E = mc² is the most famous equation of all time. It has become an icon of science, of knowledge and of learning, way beyond its origins in Einstein’s work and its applications in nuclear physics. The concept of mass-energy equivalence represented by Einstein’s famous equation is actually a very simple one but the application of the formula to nuclear processes often seems to confuse students.

 

This is a brief summary of the topic and a guide to some of the pitfalls that often confuse students.

 

Mass-Energy equivalence seems to confuse a lot of A Level Physics students (and physics undergraduates!) This is perhaps due to the counterintuitive concept of Binding Energy, unfamiliar units and the use of negative numbers in otherwise simple calculations.

 

In nuclear events such as radioactive decay, fission or fusion, the mass of the particles involved changes and there is an equivalent release (or intake) of energy. It is not strictly true to say that mass has been converted into energy, rather that mass is a representation of a particular type of energy and that transfer of energy to different forms affects the particles’ mass. In practical terms, this doesn’t matter: the calculations involved are as though mass and energy were two different currencies and E=mc² the exchange rate formula.

 

 

MASS DEFECT

 

In a nuclear event, MASS DEFECT is the difference between the TOTAL mass of particles before the event and the TOTAL mass after.

 

For example, if a Uranium-238 nucleus decays by alpha radiation, the mass defect is calculated thus:

 

MASS DEFECT = [Mass of Uranium nucleus] – [Mass of daughter nucleus + mass of alpha particle]

 

Where energy is released in a nuclear event, the POTENTIAL ENERGY of the particles DECREASES and their mass decreases as a result.

 

The equation E=mc² is often written as

 

ΔE = c²Δm

 

Where:

ΔE = Energy released

Δm = Mass defect (Decrease in TOTAL particle mass)

c² = a constant equal to the speed of light squared (≈ 9 x 10¹⁶)

 

 

BINDING ENERGY

 

Potential energy of nuclei is ALWAYS negative: if nucleons (protons and neutrons) are completely separate, they do not interact so they have ZERO potential energy.

 

Work must be done to a nucleus to separate it into its constituent nucleons. I.e. The nucleons require an INPUT of energy to RAISE them to ZERO potential energy.

 

Therefore, nucleons bound within nuclei have potential energy BELOW ZERO i.e. negative. This is sometimes called a POTENTIAL WELL because if you are at the bottom of a well, you have to climb up just to reach ground level)

 

BINDING ENERGY is defined as the WORK required to completely separate a nucleus into its constituent nucleons. Binding Energy is ALWAYS a POSITIVE number.

 

Binding Energy is equal to the magnitude of the NEGATIVE potential energy of the nucleus. (If the nucleus’ potential energy is equivalent to the depth of the well you are trapped in, Binding Energy is the height of the ladder you need to escape)

 

If a nucleus decays, energy is released (mainly as kinetic energy of moving particles: α, β etc.) so the potential energy of the remaining particles is lower (i.e. a bigger negative number.) This means the BINDING ENERGY will be greater and the total mass will be less.

This leads to the somewhat confusing situation where energy is released in a nuclear event and yet the total Binding Energy increases!

 

We have to remember that although Binding Energy is always a POSITIVE number, the potential energy of the nucleus is the NEGATIVE equivalent. (If the Binding Energy of a nucleus was e.g. 3.5MeV then its potential energy must be -3.5MeV)

 

When the Binding Energy increases, it means that the daughter nucleus (after) is in a deeper potential well than the parent nucleus (before). I.e. More work is required to separate it into its constituent parts.

 

An increase in Binding Energy is consistent with energy being released because an INCREASE in Binding Energy means that more energy is now required to separate the nucleons because the (daughter) nucleus has LOWER (more negative) potential energy than the parent nucleus: the decrease in nuclear potential energy is equal in magnitude to the energy released in the interaction.