Chemistry · Chemistry

Relativistic, quantum, nuclear, and particle physics · Nuclear physics

Atomic nuclei and the stability of isotopes

Inside any nucleus with more than one proton there is a constant battle: the electrical repulsion between protons, confined to a distance of barely $10^{-15}\ \text{m}$, should blow the nucleus apart in a fraction of a second. The fact that most of the matter in the universe is stable shows that there is another force, even stronger at those distances, capable of winning that battle. Understanding when it wins and when it loses is understanding why some nuclei are stable and others radioactive. In 1932 John Cockcroft and Ernest Walton were the first to split the nucleus using artificially accelerated particles, and the process released energy. Later, scientists would build on this work to understand nuclear energy. By the end of this page you will see why.

Composition of the nucleus: protons, neutrons and isotopes

When J. J. Thomson and Francis Aston analyzed beams of positive ions of neon gas, they discovered that they were made up of atoms that were chemically identical but had two different masses ($20\ \text{amu}$ and $22\ \text{amu}$): how is it possible for atoms of the same element not to weigh the same?

Nuclei with the same $A$ but a different $Z$ are called isobars, and nuclei with the same $N$ but a different $Z$ are called isotones.

Practice★☆☆

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

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

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

Solve

Practice★☆☆

Carbon has three natural isotopes: carbon-12, carbon-13 and carbon-14, all with $Z = 6$. a) Explain what it means for two nuclei to be isotopes. b) Calculate the number…

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

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

Solve

Practice★★☆

Consider these five nuclides: ${}^{14}{6}\text{C}$, ${}^{14}{7}\text{N}$, ${}^{12}{6}\text{C}$, ${}^{16}{8}\text{O}$ and ${}^{15}{7}\text{N}$. a) Calculate the number of…

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

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

Solve

Sizes, units and nuclear density

A baseball fits in your hand, yet a teaspoon of nuclear matter would outweigh a mountain: how can the nucleus be so small and so heavy?

Mass, size and density of the aluminum-27 nucleus

An atom has a radius of about $10^{-10}\ \text{m}$ and its nucleus about $10^{-15}\ \text{m}$. Compare their volumes, then find the density of aluminum-27 ($A = 27$, $A^{1/3} = 3$).

The radii differ by a factor of $10^{5}$. Volume goes as the cube of the radius, so the atom's volume is about $(10^{5})^3 = 10^{15}$ times bigger: the atom is almost all empty space.

Mass: $27 \times 1.66 \times 10^{-27} = 4.48 \times 10^{-26}\ \text{kg}$. Radius: $R = 1.2 \times 10^{-15} \times 3 = 3.6 \times 10^{-15}\ \text{m}$.

Volume: $V = \tfrac{4}{3}\pi R^3 = \tfrac{4}{3} \times 3.14 \times (3.6 \times 10^{-15})^3 = 1.95 \times 10^{-43}\ \text{m}^3$.

Density: $\rho = m/V = 4.48 \times 10^{-26} / 1.95 \times 10^{-43} \approx 2.3 \times 10^{17}\ \text{kg m}^{-3}$. This is about the same for every nucleus.

Practice★☆☆

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. a) Calculate…

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.

a) Calculate how many times larger the radius of the atom is than that of the nucleus.
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.
c) In light of the result, explain why it is said that the atom is mostly empty space.

Solve

Practice★★☆

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:…

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: $226\ \text{g/mol}$.

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

Solve

The strong interaction and the band of stability

If you bring two protons together to nuclear distances ($10^{-15}\ \text{m}$), the electrical force of mutual repulsion is around $230\ \text{N}$ (about the weight of a 50-pound dog, roughly 220 N, on a single proton pair!): how can dozens of them live packed together in such a tiny space without the nucleus flying into pieces?

The strong interaction beats electrical repulsion, but only at short range

The electrical repulsive force between two protons $10^{-15}\ \text{m}$ apart is enormous (of the order of hundreds of newtons for a single pair of particles), but the strong nuclear interaction between neighboring nucleons is even more intense at that distance, and it holds the nucleus together. Its range, however, is extremely short: at distances of more than a few $10^{-15}\ \text{m}$ it almost completely vanishes, whereas electrical repulsion, which has infinite range, keeps acting between all the protons at once.

Why? The strong interaction falls off very rapidly with separation (short range) and acts only between immediate neighboring nucleons, whereas electrical repulsion has infinite range and each proton repels all the other protons in the nucleus.

Practice★☆☆

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…

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.

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

Solve

Practice★☆☆

Of the nearly $3000$ known nuclides, barely $250$ are stable; the rest end up decaying sooner or later. a) Explain what it means for a nucleus to be stable and what…

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

a) Explain what it means for a nucleus to be stable and what happens to an unstable nucleus.
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.

Solve

Practice★★☆

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…

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:

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

Solve

Practice★★★

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…

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).

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

Solve

Mass defect and binding energy

If you weigh a helium-4 nucleus in a mass spectrometer you get $4.0015\ \text{amu}$, but if you add up the masses of its nucleons separately (two free protons and two free neutrons) you get $4.0320\ \text{amu}$: where has the missing mass gone when they join together?

The stability curve: the peak at iron

Why fusion and fission release energy

Light nuclei, to the left of the peak, increase their binding energy per nucleon when they fuse: the fusion of light nuclei releases energy. Very heavy nuclei, to the right of the peak, increase their binding energy per nucleon when they break into smaller fragments: the fission of heavy nuclei also releases energy. Iron-56 is at the peak and is among the most stable nuclei there are: fusing or splitting it moves to a lower binding energy per nucleon, so either process consumes energy instead of releasing it. A massive star whose core has turned to iron has no energy source left and its core collapses.

Why? As they evolve towards products with a greater binding energy per nucleon, the nucleons become more tightly bound and the average mass per nucleon decreases; the total mass difference is released as kinetic energy and photons.

Back to the beginning: The strong nuclear interaction wins the battle when the $N/Z$ ratio lies within the band of stability (between $1$ and $1.5$); if a nucleus accumulates too many protons ($Z > 83$; bismuth-209 is only very nearly stable, with an enormous half-life) or departs from that ratio, the electrical repulsion or the energy imbalance destabilizes the nucleus and it emits radiation to reach a more tightly bound configuration. The accelerated protons of Cockcroft and Walton split lithium into two helium nuclei, which lie nearer the peak of the curve, so the mass lost was released as energy.

Summary

A nucleus is described by $Z$ (protons), $N$ (neutrons) and $A = Z+N$; isotopes share $Z$ and differ in $N$.
The strong interaction beats the electrical repulsion between protons, but only at very short range ($\sim10^{-15}\ \text{m}$); stable nuclei follow a band of stability with $N/Z$ increasing from $1$ to $1.5$.
The mass defect $\Delta m$ (the mass of the nucleus is less than the sum of its nucleons) is transformed into binding energy, $E=\Delta m\cdot c^2$.
The binding energy per nucleon measures true stability; it has a maximum around iron-56 ($8.8\ \text{MeV}$).
The fusion of light nuclei and the fission of very heavy nuclei release energy because, in both cases, they bring the result closer to the peak of the curve; iron releases no energy through either process.

Let's practice! 3 medium exercises in this section

★★☆ Consider these five nuclides: ${}^{14}{6}\text{C}$, ${}^{14}{7}\text{N}$,…★★☆ The $\frac{N}{Z}$ ratio of stable nuclei is approximately $1$ for light nuclei and…★★☆ Marie Curie isolated radium in $1898$, and for decades the activity of $1\ \text{g}$ of…

Let's practice! 1 hard exercises in this section

★★★ Oxygen has only three stable isotopes: ${}^{16}{8}\text{O}$, ${}^{17}{8}\text{O}$ and…

See all the exercises in this section