3 September 1996

As chemists, we take conservation laws for granted. Our very stoichiometry depends upon conservation of mass (indeed atoms!), and our thermal and spectral toys are predicated upon the conservation of energy in its many forms. But that image above is a niggling reminder that Einstein would have us conserve matter+energy instead of either alone. The picture is of the

In the reactor, unstable nuclei radiodecay into their fission products the mass of which falls short of the starting nucleus. We're not disturbed that mass isn't conserved; indeed,

Surely the world's most recognized scientific equation isn't Newton's F=ma but rather Einstein's

E=m

More to the point, one important component of the energy Einstein needs to conserve is

So the expression Einstein

Clearly, when p=0, we recover what we expect, but, when it's not, the rest may look foreign to chemists. It isn't. They may not know that they know it yet, but it draws together things chemists take for granted in an interesting way.

And electrons can pull off that stunt only because they are

We chemists are confident that our chemical energies never approach those for which that mass dilation would be a problem, but we're wrong. Remember the reason we aren't struck by the moon (except in bad romances) is because its angular momentum about us could not be conserved if its distance to us collapsed to zero. The same

The problem is that potential energy attraction between the electron and proton is indefinitely large at infinitesimal distances. That means the electron could go as fast as it pleases, rushing toward the nucleus. Not so, says Einstein. Thus P.A.M. Dirac takes the trouble to rederive the Schrödinger Equation relativistically to see why not. And lo, the s electron isn't entirely angular momentum free! Dirac's equations show a residual "spin" sufficient to make the s electron miss the nucleus and save the Speed of Light for...well...Light.

And that spin ½ makes all electrons

And Einstein derived E = [ m

which works for both mass and light. But we also know that light has a frequency as well as a wavelength and that the two are related (as for any wave) to the speed (of light, c).

which is really nothing more obscure than d/t=v, the standard expression for speed (except that the frequency needs to be viewed in its "cycles PER second" aspect as the inverse of the oscillation period "SECONDS per cycle" to make it obvious that it's 1/t).

Substituting into deBroglie's relation gives us the expression Einstein would've written down instinctively as the momentum of light, viz.,

Which makes pc the

So for a zero rest mass photon, E = pc, or Planck's constant times frequency, just as we knew it must! But this great conservation expression, E = [ m

Sort of.

That momentum, p, incorporates mass, m, and must therefore be subject to dilation near lightspeed. If we substitute p = mv and the mass dilation law into the above, we get:

And only now do we concede that our speeds of interest are tiny, v <<< c. That makes v/c

And with great relief we recognize kinetic energy emerge in the limit where mass dilation doesn't.

How long does the relief last after we notice that

Just as the massive potential energies of the nucleus when rearranged in a radiodecay make significant inroads on the conservation of matter (in its tradeoff with nuclear energy release), so too do even the gentler potentials of electrostatic attraction (of electron to proton, say) extract their attogram of flesh. But that's just it; chemical potentials are so trivial compared to nuclear that rest mass loss is completely invisible. So we chemists are entirely entitled to cling to conservation of mass

These potentials are no less real for this construction. The mass deficits in (non-radio-) chemistry have no impact on our expressions of kinetic energy beyond those modelled by the concepts of potential energy. Trust me: Einstein wouldn't have given up potential energy as a tool for a second!

Chris Parr University of Texas at Dallas Programs in Chemistry, Room BE3.506 P.O. Box 830688 M/S BE2.6 (for snailmail) Richardson, TX 75083-0688

Voice: (972) 883-2485 Fax: (972) 883-2925 BBS: (972) 883-2168 (V.34) or -2932 (V.32bis) Internet: parr@utdallas.edu (Click on that address to send Dr. Parr e-mail.)

Last modified 17 July 2009.