Exercises: Atomic nuclei and stability of isotopes

The potassium in bananas contains a small fraction of the isotope potassium-40, whose nucleus has atomic number $Z = 19$ and mass number $A = 40$.

1. a) State how many protons and how many neutrons this nucleus has.
2. b) Write its symbol using ${}^{A}_{Z}\text{X}$ notation.
3. c) How many electrons does a neutral atom of potassium-40 have? Justify your answer.

In a helium nucleus, two protons are confined in a region about $10^{-15}\ \text{m}$ in diameter. As both have positive charge, they repel each other electrically with considerable force; however, the nucleus does not break apart.

1. a) Explain what the strong nuclear force is and why it manages to keep the nucleus bound together.
2. b) Explain why, even with that force existing, nuclei with too many protons end up being unstable.

Carbon has three natural isotopes: carbon-12, carbon-13 and carbon-14, all with $Z = 6$.

1. a) Explain what it means for two nuclei to be isotopes.
2. b) Calculate the number of neutrons in the nucleus of each of the three isotopes.
3. c) Do all three isotopes have the same chemical properties? Explain your answer by considering their electrons.

Of the nearly $3000$ known nuclides, barely $250$ are stable; the rest end up decaying sooner or later.

1. a) Explain what it means for a nucleus to be stable and what happens to an unstable nucleus.
2. b) In light stable nuclei, the number of neutrons $N$ and the number of protons $Z$ are almost equal: state the approximate value of the ratio $\frac{N}{Z}$ in these nuclei.

The radius of an atom is on the order of $10^{-10}\ \text{m}$, while that of its nucleus is on the order of $10^{-15}\ \text{m}$, as shown in the figure.

1. a) Calculate how many times larger the radius of the atom is than that of the nucleus.
2. b) The volume of a sphere is proportional to the cube of the radius: estimate how many times larger the volume of the atom is than that of the nucleus.
3. c) In light of the result, explain why it is said that the atom is mostly empty space.

Consider these five nuclides: ${}^{14}_{6}\text{C}$, ${}^{14}_{7}\text{N}$, ${}^{12}_{6}\text{C}$, ${}^{16}_{8}\text{O}$ and ${}^{15}_{7}\text{N}$.

1. a) Calculate the number of protons and neutrons in each one.
2. b) State which pairs are isotopes (same $Z$) and which pair is isobars (same $A$).
3. c) Identify two nuclides that have the same number of neutrons (isotones).

The $\frac{N}{Z}$ ratio of stable nuclei is approximately $1$ for light nuclei and increases to about $1.5$ for the heaviest ones, as shown in the figure.

Calculate the $\frac{N}{Z}$ ratio for each of these nuclides and predict, by comparing with this trend, whether they are expected to be stable or unstable:

1. a) ${}^{12}_{6}\text{C}$
2. b) ${}^{14}_{6}\text{C}$
3. c) ${}^{206}_{82}\text{Pb}$
4. d) ${}^{238}_{92}\text{U}$

Marie Curie isolated radium in $1898$, and for decades the activity of $1\ \text{g}$ of radium-226 served as a unit of measurement, the curie. Data: $N_A = 6.02 \times 10^{23}\ \text{mol}^{-1}$ and molar mass of radium-226: $226\ \text{g/mol}$.

1. a) Calculate how many radium-226 nuclei are in a sample of $1\ \text{mg}$.
2. b) Calculate how many nuclei are in $1\ \text{g}$, the quantity that defined the curie.

Oxygen has only three stable isotopes: ${}^{16}_{8}\text{O}$, ${}^{17}_{8}\text{O}$ and ${}^{18}_{8}\text{O}$. Isotopes with a neutron excess tend to decay by beta-minus emission ($\beta^{-}$: a neutron converts into a proton), and those with a neutron deficit, by beta-plus emission ($\beta^{+}$: a proton converts into a neutron).

1. a) Calculate the $\frac{N}{Z}$ ratio of ${}^{14}_{8}\text{O}$ and ${}^{21}_{8}\text{O}$ and compare it with that of ${}^{16}_{8}\text{O}$.
2. b) Predict which type of decay each of these two unstable isotopes will undergo, justifying your answer.
3. c) Write the equation for each process and identify the daughter nuclide formed in each case.