# How much energy is there in an atom

## Equivalence of energy and mass

The formula\ (\ boldsymbol {E = mc ^ 2} \) plays an important role not only for physicists. For example, the energy of the sun and thus the basis of our life is based on this connection, and electricity from nuclear power plants would also be unthinkable without the equivalence of energy and mass.

\ (E = mc ^ 2 \) is one of the most well-known equations in modern physics. It has long since ceased to be found only in specialist books, but also adorns stamps or T-shirts and even served as a template for a sculpture. This formula has recently become a hundred years old: Einstein discovered it in his miracle year 1905 and published it as the fifth of his groundbreaking works - on just three pages. For the Einstein year 2005

The equation \ (E = mc ^ 2 \) links the massm of a body - as can be measured, for example, with a kitchen scale in the earth's gravitational field and stated in kilograms (kg) - with the energy equivalent to this massE.How it is used in the form of electrical energy, combustion energy for heating, kinetic energy and so on, and stated and paid for in joules (J) or kilowatt hours (kWh).c stands for the speed of light, one of the universal natural constants, which is a good approximation of 300,000 kilometers per second.

Equivalent means that any massm a well-defined energyE. corresponds, but also that any energyE. a well-defined massm corresponds to. For example, if a lamp with an output of 20 watts lights up for one hour, the energy it consumes would have a mass of 8 × 10-13 kg equal; this amount may seem tiny to us, but a hydrogen atom, for example, weighs much, much less, namely just 1.67 × 10-27 kg. Even a uranium atom weighs only 395 × 10-27 kg.

### Energy-rich coins

The everyday masses for usm on the other hand, there are enormous energies: a euro piece weighs just seven grams or 7 × 10-3 kg. According to Einstein's energy-mass relationship \ (E = mc ^ 2 \) this coin would have an energy of 6.3 × 1014 J equivalent! For comparison: The total energy consumption of the Federal Republic on an average day is in the order of 40 petajoules, i.e. 40 × 1015 J. Thus, it would only take sixty one euro coins "converted" into energy daily (or around 75 millionths of a euro cent per inhabitant per day) and all conventional power plants could be switched off - no longer dependent on oil, gas and coal. Grafenrheinfeld nuclear power plant

But how can you release this energy? If it were so easy to do, the energy of a euro piece would have an almost horrific and destructive power to destroy humanity. But, nature has arranged it wonderfully wise: While the chemical energy contained in explosives can be released relatively easily by ignition, the conversion of the mass energy \ (E = mc ^ 2 \) only succeeds by means of atomic processes and macroscopically only partially, For example, through atomic collisions with high kinetic energy, the annihilation of electrons, mesons and other elementary particles, the fusion of protons and neutrons to form heavier nuclei, the fission of very heavy nuclei such as uranium or plutonium and other atomic or subatomic processes; some scientific and / or technical effort is therefore necessary.

Although this is everyday life in the laboratory today, it is unsuitable for abuse - for example in the form of destructive weapons. Because the individual processes are macroscopically insignificant, the energy released is tiny. The above-mentioned hydrogen atom or proton provides just 1.5 × 10-10 J or 4.2 × 10-17 KWh. Even a uranium atom would only be 3.6 × 10-8 J or about 10-14 kWh. And an irradiated electron (about 10-30 kg) steers to our heating stove just 8.2 × 10-14 J at. But the amount counts! One kilogram of uranium consists of the enormous number of around 2.5 × 1024 Atoms. Their total energy equivalent is 1017 J - Germany's entire daily energy consumption would be secured by less than half a kilogram of uranium if we could release this mass energy.

### Source of warmth and life

The sun has a luminosity of 3.8 × 1026 Watt. So it emits an energy of 3.8 × 10 every second26 J from. According to the energy-mass equivalence, the sun loses 4.2 million tons of mass! Every second! And yet: what is that in view of today's solar mass of around 2 × 1030 kg? This could continue at the current intensity for billions of years.

Where does the sun get this radiant energy from? The gravitational energy turns out to be much too small, not enough. Chemical reactions, such as the burning of coal, cannot be either - among other things, because at the sun's temperature of several million degrees all atoms are completely ionized, i.e. chemical processes no longer take place. One explanation is the energy released by the fusion of atomic nuclei. Look inside the sun

Because of the large proportion of ionized hydrogen in the sun, i.e. protons, their fusion to form helium would be the most important fusion process. The two physicists Subrahmanyan Chandrasekhar and Hans Bethe developed the associated mechanism, the so-called proton-proton reaction, in the 1930s. The further fusion of helium to carbon - called the Bethe-Weizsäcker cycle after its discoverers - currently only makes a small additional contribution. To put it simply, four protons with a total mass of four times 1.673 × 10 merge in different fusion reactions that also produce neutrons-27 kg, so 6.692 × 10-27 kg, to a helium nucleus. This is simplified to the extent that two of the fusing protons become neutrons when positrons are emitted, since a helium nucleus consists of two protons and two neutrons.

The helium core has a mass of 6.647 × 10-27 kg and thus astonishingly a slightly smaller mass than the four protons combined from which it was formed. The mass missing after the fusion, the so-called mass defect, is \ (\ Delta m = \) 0.044 × 10 in this case-27 kg. When four protons are fused to form a helium nucleus, the total energy is \ (\ Delta E = \ Delta m \; c ^ 2 = \) 3.96 × 10-12 J released. In an energy measure more common for nuclear physicists, this is 24.7 megaelectron volts (MeV) or 6.18 MeV per nucleon. In the sun, such fusions happen about 10 every second38-times!

All of these numbers are unimaginably tiny or overwhelmingly huge, almost "extraterrestrial". Because neither the world of atomic nuclei nor that of the sun fit into the human dimension of the earth. But physics can handle it. While merging light nuclei generates energy, this is no longer the case with heavy atomic nuclei. This is because these contain increasingly more protons and are therefore increasingly positively charged. Because of the increasing mutual electrical repulsion, the heavier nuclei become unstable.

### Nuclear fission and chain reaction

The medium-sized atomic nuclei, such as the iron nucleus, are most strongly bound57Fe with a binding energy of approximately 8.77 MeV per nucleon. The much heavier uranium nucleus235U only has a binding energy of 7.59 MeV per nucleon. Splits235U by bombardment with a neutron, for example in barium142Ba, krypton92Kr and two neutrons, one has a mass loss of \ (\ Delta m \) = (390.300 + 1.675) - (235.658 + 152.647 + 2 × 1.675) = 391.975 - 391.655 = 0.321, all numbers in multiples of 10-27 kg. This loss of mass leads to an energy gain of

\ (\ Delta E = \ Delta m \; c ^ 2 = 2.88 \ times 10 ^ {- 11} \; \ text {J} \) Binding energy per nucleon

In addition to the net energy gain, the reaction also produces two additional neutrons. These can be used to repeat a splitting process with a new one235U-core lead to a so-called chain reaction. After splitting all 2.6 × 1021 Atoms of one gram of uranium235So U wins 7.5 × 1010 J or around 20,000 kWh or 20 MWh. A coal-fired power plant with a typical output of 1000 MW supplies the same energy in 1.2 minutes - and a modern one
2.2 MW wind turbine in nine hours.

If you look at the elementary processes involved in the fusion of light nuclei or in the splitting of very heavy nuclei, the energy gain is roughly the same - equally tiny. And in both cases the mass is only partially converted into energy. Again, it is the huge number of atoms involved, which leads to an overall very large gain in energy. The technical implementation in the fusion reactor of the future or in the nuclear (fission) reactor of the present is, however, very different - different degrees of difficulty, with different side effects from radioactivity. But the sun does this far from the earth with life-giving success and without danger. In all cases it is based on the simple mass-energy equivalence \ (E = mc ^ 2 \)!

In an in-depth article, you will learn how this equation can be derived from relativistic physics with a little thought.