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Binding Energy and Mass Defect

We have been nibbling around the edge of this for so longNow we are going to actually calculate howmuch energy there is present in the mass (or lack thereof) of atoms and the nuclei. In order to understand howthis works, we need to realize how strong the strong (nuclear) force really is. Lets look at this: In the nucleusof a normal carbon atom there are 6 protons and 6 neutrons. Well, from studying static electricity we know thatif two (or more) like charged particles are near each they will repel. As a matter of fact, the distance betweenany two protons in the nucleus of the carbon atom is approximately 2.010 -15 meters. Using Coulombs Law

=

221

d qkq F e to determine the electrostatic force of repulsion between them, we find that there is a force of 58

Newtons acting on them. That doesnt seem like much when compared to our masses, but remember, the massof a proton has an order of magnitude of 10 -27 kg. That means the 58 N would accelerate the protons apart at anacceleration with an order of magnitude of 10 28 m/s 2. Remember, the order of magnitude of the accelerationdue to gravity is only 10 1! This means that in the nucleus, the electrostatic force want to push protons awayfrom each other with an effect that is 1 billion billion billion times greater than the effect of gravity on a 1kilogram mass!

OK, why dont they fly apart? The Strong (nuclear) force. This force acts over very short distances (i.e.the diameter of a nucleus), but in that distance it is approximately 1 billion times stronger than the electrostaticforce, AND it can attract not only like charged objects (two protons) but also attracts uncharged and charged

particles together (meaning electric charge is unrelated to the magnitude of the strong (nuclear) force.) Thestrong (nuclear) force is so strong that it physically crushes the protons and neutrons together so hard that theyin effect overlap and lose mass.

As we have learned so far, you cant really lose mass. If there is any measurable decrease in the mass of the nucleus of an atom, there must be a corresponding increase in the energy. In this case, the energy is boundup in the strong nuclear bonds carried by gluons.

Mass Defect: this is the difference between the actual nuclear mass of the atom and the total mass of all thenucleons in the atom. The amount of this difference directly corresponds to the amount of potential energy

bound up in the bonds of the nucleus holding the nucleus together.For example: A helium nucleus has an actual nuclear mass of 4.00283 u. Yet, if you add up the total mass

of all the nucleons (m p= 1.007825 u and m n= 1.008665 u), their total mass is 4.032980 u. The Mass Defect isfound by subtracting the total mass of the nucleons from the actual nuclear mass:

uuu 03075.0032980.400283.4 = This mass defect represents the amount of energy that would be needed to pull apart the nucleus of the

helium atom. This energy is referred to as binding energy .

Binding Energy: the amount of energy required to pull a nucleon out of the nucleus of an atom or to pull apartthe nucleus from its current structure. This amount of energy is found by knowing the mass defect of thenucleus of the atom, and using Einsteins famous equation to covert it into units of energy. This is a form of

potential energy.For the Helium atom used above, the mass defect of 0.03075 u corresponds to a particular amount of

energy. First convert the mass into kilograms (1 u = 1.6710 -27 kg): -5.053110 -29 kg. This is the mass for Einsteins equation:

Joules smkg mc E 1228272 1053.4)/1000.3)(100531.5( === That doesnt seem like a lot until you convert it into electron-volts (1 eV=1.610 -19 J):

-28.3 MeV (2.8310 8 eV)When you compare that to the amounts of energy needed to ionize atoms (10s and 100s of eV) that

represents a HUGE amount of energy. The negative sign indicates that this is the amount of energy needed to be added to the nucleus to break it apart, or bring the energy of the atom to zero.

This can be done for any isotope of any element. The greater the mass defect of the atom the tighter thenucleus is bound together. Still, there is a balance the needs to be maintained. For a given nucleus size, if the

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mass defect is too great, it cause the nucleus to be unstable and prone to radioactive decay. If the mass defect istoo small, the same effect occurs.

Mass/Energy Equivalence: since the mass of a particle represents a certain amount of energy we candetermine the amount of energy that the particle would release if it were annihilated (say in a matter/antimatter interaction!) The protons mass, for example, is often referred to by is electron-Volt equivalent in energy. Howmuch energy is that?m p= 1.007825 u, so convert this to kilograms: m p= 1.6810

-27 kg. Substitute this into Einsteins famous

equation and you get 1.5110 -10 Joules, which corresponds to 947 MeV.

The accepted conversion of 1 universal mass unit to electron-Volts is 1 u = 931 MeV (9.3110 8 eV)

Notice that if we simply multiplied the mass defect of the helium atom (-0.03075 u) times the conversion factor above, we get -28.6 MeV (essentially the same value due to round off errors).