## Einstein’s Mass Energy Relation

Subject: Physics

#### Overview

The law of energy conservation and mass conservation are no more independent laws, but a single law called a law of mass-energy conservation. This note provides us an information on Einstein’s mass energy relation.

#### Einstein’s Mass-Energy Relation

Einstein proved that energy and mass are related to each other by his theory of relativity. According to this theory, the energy equivalent, ΔE of mass Δm is given by

$$\Delta E =\Delta m c^2$$

where C= speed of light = 3 × 108 ms-1 . This equation represents Einstein’s mass-energy relation.

If we give ΔE energy to some matter, then by the above relation, its mass will increase by Dm given by

$$\Delta m = \Delta E/c^2$$

Since c is very high, Δm will be very small. In view of the mass-energy relation, the law of energy conservation and mass conservation are no more independent laws but a single law called a law of mass-energy conservation.

#### Atomic Mass Unit in Terms of Energy

According to Einstein’s mass energy relation E = mc2, the energy equivalent to 1 amu is given by,

\begin{align*} E &= 1.66 \times 10^{-27} \times (3\times 10^8)^2 j = 1.49 \times 10^{-10} \: J \\ \text {But} \: 1.6 \times 10^{-19} \:J = 1 \: eV, \text {so} \\ E &= \frac {1.49 \times 10^{10}}{1.6 \times 10^{-19}} \: eV \\ &= 0.931 \times 10^9 \: ev \\ &= 931\: Mev \\ \therefore \text {Energy equivalent of} \: 1\: amu &= 931\: M\:ev \\ \end{align*} It can be proved that the energy equivalent of the mass of an electron, proton and neutron are respectively given by\begin{align*} \\ m_e &= 0.511\: MeV \\ m_p &= 938.279 \: MeV \\ m_n &= 939.573\: Mev \\ \end{align*}
Types of Nuclei

There are different types of nuclei based on the number of protons or total number of nucleons present in them.

Isotopes

Isotopes are the nuclei having the same atomic number Z, but different mass number A. isotopes of an element have identical chemical behavior and differ physically only in mass.

Isotopes of some elements are the following:

1. Hydrogen: 1 H1, 1 H2, 1 H3
2. Helium: 2 He3, 2 He4
3. Carbon: 6 C14, 6 C12
4. Uranium: 92 U238, 92 U235
5. Sodium: 11 Na22, 11 Na23, 11 Na24

Isobars

Isobars are the nuclei having the same mass number but a different atomic number. These are the nuclei of different elements having different physical and chemical properties. Some of the examples of isobars are:

1. 1 H3 and 2 He3
2. 7 N14 and 6 C14
3. 8 O16 and 7 N16
4. 20 Ca40 and 18 Au40

Isotones

Isotones are the nuclei having the same neutron number. Some examples of isotones are:

1. 1 H3 and 2 He4 (N = 2)
2. 7 N17 , 8 O18 and 9F19(N = 10)

#### Binding Energy and Mass Defect

The binding energy of a nucleus is the minimum energy required to disrupt the nucleus into its constituent particles. The rest mass of nucleus of a stable atom is always less than the sum of its constituent nucleons i.e. protons and neutrons in free State. The difference between the sum of the masses of nucleons constituting a nucleus and the rest mass of the nucleus is called mass defect, denoted by Dm. The energy equivalent of this mass defect is called the binding energy of the nucleus.

\begin{align*} B.E. &= \Delta mc^2 \\ \text {If M is the rest mass of a nucleus} \: _ZX^A, \text {and} m_p, m_n \\ \\ \end{align*} and M are expressed in kilogram, then its binding energy is given by \begin{align*}\\ B.E. &= [Zm_p + (A – Z)m_n – M]c^2 \:J \\ \end{align*} where Z is number of protons, A is mass number and\begin{align*}\: (A – Z) \:\text {is number of neutrons.} \\ \end{align*} If these masses are measured in atomic mass unit, then,\begin{align*}\\ B.E. &= [Zm_p + (A – Z)m_n – M]\times 931\: MeV. \\ \end{align*}

##### Binding Energy per Nucleon or Average Binding Energy

It is the total binding energy of a nucleus divided by its mass number A.

$$\therefore \vec {B.E.} = \frac {\text {Total Binding Energy}}{\text {Mass Number}} = \frac {B.E.}{A}$$

It determines the measure of the stability of the nucleus against disintegration.

The graph rises steeply at first and then gradually until it reaches a maximum value of 8.79 M ev corresponding to 26Fe56. Then it drops slowly to 7.6 M ev at the highest mass number of 92U238. After mass number 120, the binding energy per nucleon starts decreasing and drops to 7.6 M eV for uranium. This low value of binding energy per nucleus is unable to have control over the coulomb’s repulsion between the large numbers of protons. Such nuclei are unstable and are found to disintegrate emitting α- or β- particles. All such nuclei which undergo α and β decay are called radioactive nuclei.

Nuclear Forces

The atomic nuclei are composed of protons and neutrons. As proton carry a positive charge, the electrostatic force of repulsion between them should cause a disruption of the nucleus. The gravitational force of attraction is too weak to account for the observed binding energy of nuclei. Thus, there must be certain forces that bind the nucleons into the nucleus. These strong attractive forces acting between the nucleons that bind them into the nucleus are called nuclear forces.

Reference

Manu Kumar Khatry, Manoj Kumar Thapa, Bhesha Raj Adhikari, Arjun Kumar Gautam, Parashu Ram Poudel.Principle of Physics. Kathmandu: Ayam publication PVT LTD, 2010.

S.K. Gautam, J.M. Pradhan. A text Book of Physics. Kathmandu: Surya Publication, 2003.

##### Things to remember

The law of energy conservation and mass conservation are no more independent laws, but a single law called a law of mass-energy conservation.

Isotopes are the nuclei having the same atomic number Z, but different mass number A.

Isobars are the nuclei having the same mass number but a different atomic number.

Isotones are the nuclei having the same neutron number.

The energy equivalent of this mass defect is called the binding energy of the nucleus.

The strong attractive forces acting between the nucleons that bind them into the nucleus are called nuclear forces.

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