E.3.1 Stability and binding energy

Understanding Isotopes, Nuclear Stability, and the Strong Nuclear Force

1. Imagine you're holding a helium balloon.
2. The helium inside is made of atoms, and at the center of each atom lies a tiny nucleus containing protons and neutrons.

But here's a curious question: what keeps those positively charged protons from flying apart due to their mutual repulsion? And why do some nuclei stay stable for billions of years while others decay in seconds?

What Are Isotopes?

• Every atom is defined by its number of protons, denoted as $Z$, which determines the element.

• However, not all carbon atoms are identical—they can have different numbers of neutrons, $N$.
• The total number of protons and neutrons, called the nucleon number, is denoted as $A$, where:

$$A=Z+N$$

• Atoms of the same element with different numbers of neutrons are called isotopes. For example:
  • Carbon-12 (${}_6^{12}\text{C}$): 6 protons and 6 neutrons.
  • Carbon-14 (${}_6^{14}\text{C}$): 6 protons and 8 neutrons.

Key Properties of Isotopes

• Same chemical properties: Since chemical behavior depends on the number of electrons, and isotopes have the same $Z$, their chemistry is identical.
• Different physical properties: The difference in neutron number affects properties like mass and nuclear stability.

Binding Energy Per Nucleon: A Measure of Stability

1. When you assemble protons and neutrons to form a nucleus, the total mass of the nucleus is slightly less than the sum of the individual masses of its protons and neutrons.
2. This "missing mass" is called the mass defect, and it is converted into energy according to Einstein’s famous equation:

$$E=m{c}^2$$

This energy, known as the binding energy, is what holds the nucleus together. The binding energy per nucleon is a key indicator of nuclear stability.

$$\text{Binding Energy per Nucleon}=\frac{\text{Total Binding Energy}}{\text{Number of Nucleons}}$$

The Binding Energy Curve

If we plot the binding energy per nucleon against the nucleon number $A$, we observe:

1. Rapid increase for small nuclei: Light nuclei like helium ($A=4$) have relatively low binding energy per nucleon.
2. Peak stability near iron ($A\approx 60$): Iron-56 has one of the highest binding energies per nucleon, making it extremely stable.
3. Gradual decrease for heavier nuclei: As $A$ increases beyond 60, the binding energy per nucleon decreases, making larger nuclei like uranium less stable.

Mass-Energy Equivalence and Energy Release

1. Einstein’s equation, $E=m{c}^2$, reveals the deep connection between mass and energy.
2. In nuclear reactions, mass is not conserved in the traditional sense—some of it is converted into energy.
3. This is why nuclear reactions release such enormous amounts of energy compared to chemical reactions.

Energy from Mass Defect

The energy released in a nuclear reaction can be calculated using the mass defect ($\mathrm{\Delta }m$):

$$Q=\mathrm{\Delta }m\cdot c^2$$

where:

• $Q$ is the energy released,
• $\mathrm{\Delta }m$ is the difference in mass between the reactants and products,
• $c$ is the speed of light ($3.00\times {10}^8\text{m/s}$).

The Strong Nuclear Force: The Glue of the Nucleus

Key Characteristics

1. Attractive and short-ranged: The strong nuclear force is incredibly strong at distances of $\sim {10}^{-15}\text{m}$ (the size of a nucleus) but drops off rapidly beyond this range.
2. Independent of charge: Unlike the electromagnetic force, the strong nuclear force acts equally between protons and neutrons.

Reflection and Applications

Understanding isotopes, binding energy, and the strong nuclear force is essential to explaining phenomena like nuclear stability, radioactive decay, and energy release in nuclear reactions.