# How to Calculate Nuclear Binding Energy

All matter is made of atoms. In an atom exist sub-atomic particles called neutrons, protons and electrons. We know that the protons and neutrons are held together in a core called the nucleus. Do you see that it needs energy? Protons are positively charged and to hold these particles at the same charge, energy is required. Nuclear binding energy originates from this concept.

## Mass Defect and Binding Energy

It has been observed that the sum of individual masses of the components of the nucleus is higher than that of the atom’s nucleus. This means some mass is lost in the process of binding. This mass difference also known as the mass defect, which is directly related to the binding energy by the famous Einstein equation E= mc2, where E = nuclear binding energy, m = mass difference or mass defect, and c = speed of light

## How to Find Binding Energy

We shall now look at how to calculate nuclear binding energy of an atom. Let’s take 12C.

Step 1: Calculte the mass defect (m)

Use the information given in the table.

 Particle Mass (kg) Mass (u) Mass (MeV/C2) 1 a.m.u 1.660540 x 10-27 kg 1.000 u 931.5 MeV/c2 neutron 1.674929 x 10-27 kg 1.008664 u 939.57 MeV/c2 proton 1.672623 x 10-27 kg 1.007276 u 938.28 MeV/c2 electron 9.109390 x 10-31 kg 0.00054858 u 0.511 MeV/c2

12C has 6 protons and 6 neutrons and 6 electrons.

Mass defect       = Total mass of individual sub-atomic particles – Mass of the atom

= 6 * 1.008664 u + 6 * 1.007276 u + 6 * 0.00054858 u – 12.000 u

= 0.098931 u

Step 2:  Now, apply the E= mcequation to calculate nuclear binding energy.

Binding energy = (1.660540 x 10-27 kg (per a.m.u) * 0.098931) * (3*10 8 ms -1)2

Or could simply calculate nuclear binding energy directly by converting it into MeV by,

= 0.098931 u * 931.5 MeV/u

= 92.15 MeV

## Convert Energy Unit Joule into MeV

Why have we bothered to convert the usual energy unit Joule into MeV?

Binding energies are conventionally expressed as MeV because it is a very large energy, and expressing in eV shows how charge plays a role in the binding energy formation.

Other conventions:

a)      Expressing nuclear binding energy as energy per mole

Once the binding energy is found “per atom” simply multiplies the value by Avagadro’s Number which is 6.022 x 1023 mol-1.

If the binding energy was expressed in Joules, you might want to convert it to kJ because the value is higher. To do so, divide the answer by 1000.

b)      Expressing binding as MeV per nucleon

Nucleons are the particles which make up the nucleus (protons and neutrons). Once the binding energy is found simply divide the value by the number of nucleons present in the atom. In 12 C case it should be divided by 12.

Why is this important?

Nuclear binding energy is extremely important to know in nuclear reactions. When nuclear reactions take place these binding energies are the key factors which control or holds their probability of occurring.