Mass Defect and Nuclear Binding Energy - Simplified Notes

 Nuclear Binding Energy

Nuclear binding energy is the energy required to break the nucleus into its constituent particles. (OR) It is the energy released when protons and neutrons come together to form a nucleus.

Lets say we have He-4 atom. This has 2 protons and two neutrons in its nucleus. These protons and neutrons are present as one mass in the nucleus and are stuck together something called nuclear strong force (we will talk about this later). 

So, the nuclear binding energy is the amount of energy required to separate these protons ans neutrons so that they aren't together.

Now suppose we have 2 protons ans 2 neutrons . We want to form a nucleus  by combining these particles. To combine them, they are brought close to each other. (we will talk about the formation of a nucleus under nuclear strong force). When these particles combine, they release a certain amount of energy. This energy released is known as nuclear binding energy.

Mass Defect

Mass defect is the difference of mass of the the composite particles and the overall mass of the atom.

This means that when we predict the mass of an atom by adding the mass of the constituent particles (protons and neutrons), we find that the mass is less than the actual mass when we weigh the atom.

It will become clearer after the following example:

Calculate the mass of a He-4 atom.

we know that He-4 atom consists of 2 protons and 2 neutrons.

mass of 1 proton= 1.00727647 u. (atomic mass units)

mass of 1 neutron= 1.008664 u.

(these absolute masses were obtained by weighing the particles)

∴ total mass= 2.(1.00727647) + 2.(1.008664)

= 2.01455294 + 2.017328

= 4.03188094 u.

This is what we predicted the mass would be of a He-4 atom. Lets call this the predicted mass. 

When scientists weighed 1 atom a He-4 atom they found that its mass was 4.002602 u! Lets call this the actual mass.

We notice that there is a difference between the predicted mass and the actual mass of an He-4 atom. This difference between the predicted mass and the actual mass is known as the mass defect.

∴ predicted mass - actual mass = mass defect

 ∴ 4.03188094 u - 4.002602 u = mass defect

 ∴ 0.02927894 u = mass defect.

Hence we can say that the mass of reactants > mass of product formed.

How is this possible?

This difference in mass is because when the particles combine, a part of the reactants mass gets converted into energy. This energy is the nuclear binding energy.

∴ Nuclear binding energy and mass defect are kind of the same things. 

We said that part of the reactants mass got converted into energy. Einstein gave the famous equation E=mc^2 

where E is the energy equivalence of the mass

m= mass of substance

c = speed of light in vacuum (3 x 10^8 m/s)

Back to our problem, we had  0.02927894 u of mass that had got converted into energy. Lets calculate how much this energy actually is by using Einsteins equation.

1u = 1.6 × 10^ -27 Kg

∴ 0.02927894 u = 1.66054 × 10^-27 ×0.02927894 Kg
                        = 0.046846304 × 10^-27 Kg
                        = 4.6846304 × 10^-29 Kg

Now put this value in E=mc^2:

∴ E= (4.6846304 × 10^ -29) × (3 × 10^8)^2
    = 4.6846304 × 10^-29 × 9 × 10^16
    = 4.21616736 × 10^-13 J

This is the amount of energy released when a He-4 nucleus is formed (or) It is the amount of energy required to break a He-4 nucleus into its constituent particles (2 protons and two neutrons). This energy is called Nuclear Binding 
Energy.