Inside the nucleus: size, mass number and isotopes

A tiny core with almost all the mass

Imagine the atom as a stadium. The nucleus is then a pea at the centre, and the electrons are far away in the seats. Yet the pea holds more than $99.9\ \%$ of the mass of the atom. In Rutherford's scattering experiment, which you may have met in the lesson on the nuclear atom, a few alpha particles bounced straight back. That told us the positive charge and the mass sit in a very small space.

How small? A distance of $10^{-15}$ m is called one femtometre, written 1 fm (physicists also call it one fermi). An atom is about $10^{-10}$ m wide, so a nucleus is about $10^5$ times narrower than the atom around it.

What a nucleus is made of

A nucleus is built from two kinds of particles, the protons and the neutrons. Together we call them nucleons. We use three numbers.

• Atomic number $Z$: the number of protons. It fixes the element. NCERT says atomic number. A-Level calls it the proton number.
• Neutron number $N$: the number of neutrons.
• Mass number $A$: the number of nucleons, so $A = Z + N$. A-Level calls it the nucleon number.

We write a nuclide as ${}^{A}_{Z}\text{X}$. For example ${}^{56}_{26}\text{Fe}$ has $26$ protons and $56 - 26 = 30$ neutrons. The proton has a charge of $+e$ and the neutron has no charge. Their masses are almost equal: $1.007276\ \text{u}$ for the proton and $1.008665\ \text{u}$ for the neutron. An electron is about $1836$ times lighter, so the mass of an atom is nearly all in its nucleus.

The atomic mass unit

The kilogram is too big for single atoms. We use the unified atomic mass unit, u. It is defined as $\dfrac{1}{12}$ of the mass of one neutral atom of carbon-12. In kilogram,

By this definition a carbon-12 atom has a mass of exactly $12\ \text{u}$. Every other atomic mass is measured against it. The mass number $A$ is a whole number close to the atomic mass in u.

Isotopes, isobars and isotones

Nuclei with the same $Z$ but different $N$ are isotopes. They are the same element, so they have the same chemistry. Hydrogen has three: ${}^{1}_{1}\text{H}$, ${}^{2}_{1}\text{H}$ (deuterium) and ${}^{3}_{1}\text{H}$ (tritium).

Two related words are easy to mix up.

• Isobars have the same $A$ and different $Z$. Carbon-14 and nitrogen-14 are isobars.
• Isotones have the same $N$ and different $Z$. Carbon-13 and nitrogen-14 both have $7$ neutrons.

A chemical element in nature is usually a mixture of isotopes. Its average atomic mass is weighted by how common each isotope is. Chlorine is $75.8\ \%$ chlorine-35 ($34.969\ \text{u}$) and $24.2\ \%$ chlorine-37 ($36.966\ \text{u}$). So the average is $0.758 \times 34.969\ \text{u} + 0.242 \times 36.966\ \text{u} = 35.45\ \text{u}$. This is the number you see in the periodic table. No single chlorine atom has this mass.

The size of a nucleus

Scattering experiments with fast electrons and alpha particles show that the radius of a nucleus grows with the mass number like this:

Here $R_0$ is a constant. NCERT uses $1.2$ fm. Some books use values from $1.1$ to $1.4$ fm, depending on how the radius is measured.

Where does the cube root come from? Suppose each nucleon takes up the same small volume, as marbles in a bag do. Then the volume of the nucleus is proportional to $A$. The volume of a sphere is $\tfrac{4}{3}\pi R^3$, so $R^3 \propto A$ and $R \propto A^{1/3}$.

The picture shows the radii of five nuclei. Notice how slowly the radius grows. A uranium nucleus has $238$ times as many nucleons as hydrogen, but it is only $6.2$ times wider.

Worked example: radius of two nuclei

Find the radius of aluminium-27, and the ratio of the radii of two nuclei with $A = 64$ and $A = 8$.

1. Aluminium. $R = 1.2\ \text{fm} \times 27^{1/3} = 1.2\ \text{fm} \times 3 = 3.6\ \text{fm}$.
2. The ratio. $\dfrac{R_{64}}{R_8} = \left(\dfrac{64}{8}\right)^{1/3} = 8^{1/3} = 2$. The constant $R_0$ cancels, so we do not need its value.

Nuclear matter has one density

The mass of a nucleus is about $A$ times $1\ \text{u}$. Its volume is $\tfrac{4}{3}\pi R_0^3A$. The factor $A$ cancels in the density:

This is true for every nucleus. It is about $2 \times 10^{14}$ times the density of water, which is $1000\ \text{kg/m}^3$. One cubic centimetre of nuclear matter would have a mass of $2.3 \times 10^{11}$ kg. A teaspoon of it, $5\ \text{cm}^3$, would weigh about a billion tonnes. Neutron stars are made of matter at about this density.

Slide the mass number from $1$ to $240$. By what factor does the radius change? By what factor does the density change?

Practise with new numbers

Try it in Python

The program finds the radius and the density of five nuclei, from hydrogen to uranium.

Common mistakes

• Using $R \propto A$. The radius goes as $A^{1/3}$. It is the volume that is proportional to $A$.
• Saying heavy nuclei are denser. All nuclei have almost the same density. A heavy nucleus is only larger.
• Mixing up isotopes, isobars and isotones. Same $Z$: isotopes. Same $A$: isobars. Same $N$: isotones.
• Calling the mass number the mass. $A$ is a count of nucleons. The atomic mass is in u and is close to $A$, but not equal to it.
• Forgetting that fm is $10^{-15}$ m. Convert to metre before you find a volume or a density.

Remember these

• $A = Z + N$. The nuclide ${}^{A}_{Z}\text{X}$ has $Z$ protons and $A - Z$ neutrons.
• $1\ \text{u} = \dfrac{1}{12}$ of the mass of a carbon-12 atom $= 1.660539 \times 10^{-27}$ kg.
• Isotopes: same $Z$. Isobars: same $A$. Isotones: same $N$.
• $R = R_0A^{1/3}$ with $R_0 = 1.2$ fm, and $1\ \text{fm} = 10^{-15}$ m.
• Nuclear density is about $2.3 \times 10^{17}\ \text{kg/m}^3$ for every nucleus.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. How many protons and how many neutrons are there in ${}^{238}_{92}\text{U}$?

2. The radius of a nucleus of mass number $A$ is $R$. What is the radius of a nucleus of mass number $8A$?

3. Why is the density of a nucleus the same for hydrogen-2 and for lead-208?

4. Name the pair: ${}^{14}_{6}\text{C}$ and ${}^{14}_{7}\text{N}$. Name the pair: ${}^{35}_{17}\text{Cl}$ and ${}^{37}_{17}\text{Cl}$.

References

• NCERT Physics Part II (Class 12, rationalised edition), Chapter 13 Nuclei, on the composition and size of the nucleus.
• OpenStax University Physics Volume 3, Chapter 10 Nuclear Physics, on the properties of nuclei.