Mass Energy Equivalence: Matter-Antimatter Annihilation, Equations & Derivations

Mass Energy Equivalence principle states that mass of a system and its energy are the same property in a physical system. Anything with a mass will have an equal amount of energy and vice versa. Einstein gave the mass energy equivalence relation as: - E=mc². This relation helps in understanding nuclear masses and the interaction of nuclei with each other.

Key Terms: Mass Energy, Alpha Decay, Equivalence, Gamma Decay, Annihilation, Nuclear energy, Chemical Energy, Einstein’s equation

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What is Mass Energy Equivalence?

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Mass energy equivalence is defined as the relation between energy and mass in which the two values differ only by the units of measurement and by a constant. This implies that even though the changes occur, the total mass of a system, the momentum, and the total energy remains constant.

The equivalence of energy and mass is an outcome of Einstein's theory of special relativity. It simply means that the mass can be transformed into energy and vice-versa. Mass and energy are identical according to Mass Energy equivalence. The discovery of mass energy equivalence is proved to be crucial to the development of fission reaction and atomic fusion.

Mass energy equivalence is expressed by the following formula: 

E = mc²

Mass Energy Equivalence 

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Matter- Antimatter Annihilation

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Matter and antimatter are collections of particles that create particle pairs that have the same mass but opposite electric charge.

Examples: Electron (negative charge) and a Positron (positive charge), or a Proton (positive) and an Antiproton (negative).

When matter and antimatter collide, particles destroy each other. This causes a massive energy release. Depending upon the colliding particles, new and different particles may also be produced (like neutrinos and various flavours of quark).

Every new particle will have a lower mass than those in the original collision. This happens due to law of conservation of energy and Einstein’s equation E=mc². Some of the energy goes into heat and light while some into forming the new particles. 

Products from Matter-antimatter Collision 

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What is Annihilation?

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Annihilation is a process that is found mostly when the subatomic particles collide with their antiparticle. All of this energy is released in form of electromagnetic waves. This process is very uncommon because there is not much antimatter around us.

Let us consider electrons, we do not have anti-electrons for it. After looking at this process, we will find that a position, which is electron and antielectron, annihilate and come together and produce energy in form of a gamma photon. Gamma means photons which simply means energy.

The mass of a positron or an electron is 9.11 × 10^-31 kg.

Annihilation

The energy released will be E = mc² from positron 2mc² and from electron + mc²

Thus, it turns to be 1.64 × 10^-13 joules. It is not very much, but it was a positron and electron. We get 10¹⁰ joules if instead of one pair of electron-positron, a mol of them are annihilated. It is a lot of energy. It turns out that every release of single energy can be understood in terms of this mass energy equivalence.

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Chemical Energy

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Chemical energy is stored in bonds of chemical compounds, like atoms and molecules. Water is a molecule that is formed by taking an oxygen atom and two hydrogen atoms and thus it makes two bonds. These bonds cost 918 kilojoules per mol of energy. It implies that there are 1.5× 10^-18 joules of energy bond per molecule.

Here, we will find that mass of the water molecule is a bit less than two times of mass of hydrogen, plus the mass of oxygen. The difference in mass is known as the mass defect and is related to the mass of binding energy. The change in the mass is the energy and is given by:

Where,

• Δm = change in mass
• c = velocity of light

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Nuclear Energy

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Nuclear Energy is the energy present in the nucleus or core of an atom. For example, Let us consider the alpha decay of Uranium 238:

Uranium 238 decays into helium 4 and thorium 234. If we add the mass of this thorium nucleus and helium nucleus, we will get a mass smaller than the mass of the uranium nucleus.

Relative mass, mr = mm 

The relative mass is about 2 × 10^-5. It remains small but its five orders of magnitude are huge than released by the chemical energy. Nuclear reactions release a lot of energy than chemical reactions. The fraction of mass that is released in form of energy is 1 lakh times greater.

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Neutron Star

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Neutron star is the densest object that can be observed by astronomers directly. They crush half a million times Earth's mass into a sphere about 12 miles across. Black holes and neutron stars release the largest amount of mass energy other than matter-antimatter annihilation. We can get relative releases of the energy with a neutron star of order 7% (7× 10^-2). For some rotating black holes, we can get the energy almost up to half, 42%.

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Einstein's Mass Energy Relation

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Einstein's Mass Energy Relation states that energy and mass (matter) are interchangeable. They are different forms of the same thing. Mass energy equivalence implies that even in a stationary position, every object possesses certain energy. Thus, mass energy equivalence states that all massive objects have some amount of corresponding intrinsic energy.

A stationary body doesn't have kinetic energy. It is motionless and has no momentum. The mass and energy are equal and only differ by a constant. It possesses only probable chemical, thermal and potential energy. The sum of all these energies is small than the square of the speed of light and the product of the mass of an object.

By using the theory of relativity, Einstein describes the relationship between mass and energy. Thus, this equation is a Mass energy equation of Einstein and is expressed as:

Where,

• E = the equivalent kinetic energy of an object,
• m = mass of an object, and
• c = velocity or speed of light in a vacuum c ≅ 3×10⁸ m/s

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Derivation of Einstein's Equation

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The two derivations of Einstein's Equation are explained below:

Derivation 1

To derive Einstein's mass energy equation, the simplest method is as follows,

Consider an object that moves at a speed approximately equal to that of the light. A uniform force acts on it. The momentum and energy are induced in it, because of the applied force.

As the force is constant, the increase in the momentum of an object is equal to the product of mass and velocity of the body.

We know that:

E= F x c -------(a)

Also,

The momentum gained = Force x Duration

As Momentum = mass x velocity,

The momentum gained = m x c

Therefore, Force= m x c ---------(b)

By combining the equation (a) and (b) we get,

E= mc²

Derivation 2

whenever an object is in speed, it seems to get heavier than before. The following equation gives the increase in mass because of speed.

Where,

• m = mass of an object at traveling speed
• m₀ = mass of an object at a stationary position
• v = speed of an object
• c = speed of light

We know that an object possesses kinetic energy in motion and it is given by,

E= ½ (mv²)

The total energy that is possessed by an object is approximately equivalent to kinetic energy and due to speed, there is an increase in mass.

E≅ (mc²) + ½ (mc²)

E - (mc²) = ½ (mv²), for the small v/c

E= Relativistic kinetic energy + mc²

The relativistic kinetic energy depends on the speed of the object and kinetic energy. The equation can be simplified by setting the speed of the object as 0. Therefore, the equation becomes,

E= 0 + mc²

E= mc²

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Applications of Einstein’s Equation

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The application of Einstein's equation is given in the below points:

• Einstein's theory was used to know fusion reactions and nuclear fission. It was revealed that a big amount of energy is liberated during fusion processes and nuclear fission using the formula. In creating nuclear weapons and nuclear power this phenomenon is used.
• The equation is used to find out binding energy in an atomic nucleus. Binding energy is calculated by measuring and subtracting the masses of various nuclei from the sum of masses of neutrons and protons. The measurement of binding energy is used to calculate the energy that is released during the nuclear reactions.
• During the chemical reactions, Einstein’s equation is used to find out the change in mass. Formation and breakage of new bonds take place whenever there is a chemical reaction. Change in mass takes place during the exchange of molecules. Einstein’s equation for chemical energy can be written as E= Δm x c² where Δm is the change in mass.
• The radioactivity of various elements is based on this theory. Radioactivity produces gamma rays, x–rays. The same principle is used in many radiotherapies equipment.
• To understand the effect of gravity on the moon, planet, all-stars, and also to measure the age of fossil fuels. 
• The equation is used to understand the universe, the age of planets, and their constituents.

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Things to Remember

• The relation between energy and mass in which the two values differ only by the units of measurement and by a constant is known as Mass energy equivalence.
• The equivalence of energy and mass is an outcome of Einstein's theory of special relativity.
• Mass and energy are identical according to Mass energy equivalence.
• Matter- Antimatter Annihilation is the only way to transform mass into energy.
• Mass and Energy possess only probable chemical, thermal and potential energy.
• Mass energy equivalence is expressed by the formula: E=mc².
• Einstein's theory was used to know fusion reactions and nuclear fission.
• The equation is used to understand the universe, the age of planets, and their constituents.