Mass Energy Equivalence

Before posting more about nuclear energy it would be better if our readers know the physics behind  such enormous energy release from  subatomic reactions.

Annus Mirabilis papers (Latin for Miracle Year) are the scientific papers published by Albert Einstein in the year 1905. These included four articles which served as a milestone for formulation of modern physics. His theories redefined the classical Galilean and Newtonian Physics. Thus, he gained the  sobriquet of "Father of Modern Physics".

The fourth article in his papers explained about the Mass-energy equivalence and provided the scientific world's most famous equation of all:-

E=mc^2.

This was an outcome of his Theory of Special Relativity. So, this going to the topic of  interest in the current post.

The first appearance of E=mc^2 ,here relativistic mass is considered:-

Mass-Energy Equivalence:-

Until the advent of The Special Theory of Relativity it was presumed that mass and energy were conserved separately in any kind of reaction. But it was Einstein who showed that we have to consider mass as yet another form of energy. As per the First law of Thermodynamics-Energy of the whole universe is a conserved quantity. So, if mass is a form of energy we can convert mass-energy into heat energy through nuclear reactions(fission and fusion)for power generation purposes.

E=mc^2; where m - Mass of substance

                            c - Speed of light in vacuum

The above theory has been experimentally verified in nuclear reactors and study of electrons, nucleons and other subatomic particles.This concept is important in understanding the the nuclear phenomenon in fission and fusion.

 Whenever energy is added to a system, the system gains mass, as shown when the equation is rearranged:-

1.  A spring's mass increases whenever it is put into compression or tension. Its added mass arises from the added potential energy stored within it, which is bound in the stretched chemical (electron) bonds linking the atoms within the spring.
2. Raising the temperature of an object (increasing its heat energy) increases its mass. For example, consider the world's primary mass standard for the kilogram, made of platinum/iridium. If its temperature is allowed to change by 1 °C, its mass changes by 1.5 picograms (1 pg = 1×10^−12 g).
3. A spinning ball weighs more than a ball that is not spinning. Its increase of mass is exactly the equivalent of the mass of energy of rotation, which is itself the sum of the kinetic energies of all the moving parts of the ball. For example, the Earth itself is more massive due to its rotation, than it would be with no rotation. This rotational energy (2.14×10^29 J) represents 2.38 billion metric tons of added mass.

Nuclear Binding Energy:--

We know the nucleus of any atom constitutes of protons and neutrons, so it is clear that the mass of nucleus must be equal to the sum  of  the individual masses of neutrons and protons in an unbound condition. However, experiments suggests that the mass M of nucleus is always less than this.

For instance let us consider oxygen nuclei:-

Mass of 8 neutrons = 8 x 1.00866 u

Mass of 8 protons=8 x 1.00727 u

so, as per theory Mass of atomic oxygen = 8 x 2.01593 = 16.12744 u

But through mass spectroscopy atomic  mass of oxygen is found to be 15.99493 u subtracting the mass of 8 electrons from this we get 15.99053 u.

Thus, we find that the mass of oxygen nucleus is less than the total mass of constituents by 0.13691 u,this difference in mass is termed as mass defect, given by:-

Δ M=[Zmp +(A-Z)mn]-M

This is where Einstein's mass-equivalence comes into play.As the mass of oxygen nucleus is less than the mass of constituents in an unbound state, the energy associated with oxygen nucleus is less than the energy of neutrons and protons in unbound condition.

As per modern particle physics, interaction of quarks with the gluon field gives mass to proton and neutron.About 99% of proton mass is, in fact potential energy of interaction of quarks with gluons.A quark is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form particles called hadrons, the most stable out of them are protons and neutrons, the constituents of an atomic nuclei. A gluon is an elementary particle that acts as the exchange particle  for the strong force between quarks. It is analogous to photons in the electromagnetic force between two interacting charged particles.So, it can inferred that fluctuation in gluon field gives the mass defect energy in the nuclear reactions of fission and fusion.

The mass defect  ( ΔM ) accounts for energy release according to Einsteins's mass-energy equivalence;

mass of released energy= E/c^2=m(initial) - m(final) 

Since, value of velocity of light is large ,a small defect in mass is associated with tremendous release of energy( in form of K.E  of fission  fragments and radiations; both beta and gamma)which carries the missing mass after the the reaction.

If one wants to split this oxygen atom, they have to supply this extra energy E= Δ Mc^2  .This is called as  the binding energy of an atom.If we bring together neutrons and protons to form a nucleus this energy will be release, as in nuclear fusion process.Nuclear binding energy is a convenient way of  determining how well a nucleus is bonded together.But Binding energy per nucleon is more useful in measuring this, and provides information about stability of nucleus.

Given by,

E_bn=E_b/A

It can be defined as the mean energy per nucleon required to split a nucleus into it's constituents.

From the above plot of average binding energy per nucleon vs number of nucleons we can draw the following observations:-

1. The Binding energy per nucleon is nearly constant from 30<A<170.
2. The value is both lower for A<30 and A>170.
3. And some anomaly from He to Oxygen, where the plot is not smooth .

Inferences:-

• The nuclear forces holding together nucleons is attractive and stronger than Coulomb force and gravitational forces, and is sufficiently strong to produce a binding energy in the range of MeV.
• The constancy in the region 30<A<170 is due the short range of Nuclear forces(~fm=10^-15) and is powerfully attractive in this range.If a nucleon can have a maximum of p neighbours in within the range of nuclear force,it's binding energy will be proportional to p.Consider a particular nucleon within a  nucleus the nuclear force will  only affect the nucleons within the range and if a nucleon is at a distance more than the range it will have no effect on the Binding energy of nucleon under consideration.Let the binding energy of nucleons be pK, where K is a constant having dimensions of energy.If we increase the nucleons (A) they will not affect the binding energy of nucleons inside. Since, most of the nucleons reside inside the nucleus and not on the surface, thus the change in binding energy will be small. The property of nuclear forces, that given nucleons only influence the nucleons close to or in the range of nuclear force is also called as Saturation of nuclear force.
• Suppose we consider a heavy nucleus (A=240),as we can see from the figure it has low binding energy per nucleon as compared to a nucleus with A= 120. So, if the first nucleus breaks into two nuclei of A=120, it will get more tightly bound.From this we know that the energy associated with A=120 is less than that of A=240, hence energy will be released.It has important applications in power generation sector through Nuclear fission.
• Now, lets consider two light nuclei (A<10) which have low Binding energy per nucleon fuse together to form a heavier nucleus.The binding energy per nucleon of fused nucleus is more than the lighter nuclei. Hence, more tightly bound than the initial system and energy will be released.This the concept of Nuclear fusion energy ,the process that fuels the Stars in our Universe.

Hence, the energy release in nuclear reaction is due to the mass defect in transition to more tightly bound nucleus.And this release of energy in nuclear reactions like fission and fusion can be engineered into reactors for power generation purposes.