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.

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.

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.

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.

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.

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.

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 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.

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.

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.