We have talked about the fact that the nucleons (protons and neutrons) are bound together inthe nucleus by the strong force. What is note obvious is that the strength of the attraction is reflected in a remarkable way in the mass of a given nucleus. To begin, let's remember the Einstein equation:

E = mc²

This equation relates mass to energy so that we can talk about anything having mass as being equivalent to a given amount of energy. Let's put some numbers into this and see what we can learn.

What is the energy equivalent of one gram of matter? We are going to solve this equation for energy in units of Joules or Newton-meters, where Newtons are Kg x m/sec². To have the right units, we must recall that 1 g = 1 x 10^-3 Kg. The speed of light is 3 x 10⁸ m/sec. Putting this into the equation yields:

E(N-m) = (1 x 10^-3 Kg)(3 x 10⁸ m/sec)² = 2.0 x 10¹³ N - m

Now, to put this into perspective, it takes about 334 kJ to raise the temperature of one liter of water from 20 ^oC to 100 ^o C, which is roughly room temperature to boiling. One gram of matter has enough energy to heat 2.7 x 10⁸ liters which is quite a lot of water.

Scientists who work in the nuclear community like to use a special unit for energy called the electron volt. Simply, this is the energy an electron gains if it moves 1 cm through a 1 volt field. To give you a rough idea, it takes about 14 electron volts to remove an electron from a hydrogen atom. We can show that the energy equivalent of one atomic mass unit (or Dalton) is 931.4 MeV. This is roughly the energy equivalent of the proton.

An electron has a mass which is roughly 1/1870 that of the proton. If an electron and a positron annihiliate the combined mass would be 2 x 1/1870 of a proton or 1.02 MeV. It is known that when this electron-positron annihilation occurs two photons are released, each with 0.51 MeV of energy. Each photon then carries off the energy equivalent of one electron.

Nuclear Mass Defects

We know that a proton has a mass of 1.00728 Daltons and a neutron has a mass of 1.00867 Daltons. When we put these two particles together to form a deuterium nucleus we know that the nucleus has a mass of 2.014102 Daltons. If we had simply added the masses of the proton and the neutron we'd have gotten 2.01595 Daltons. There seems to be 0.00185 Daltons missing here. Now that doesn't seem like much, but if we recall that each Dalton is equivalent to 931.4 MeV, this small missing mass is equivalent to 1.72 MeV!

Taking a more dramatic example, the helium nucleus has two protons and two neutrons which should add up to be:

2 x 1.00728 + 2 x 1.00867 = 4.03190 Daltons.

The actual mass of a helium nucleus is known to be 4.00150 Daltons for a mass defect of 0.03040 Dalton or an energy of 28.3 MeV.

Clearly the masses of nuclei don't seem to be adding up to be equal to the total masses of the particles that go into them. What is the reason? It turns out that the missing mass is equivalent to the force of attraction holding the nuclear particles together. The mass seems to go away, but in its place we get binding energy. If we think in terms of conservation of mass and energy, we see that we can account for everything.

It is conventional to plot the binding energy per nucleon (which we get by simply dividing the energy we got above by the number of nuclear particles) versus the atomic mass of the nucleus. Such a curve is presented below. Notice that iron has the highest binding energy per nucleon and that this binding energy drops off as the mass of atoms increases after iron. Recall that stars can only fuse atoms together up to iron, which is the most stable nucleus. Heavy atoms such as uranium have relatively low binding energies per nucleon and if they split apart the binding energy of their daughter atoms is actually higher. The difference in energy shows up as heat which is how we get power out of the fission of uranium.

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