EN | FR # E=MC²

## The Einstein formula, a relation between mass and energy Mass-energy equivalence: E=mc2
This pair of scales would show that an alpha particle is lighter than the two neutrons and two protons that make it up. This ‘mass defect’ can be translated into an energy difference by multiplying by c2. The resultant energy is known as the binding energy: in the case of two protons and two neutrons, 28 million electronvolts. This substantial value of binding energy represents less than 1% of the internal energy (or mass) of the alpha particle. Jean_Marc Faday (CNRS/Images-Média)

The energy released by a chemical reaction is generated by variations in the so-called ‘internal’ energies. A good example is the combustion of carbon: C + O² => CO²

The calories released by logs burning in a chimney are the result of an internal energy loss between the initial state of carbon and oxygen and the final state of carbon dioxide. Where do these ‘internal’ energies come from?

The E in Einstein’s famous equation E=MC² represents a value of internal energy:

This energy (sometimes referred to as the mass energy) is equivalent to the mass m of an object multiplied by the square of the speed of light c. This speed, equivalent to some 300 million metres per second, is one of the largest and most impressive of the natural constants, and the multiplying factor of c to the square is therefore enormous. It only takes minuscule changes in the value of m to generate the energies we see in our chimneys, the engines of our cars, or even in our factories. Bound nucleons lose weight
The English physicist Francis Aston was the first to measure the mass of nuclei. Dividing this mass by the number of nucleons present allows for a calculation of the average mass of a bound nucleon. Aston’s curve shows that belonging to a nucleus makes a nucleon lose just under 1% of its mass. This loss, when multiplied by the speed of light squared, comes out an energy loss which can surpass 8 million electronvolts. This is the energy that on average needs to be spent to rip a nucleon away from the nucleus: the binding energy. The proportionality between mass loss and binding energy is general: in the case of a nucleus, the mass lost is low but detectable, whereas in chemical phenomena involving energies of a few electronvolts the masses defects are too small to measure. IN2P3
In the example above, if we were to weigh the total mass of carbon and oxygen present before the reaction and compare it to the total mass of carbon dioxide produced, we would see that a tiny amount of mass is lost in the reaction. This mass difference is too small to be noticed by even the most sensitive weighing apparatus, but when multiplied by the speed of light squared it is capable of generating that friendly heat we gather round the fireplace to enjoy.

In the case of radioactive disintegrations and nuclear phenomena, the liberated energies are several hundreds of thousands or even millions of times larger than in chemical reactions. At these kinds of energies, the mass defects become perceptible. The mass difference between an alpha particle and the protons and neutrons that make it up is almost 1%. A uranium nucleus that undergoes fission therefore loses just under one thousandth of its mass. Speed of relativistic particles
T= 1/2 mv2, the well-known formula for kinetic energy, is no longer valid for particles traveling at speeds in excess of ten thousand kilometers per second (also known as relativistic speeds). This is the case for those electrons produced by beta decay whose speeds approach (without ever surpassing) the speed of light in vacuo (300,000 km per second). According to classical physics, no such speed limit should exist. The experimental observation that particles never travel faster than light is therefore a good confirmation of the validity of the theory of relativity IN2P3

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