. b) For two nuclei to fuse they must approach within about $10^{-15}\ \text{m}$: explain what force must be overcome and why it is necessary to heat the gas to above $10^{8}\ \text{K}$.

Alpha radiation is the most ionising of the three and yet the least penetrating; gamma is the most penetrating and the least ionising.

1. a) Explain what it means for radiation to be ionising.
2. b) Reason why alpha radiation, with its high ionising power, barely penetrates the outer layer of skin.
3. c) Americium-241 in smoke detectors emits alpha radiation: explain why it poses no danger while it remains inside its casing, but would be dangerous if inhaled or ingested.

A cobalt-60 source in a radiotherapy unit has a half-life of $5,27$ years and an initial activity of $3,7 \cdot 10^{11}\ \text{Bq}$.

1. a) Express $12,5\ \%$ of the initial activity as a power of $\frac{1}{2}$.
2. b) Calculate how many half-lives must elapse for the source to be reduced to that value and the total time this represents.
3. c) Calculate the activity of the source at that moment.

The activity of carbon-14 in living organisms is about $15$ decays per minute per gram of carbon. In a prehistoric human tissue, an activity of just $1$ decay per minute per gram is measured; take it as $\frac{1}{16}$ of the initial activity. The half-life of carbon-14 is $5730$ years.

1. a) Calculate the age of the tissue.
2. b) Estimate the activity we would measure in a sample about $60000$ years old (about $10$ half-lives) and explain why the carbon-14 method ceases to be reliable for such great ages.

In the uranium radioactive decay series, the radium-226 nucleus (${}^{226}_{88}\text{Ra}$) eventually transforms into lead-206 (${}^{206}_{82}\text{Pb}$) by emitting only alpha and beta-minus particles. Observe the diagram:

1. a) Each alpha emission reduces $A$ by $4$ units and each beta-minus emission does not change it: calculate how many alpha particles are emitted in total to go from $A = 226$ to $A = 206$.
2. b) Taking into account that each alpha particle reduces $Z$ by $2$ units and each beta-minus increases it by $1$, calculate how many beta-minus emissions are needed to end at $Z = 82$.
3. c) Verify that the final balance of the two conservation laws is consistent.

In the laboratory, a radioactive source of an unknown isotope is studied: it is found that in $24$ days its activity has been reduced to a quarter of its initial value.

1. a) Express the remaining fraction as a power of $\frac{1}{2}$ and deduce how many half-lives have elapsed.
2. b) Calculate the half-life of the isotope.
3. c) Predict what fraction of the initial activity will remain after another $24$ days, and verify it with a calculation.

In each fission of a uranium-235 nucleus, about $200\ \text{MeV}$ is released. Data: $1\ \text{eV} = 1,6 \cdot 10^{-19}\ \text{J}$, $N_A = 6,02 \cdot 10^{23}\ \text{mol}^{-1}$ and molar mass of uranium-235: $235\ \text{g/mol}$.

1. a) Express the energy released in a single fission in joules.
2. b) Calculate how many nuclei there are in $1\ \text{kg}$ of uranium-235.
3. c) Calculate the total energy released when this kilogram undergoes complete fission and compare it with petrol, which releases about $4,6 \cdot 10^{7}\ \text{J/kg}$: how many kilograms of petrol would be needed to equal it?

In the Sun, the net balance of the proton–proton chain is the fusion of four protons to form a helium-4 nucleus, with a mass defect of $\Delta m = 0,0287\ \text{u}$ for each helium nucleus formed. Data: $1\ \text{u} \cdot c^2 = 931,5\ \text{MeV}$, $1\ \text{eV} = 1,6 \cdot 10^{-19}\ \text{J}$ and power radiated by the Sun: $P = 3,8 \cdot 10^{26}\ \text{W}$.

1. a) Calculate the energy released when each helium nucleus is formed, first in $\text{MeV}$ and then in joules.
2. b) Calculate how many helium nuclei are formed every second inside the Sun to sustain this power output.
3. c) Reason, knowing that the mass of the Sun is about $2 \cdot 10^{30}\ \text{kg}$, why it has been able to shine like this for billions of years.

An amateur paleontologist proposes using carbon-14 to date a dinosaur femur that is $66$ million years old. The half-life of carbon-14 is $5730$ years and that of potassium-40 is $1,25 \cdot 10^{9}$ years.

1. a) Calculate how many carbon-14 half-lives these $66$ million years correspond to.
2. b) Estimate what fraction of carbon-14 would remain in the bone and explain why the method proves useless in this case.
3. c) Reason what condition the half-life of an isotope must satisfy in order to date a sample of a given age, and justify why long half-life isotopes such as potassium-40 are used for rocks millions of years old.

In the alpha decay of radium-226 into radon-222, energy is released because the mass of radium is greater than the sum of the masses of the products. Data: $m(\text{Ra}) = 226,0254\ \text{u}$, $m(\text{Rn}) = 222,0176\ \text{u}$, $m_{\alpha} = 4,0026\ \text{u}$ and $1\ \text{u} \cdot c^2 = 931,5\ \text{MeV}$.

1. a) Calculate the mass defect of the process, in u units.
2. b) Calculate the energy released in each decay, in $\text{MeV}$.
3. c) Explain where this energy goes and reason why practically all of it appears as kinetic energy of the alpha particle.