Re: Aren't mass-energy and potential energy redundant?

Mr. Harmon,

I would use the term "equivalent" rather than "redundant" when referring to
mass and energy. Lets look at your example of the formation of helium. To
keep it simple, lets assume the building blocks are simply two deuterium
atoms, each with one proton and a neutron, so we end up with the same number
and type of nucleons with which we started. As you point out, the mass of
a proton doesn't change. Neither does the mass of a neutron. Therefore, if
you add up the masses of the constituent particles, the helium nucleus is
identical to two deuterium nuclei, and all the energy released in the fusion
process can be accounted for by the difference in the binding energies of
the two types of atom. The nuclear forces holding deuterium together are
stronger than those in helium, and it is that extra potential energy that
gets released. Your analysis is correct, and that is a completely consistent
way of looking at the problem.

Now, think about what we are talking about when we discuss the "mass" of
a particle. As you are aware, "mass" in physics is "inertial mass," meaning
that it is a measure of a particle's response to an applied force: m=F/a.
You apply a known force to a deuteron, measure the acceleration, and
calculate its inertia. Then do the same thing for a helium nucleus (also 
called an alpha particle). What you'll find is that 2*m(D) > m(He). The 
helium nucleus has slightly less inertia than two deuterium nuclei added 
together. So, from this measurement one would conclude that the extra 
energy released in the fusion process is due to the reduction in the mass 
of the system. This is also correct, and a completely consistent way of
looking at the problem.

How do we bring these two apparently conflicting views into harmony?
Einstein showed the way: E=mc^2. Mass and energy are equivalent. Applied to
this example, what that means is that the nuclear potential energy holding
the helium atom together HAS INERTIA! When you want to calculate the mass
of a helium nucleus, you can't just add up the masses of two protons and two
neutrons. You'll get the wrong answer. You have to include in your 
calculation the added inertia contributed by the nuclear potential energy.
The change in mass is not, as you suggested, a change in the mass of the
nucleons. It is instead a change in mass of the *nucleus*. The component
nucleons are unchanged.

The law of Conservation of Mass that we all learned in high school physics
is not, strictly speaking, valid. It is a good approximation when dealing
with relatively weak forces like electricity and gravity, because the
energies associated with these forces are too small to have a significant
impact on the inertia of a system. In the nuclear realm, the strong force
dominates and it makes a measureable difference. Conservation of mass must
be replaced by conservation of energy, where rest-mass is simply another
form of energy. Roughly speaking, the binding energy for larger nuclei,
with 12 nucleons or more, is 8.5 MeV/nucleon. For comparison, the rest
mass of an electron is merely 0.511 MeV. The binding energy of that
electron is only 10 eV or so, a million times smaller than the nuclear
binding energy.

Einstein's equation doesn't just mean that energy can convert to mass and
vice versa. It means that energy has/is inertia. It resists acceleration. 
The more energy a particle has, the harder it is to accelerate, whether 
that energy is nuclear potential energy, "mass energy," or kinetic energy.

As I said at the beginning. Rather than use "redundant," I prefer
"equivalent." I hope this cleared up the distinction for you, and gave you
some ammunition for the next time you face your AP Physics class. Those 
kids can be relentless. :-)

Jay