What is E=mc² all about? (Equivalence or Mechanism?)

E=mc² (an equivalence or a mechanism)

This is Einstein's famous equivalence relationship, showing that mass and energy are fundamentally related.

Mass is a form of energy and a body at rest possesses an intrinsic energy called its rest-energy.

This relationship is one of the foundations of modern physics, yet it is also one of the most misunderstood equations in science.

E=mc² is not a mechanism describing how energy can be released from mass.

The Equivalence of Mass and Energy

If you have a mass, it has an equivalent amount of energy, its inherent energy. If you have an amount of energy, it has an equivalent mass.

So, you can take any value of mass and determine its equivalent energy from this relationship and this is how rest-mass is quantified, as its energy equivalence.

For example, the mass of a proton is shown to be 938.272 MeV/c², from the equivalence relationship. In particle physics, it is common to use natural units where c=1, allowing masses to be quoted simply in MeV. So, the mass of a proton is written as just 938.272 MeV.

Common Misunderstandings of E=mc² as a Mechanism

When people talk about the colossal energy released in a nuclear fission bomb, or a thermo-nuclear fusion bomb, they tell you that these explosions are caused by the transformation of all the matter in the bomb into energy.

They then go on to explain that the reason that the explosions are so enormous is because c² is such a big number and that means, from E=mc², you get a very, very big explosion when all the matter of the bomb is converted to energy! E=mc² tells you so! If all the matter of the bomb was converted to energy, these explosions would be earth shattering – quite literally.

This popular explanation is highly misleading!

E=mc² is not the cause of fusion or fission. So, where does the vast power of these weapons of mass destruction come from, if E=mc² is not responsible?

The Mechanisms for Nuclear Fusion & Fission

Fusion is all about two low mass nuclei fusing together to produce a single nucleus.

Fission is all about a large nucleus splitting into two smaller nuclei.

Being pedantic, we are concentrating on exothermic reactions, where energy is given off in each fusion or fission event. For the record, there are also endothermic reactions, where external energy is required to make the process work.

For both exothermic fusion and fission events there are consequences.

The Consequences of Fusion & Fission

In both processes, there are inputs (2 nuclei in fusion and 1 in fission) and outputs (1 nucleus for fusion and 2 for fission).

There are three consequences for both exothermic fusion in the stars and exothermic fission in nuclear reactors as well as in fission or fusion weapons capable of destroying humanity.

Consequence 1: The total rest-mass of the input(s) is always greater than the total rest-mass of all the output(s) in a single exothermic event.

Consequence 2: The total binding energy of the output(s) is always greater than the total binding energy of the input(s) in a single exothermic event. This only refers to bound nuclei.

Consequence 3: Due to conservation of energy, there is always an output of energy from both the exothermic fusion and fission processes.

The product nucleus of fusion and the product nuclei of fission are more tightly bound than the original inputs.

This means that the final system has a lower total mass-energy. The difference appears as released kinetic energy and radiation, in accordance with conservation of energy.

Alternatively, if the binding energy of the output nucleus (or nuclei) is greater than the binding energy of the input nuclei (or nucleus), then an amount of energy equal to the difference in binding energy, must be released in accordance with the conservation of energy.

Let's take two examples to see what's involved in both a fusion and a fission event.

Example 1 - Fusion

The fusion of two helium-3 nuclei in the sun is defined as follows;

He-3 + He-3 → He-4 + p + p

Two He-3 nuclei are the reactants (inputs) and He-4 and two protons are the products (outputs).

As well as producing a helium-4 nucleus, two protons are turfed back into the sun's core after the reaction. Don't forget these two protons as they play an important part from the perspective of E=mc².

Example 2 - Fission

The induced fission of uranium-235 results in the production of highly unstable U-236, which undergoes fission into two daughters and a release of neutrons;

U-235 + n → U-236 → Ba-141 + Kr-92 + n + n + n

A thermal neutron penetrating the nucleus of U-235, produces a highly unstable parent nucleus of U-236, which undergoes spontaneous fission, producing the two daughters and three neutrons are turfed out. (Due to the chaotic nature of fission, the above is just one example of the daughters that can be produced).

U-236 is the parent (input) and Ba-141, Kr-92 and three neutrons are the products (outputs). These three neutrons also play an important part from the perspective of E=mc².

Difference in Rest-mass in examples

The difference in rest-mass (ΔM), before and after the example fusion and fission events can be calculated as follows, where RM is Rest-Mass;

ΔM(1) = 2RM(He-3) – [RM(He-4) + 2RM(proton)] For the fusion example

and

ΔM(2) = RM(U-236) – [RM(Ba-141) + RM(Kr-92) + 3RM(neutron)] For fission,

E=Δmc2 can now be used to calculate the energy equivalence of the two values of ΔM.

IMPORTANT: The rest masses of the protons and neutrons must be included in the relevant calculation to ensure correct value of ΔM(1) and  ΔM(2) are produced.

In the two examples, the decrease in rest mass ΔM and the released energy are two equivalent descriptions of the same physical process.

Difference in Binding Energy in examples

The difference in binding energies implies that the product nuclei of both fusion and fission are more tightly bound together than the input nucleus or nuclei. Using 'BE' to represent binding energy, this can be represented as;

ΔBE(1) = BE(He-4) - 2BE(He-3)   for example 1

Binding energy is only relevant to the nuclei involved in fusion.

and

ΔBE(2) = [BE(Ba-141) + BE(Kr-92)] - BE(U-236) for example 2.

Binding energy is also only relevant to the parent and daughter nuclei in fission.

The Relationship between ΔM and ΔBE

For both examples, the increase in binding energy corresponds exactly to the decrease in total rest-mass.

Using Einstein’s equivalence relationship,

ΔBE=Δmc²

The amount of released energy can be calculated either from the increase in binding energy or from the decrease in total rest-mass.

For He-4 production in the Sun, this amounts to a release of approximately 12.859 MeV.

For induced U-236 fission, this amounts to approximately 166.3 MeV.

Both fusion of light nuclei and fission of heavy nuclei move nuclear matter towards the region of maximum binding energy per nucleon near iron and nickel.

Binding energy per nucleon

The binding energy per nucleon is not a constant for all nuclei. The Atomic Mass Data Center has produced a table containing the total binding energy and the binding energy per nucleon of every element in the periodic table, including isotopes.

From this data a graph can be produced showing that there is a clear trend in binding energy per nucleon from light to heavy nuclei as shown in the following graph.

This graph clearly demonstrates how fusion and fission move nuclear matter towards the region of maximum energy per nucleon near iron and nickel.

Final Takeaways

E=mc² does not explain fusion or fission, it is simply the relationship indicating that mass is a form of energy and a body at rest possesses an intrinsic rest-energy.

The product nuclei in fusion or fission are more tightly bound than the input nuclei or nucleus, leading to an increase in total binding energy after the event.

Because of conservation of energy, the increase in total binding energy corresponds to an equivalent release of kinetic energy and radiation.

Conversely, because of the conservation of energy, the decrease in total rest-mass also corresponds to the released energy. Any calculation of rest-mass must include all products, nuclei and released protons or neutrons.

Using E=mc², ΔBE = ΔMc²

The energy of Atomic bombs comes from an increase in binding energy of billions upon billions of fission events and only a tiny portion of the total rest-mass of the bomb material is converted to energy.

Both fusion of light nuclei and fission of heavy nuclei move nuclear matter towards the region of maximum binding energy per nucleon near iron and nickel.

E=mc² is not about mechanisms in physics, it's simply about how mass relates to energy and how energy relates to mass, as Einstein intended.

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In Part 2 of A Journey into Modern Physics, the way in which binding energy can be determined from first principles, using what's called the mass defect of a nucleus, is shown in detail for a better understanding of binding energy.

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