Binding energy is an excellent concept of physics which can be defined as “a small amount of energy required for disassembling a system of particles into individual atoms”.

**In general, nucleons in the nucleus, i.e., protons and neutrons, adhere by a strong force. The binding energy helps them to be apart from each other. In this post, you will know how to calculate binding energy for different kinds of particles.**

The binding energy of any particle follows Einstein’s famous mass-energy relation, which accounts for converting mass into energy. In the below section, we have provided information regarding how to calculate binding energy of an atom, electron, and various other types.

The steps provided below have to be followed to calculate binding energy.

**At first, find the mass defect of the given system using Einstein’s mass-energy relation.****Convert the mass defect into energy representing the binding energy required to break the particle system.****Express the obtained binding energy in terms of energy per mole of atom or energy per nucleon.**

**How to calculate binding energy of an atom?**

Binding energy disperses all the particles in the system and applies to the subatomic system of atomic nuclei.

**The binding energy of an atom separates an atom into free electrons and its constituent nucleus. The atomic binding energy is carried in the presence of a photon as a mediator by the electromagnetic interaction between the electron and the nucleus.**

Let us solve an example problem on how to calculate binding energy of an atom given below.

**Find the binding energy of copper-63**

**Solution:**

The mass of the given copper-63 atom is found to be 1.66054×10^{-27}kg, and the speed of the light is 2.99×10^{8}m/s.

Thus the binding energy of copper-atom is calculated by substituting the above values in mass-energy relation as;

BE_{Cu}=(1.66054×10^{-27})(2.99×10^{8})^{2}

BE_{Cu}=1.4845×10^{-10}J.

**How to calculate binding energy per nucleon?**

Since we know that some energy is required to hold the nucleons in the nucleus, that energy is nothing but binding energy.

**The nucleons are nothing but a large number of protons and neutrons inside the nucleus. Inside the nucleus, strong repulsion will be offered between the positive charges and proton, so some energy is required to hold the proton and neutron together, which is known as the binding energy. The binding energy possessed by individual protons and neutron gives binding energy per nucleon.**

The above question is how to calculate binding energy per nucleon. To answer, steps have to be followed.

**The initial step is to check the atomic number and mass number of the given nucleus, which must be balanced in the reaction.****Next, calculate the total mass before and after the reaction.****The total mass before and after the reaction creates a difference in the mass, so this difference in the mass has to be calculated.****Once all the procedure is done, find the binding energy using mass-energy relevance.****At last, divide the obtained binding energy by the mass number of the given nucleus to get binding energy per nucleon.**

**For example, the binding energy of the ^{20}_{10}N can be calculated given that the mass defect is 0.1725amu.**

Since we know the mass defect of the given Ne nucleus, the atomic number of and the mass number of Ne are Z=10 and A=20, respectively.

BE=Δmc^{2} gives the binding energy

BE=0.1725amu×c^{2}

But 1amu=931.5MeV/c^{2}

Thus the binding energy per nucleon can be written as

BE_{N}=8.034MeV

**How to calculate binding energy of electron?**

Electrons are naturally negatively charged particles orbiting around the nucleus. Since there is a huge number of positive charges inside the nucleus, the electron orbiting around the nucleus is attracted towards the nucleus.

**Some energy has to be applied to the electron to eliminate this attraction between the nucleus and the electron. Since the surrounding nucleus is made of electrons, there will be electron-electron repulsion, making an effort unsolvable. But the spin-orbit interaction and splitting of an electron into different levels help calculate the binding energy.**

The formula of binding energy of the ground level electron is; BE=E_{0}Z^{2}

Where E_{0} is the ground-state energy and Z is the atomic number of the given nucleus.

The electron’s binding energy is also called **“Ionization potential.”**

**The electronic configuration of the given nucleus has to be known and determined from which level the electron has ejected.****Find the wavelength corresponding to the ejected state.****The difference between the electron’s energy level with the wavelength multiplied by 1.2395×10**^{-4}gives the binding energy of that electron.**Express obtained binding energy in electron volts; where 1eV=1.602×10**^{-19}J.

**How to calculate binding energy of alpha particle?**

An alpha particle from the helium nucleus is emitted by a radioactive element through the decay process.

**Alpha particle consists of two protons and two neutrons, so the energy required to separate them from the nucleus is the binding energy. The combined total mass of the nuclei of alpha particles has to be found initially.**

The formula gives the binding energy of the alpha particle following two steps.

**The first step is the difference in the mass given by the formula; Δm=mass of alpha particle-combined mass of nuclei****Then find the binding energy using Einstein’s mass-energy relation. BE=Δmc**^{2}.

**How to calculate binding energy of a metal?**

The binding energy of metal can be defined as the amount of energy required to eject an electron from the surface of the metal.

**The threshold frequency of the metal empowers the binding energy of the metal.**

Let us solve an example problem to understand how to calculate the binding energy whose threshold frequency is 4.83×10^{14}s^{-1}.

The energy and threshold frequency is related by equation E=hν, where h is the plank’s constant and ν is the threshold frequency.

Substituting the value of ν and plank’s constant, we get

E=(6.626×10^{-34})(4.83×10^{14})

E=3.20×10^{-19}

The binding energy per metal atom can be obtained by multiplying the energy by the Avogadro number. Expressed as

BE=(3.20×10^{-19})(6.023×10^{23})

BE=192.73×10^{3}J.

**How to calculate binding energy from mass defect?**

Mass defect specifies the difference between the actual mass and predicted mass of the given nucleus. The predicted mass is nothing but the masses of neutron and proton together.

**The mass of the nucleons is always greater than the actual mass of the nucleus; thus, some masses are lost by releasing a certain amount of energy. The energy thus released in the form of reduction creates a small missing mass termed a mass defect.**

The formula mentioned below gives the mass defect

Δm=(Zm_{p}+Nm_{n})-M_{A}

Where; Δm is the mass defect, Z is the proton number, N is the neutron number, m_{p} is the proton’s mass, m_{n} is the mass of the neutron, and M_{A} is the mass of the nucleus.

And finally, the binding energy is BE=Δmc^{2}.

**Solved example problems**

**Calculate the binding energy of an electron in the ground state whose spectrum is found to be in the third Balmer line at 108.5nm for a hydrogen atom.**

**Solution:**

**The binding energy of an electron in the ground level is given by**

**BE _{g}=E_{0}Z^{2}**

The wave number at the Third Balmer line is

[latex]v=\frac{E_0Z^2}{hc}\left(\frac{1}{2^2}-\frac{1}{5^2}\right)[/latex]

[latex]v=\frac{E_0Z^2}{hc}\left(\frac{21}{100}\right)[/latex]

The reciprocal of wave number gives the wavelength, thus

[latex]\lambda=\frac{1}{v}[/latex]

[latex]\lambda=\frac{hc}{E_0Z^2}\frac{100}{21}[/latex]

Substituting the values

[latex]\lambda=\frac{hc}{BE_g}\frac{100}{21}[/latex]

[latex]BE_g=\frac{hc}{\lambda}\frac{100}{21}[/latex]

Substituting h=6.626×10^{-34} and c=3×10^{8}; we get hc=1.9878×10^{-25}.

[latex]BE_g=\frac{1.9878\times10^-22}{108.5\times10^-9}\frac{100}{21}[/latex]

[latex]BE_g=\frac{8.72\times10^-18}{1.602\times10^-19}[/latex]

BE_{g}=54.45eV.

**How to calculate binding energy of an alpha particle given that the particle’s mass is 4.001265amu, the mass of the neutron is 1.00866amu, and the mass of the proton is 1.007277amu?**

**Solution:**

**The mass defect can be calculated as**

**Δm=2(m _{p}+m_{n})-M_{α}**

**Δm=2(1.007277+1.00866)- 4.001265**

**Δm=4.0318-4.001265**

**Δm=0.0306.**

Binding energy E=Δmc^{2}

[latex]E={(0.00306\times931.5\frac{MeV}{c^2})c^2}[/latex]

E=28.44MeV

**Calculate the binding energy per nucleon of [latex]_{7}^{14}\textrm{N}[/latex] nucleus and also determine the mass defect. Given the mass of the nitrogen, the nucleus is 14.00307u.**

**Solution:**

We know that mass of proton m_{p}=1.00783amu

Mass of neutron m_{n}=1.00867amu

Mass defect is given by

Δm=(Zm_{p}+Nm_{n})-M_{A}

Δm=(7×1.00783+7×1.00867)-14.00307

Δm=0.1124amu

Binding energy BE=Δm×931.5

BE=0.1124×931.5

BE=104.72MeV

Binding energy per nucleon

[latex]BE_N=\frac{BE}{A}[/latex]

[latex]BE_N=\frac{104.72}{14}[/latex]

BE_{N}=7.4806MeV.