Question

Find the binding energy of a $\alpha - $particle from the following data.
Mass of the helium nucleus$ = 4.001265amu$
Mass of proton$ = 1.007277amu$
Mass of neutron$ = 1.00866amu$
 (Take$1amu = 931.4813MeV$)

Answer 1

Hint – Begin the solution by first describing the relation between a $\alpha - $particle and $_2^4H$ (Helium) nucleus and use it to obtain mass of $\alpha - $particle. Then place the value into the equation of binding energy.

Complete step by step answer:
First let’s understand, what a $\alpha - $particle is?

A $\alpha - particle$(pronounced as alpha-particle) is a fast moving particle having two protons and two neutrons. You may have heard about it alongside $\beta - $particles (pronounced as beta-particle) and $\gamma - $particles (pronounced as gamma particles). The $\alpha - $particles are formed during $\alpha - $decay, and do not have much penetrating power, they have so less penetrating power that they can be stopped by pieces of paper.

Alpha particles are very similar to the $_2^4H$(Helium) nucleus, as both of them have two protons and two neutrons.

Considering all the values given in the question –

Mass of the helium nucleus$ = 4.001265amu$
Mass of proton$ = 1.007277amu$
Mass of neutron$ = 1.00866amu$

For finding the binding energy of $\alpha - $particles, we have to find its mass. As we know that nuclei of Helium atoms and a $\alpha - $particles have the same composition, they must also have the same weight.

Thus,

Mass of $\alpha - $particleMass of helium nucleus$ = 4.001265amu$

Binding Energy – It is the amount of energy required to separate all the components of a nucleus from each other.
We know that Binding Energy,
Now,

$\Delta m = $Mass of helium $ - 2$(mass of neutron + proton)
$ = 4.001265 - 2(1.007277 + 1.00866)$
$ = 4.001265 - 4.031874$
$ = - 0.03061amu$

Substituting this value in the equation for binding energy, we get

\[E = \Delta m{c^2}\]
$ = 0.03061 \times 931.4813MeV$
$ = 28.5126MeV$

Note - We consider $1amu = 931.4813MeV$ in most theoretical problems, but, here it is specifically mentioned in the question that we have to Take$1amu = 931.4813MeV$, so do remember it while solving such types of questions. Beware to never round this figure while attempting such questions.