Nuclear Energy

I alluded to the notion of energy conservation when talking about absorption and emission of photons. However, I need to expand the idea a little bit.

• As shown by Einstein in the early 1900s, mass and energy are to be thought of as different forms of one physical quantity. Einstein related mass and energy in his famous relation,

  E = mc²

  Note -- For a particle of mass m, we call the quantity mc², its rest mass energy.

• As a result, we must expand our notion of energy conservation to include the possibility that matter can be turned into radiation and vice versa.

In the early Universe, such transformations of mass into energy and back into mass were very common. Note in the above pictures that whenever a matter particle such as an electron, e^- comes into contact with its anti-matter twin, the positron, e⁺, they annihilate producing energy. Conversely, if energy is converted to mass (particles), the mass is always produced in matter-anti-matter pairs. This property of formation/annihilation of mass is what makes the slight (1 part in 1 billion or so) matter/anti-matter asymmetry seen in the Universe hard to understand.

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Now, for convenience, let me define an efficiency, ε, for energy production. I will use the annihilation process as my benchmark.

• The annihilation efficiency is
  • ε_ann = energy produced / (mass involved x c²)

  For annihilation radiation, ε_ann = 1

• For chemical burning

  • ε_ch ~ 35 million J (per kg) /( 1 kg x c²) ~ 5x10^-9

• For the nuclear reactions which power the Sun (4 hydrogens into 1 helium)
  • ε_nuc = 0.007, that is, only 0.7 % of the mass is converted into energy

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So, we see that the annihilation process is the most efficient and that chemical burning is the least efficient. Although, nuclear fusion is less than 1 % efficient, it is still efficient enough to account for the Sun's luminosity:

Sun's Nuclear Energy = Mass of the Sun times the speed of light squared ~2x10⁴⁷ Joules

Now the Sun is ~ 1 % efficient ===> 2x10⁴⁵ Joules are available to radiate--much more than is needed to explain the lifetime of the Sun!