Robert P. Bauman Polytechnic Institute of Brooklyn Brooklyn, New York
Can Matter Be Converted to Energy?
Several recent textbooks of chemistry provide continuing evidence of the difficulty often experienced in explaining the mass-energy concept a t an elementary level. Along with the typical statements that mass is converted to energy are found well-hedged versions suggesting that mass apparently disappears, or that the mass does not really disappear but it is now seen as energy. All of these discussions take as the basic equation the relativistic expression E = mc2
(1)
given by Einstein in 1905. Although the elementary texts seldom say so, it is generally recognized that the mass, m, appearing in eqn. (1) is the relativistic mass which is related to the rest mass, nlo, by the equation, (2)
where u is the speed of the body and e is the speed of light. Even a hasty examination of eqn. (1) by an unprejudiced mathematician would reveal that if matter is to disappear, energy must also disappear, for the proportionality constant, e2, is necessarily a positive number. Thus Einstein's equation clearly prohibits the simultaneous destruction of mass and creation of energy, or the destruction of energy and creation of mass.
and therefore a measurable increase of energy. Eqn. (1) says that it has also increased in mass.' The increase in energy has come from the kicker; hence we may conclude that the kicker now has less energy and less mass. (But kicking a soccer ball is not a very effectiveway to reduce, since the mass transferred is a small part of a picogram, or micromicrogram.) As the soccer ball bounces along the ground it gives up energy, and mass, to the earth until it comes to rest with its initial mass (minus attrition caused by surface wear). Note that by the "initial mass" we mean the rest mass, mo,of the ball a t rest with respect to the earth. If we mere to measure the mass of the ball from a moving platform while the ball is a t rest with respect to the earth, a somewhat larger mass would be found, and since the ball would have a velocity and kinetic energy, with respect to the platform, the measured energy would also be higher. A more dramatic, if less often observed, example of energy transfer is nuclear fission. The original nucleus, a t rest, has a mass ma. At some instant it flies apart, the several pieces having collective relativistic masses adding up to the original value mo,provided the masses of all fragments, including photons and neutrinos, are included. As the particles collide with surrounding matter they give up energy and mass, to the surrounding matter, eventually coming into t h e r n d eqnilibriuni
The Relativistic Interpretation
-
Some common exam~leswill illustrate thc meaning of the relativistic mass its relation to energy, Consider first a soccer ball lying a t rest, with a mass mo. When the ball is kicked it acquires a substantial speed
366 / Journal of Chemical Education
'The mass of the moving ball is given exaotly by eqn. (2), ful. m y speed v. The increase in energy is given exactly, from eqo. ( I ) , by the expression aE = me' - maca. When u is small oompared to e the last expression reduces to the eq~tationAE = 1limu2.
with the surroundings. The rest masses of the fragments add up to less than the original rest mass, but the mass of the surroundings has increased. A Non-Relafivistic Interpretation
I s there some sense in which i t would be proper to say that matter is converted to energy? We may reconsider the nuclear disintegration from the point of view of the experimentalist who is not aware of the special theory of relativity and has available an analytical balance (for determination of atomic weights) and a thermometer (for determination of the heat capacity and temperature of an isolated system). This experimentalist finds that the mass, ma, of his radioactive sample has decreased and the thermal energy, or "kinetic" energy, of the environment has increased. Careful measurements would show that the proportionality constant between the mass change and the kinetic energy change is of magnitude c2, but of course the energy change and mass change have different signs. There fore he would write A(K.E.)
= -(A%)
c2
(3)
We, in our greater wisdom, recognize that because the lefehand side refers to the environment and the righe hand side to the radioactive sample, this equation is relativistically correct and describes the transfer of mass and energy from the sample to its environment. Lacking such sophistication, and lacking the theory or experimental test to see that the radioactive sample has lost energy or that the surroundings have gained mass, our experimentalist assumes eqn. (3) to be the total story for the isolated system consisting of radioactive sample and its environment. Equation (3) is suggestively rearranged into the form A(K.E.)
+ a(m0ca)= 0
(4)
which is similar in meaning,%as well as form, to the more familiar equation relating changes of kinetic and potential energies. A(K.E.)
+ A(P.E.) = 0
That is, the moc2,in this interpretation, plays the part of a potential energy that may be converted to kinetic energy by a variety of processes. This would be a satisfactory way, for example, of thinking of the changes in a spring as it unwinds; the increase in kinetic energy of the spring or its immediate surroundings is accompanied by a decrease in rest mass of the spring. The higher rest mass of the wound spring, or the higher rest mass of the unstable nucleus, may be considered as a measure of the potential energy stored in the spring, or the nucleus. I t should be noted, however, that the non-relativistic *Eqn. (4) would be relativistically correct if the change in kinetic energy were expressed as A(me2) - A(mac"), but the meaning assigned to this term by our unsophisticated experimentalist is A('/2mvl) because he is looking at the change in energy of the molecules of the environment, which have constant rest masses during the process. 'The non-relativistic description is adequate to explain the experimental evidence given. If the non-relativistically minded experimentdilist had a convenient means of measuring the mass of the surroundings, or an independent means of measuring the energy of the nucleus before tission, he would immediately discover eqn. (1) and he led to the relativistic result, since nature follows only the relstivistio laws.
interpretation, while it is an adequate3way of representing systems a t rest, cannot be described by eqn (1). The energy terms in eqns (1) and (3) have different meanings; the mass terms have different meanings; and eqn. (3) has a minus sign that is not present in eqn. (1). If the non-relativistic description were to be preferred for introductory purposes (and this is certainly questionable) it would be necessary to discuss it solely in terms of eqn. (3). Is Energy Conserved?
In the preceding discussion it has been tacitly assumed that energy, and hence also mass, is conserved. Eqn. (1) is not inconsistent with such an assumption; it does say that if energy is conserved, then mass is also conserved, and vice versa, but there is nothing in the special theory of relativity that requires that energy be conserved. Because all experiments carried out thus far seem to be consistent with conservation of energy, this is introduced as a separate assumption, often labeled as the first law of thermodynamics. But because there is no proof of the general validity of the assumption, nor any broad underlying theory that can independently predict it, new experiments are required from time to time to verify the assumption under different sets of conditions. For example, conservation of energy was imperiled by the early measurements of 0 decay of nuclei, but was rescued by the theory of the neutrino (a theory also required for other purposes), and a t the time when it became known that parity was not conserved in weak interactions, it became necessary to perform experiments to reaffirm the conservation of energy in processes involving the weak interactions. It has been suggested that energy is not really conserved on the scale of the universe, but the postulated deviations are so small that they cannot be verified or refuted by direct measurement and are therefore important only in the realm of astronomy. Summary
There is too often a confusion between non-relativistic and relativistic interpretations of transfers of energy. In the non-relativistic picture, rest mass is converted to kinetic, or thermal, energy, as described by eqn. (3) in which a minus sign appears. (The shortcoming of this description is simply that it does not recognize that the body that has lost rest mass has also lost energy or that the body that has gained energy has also increased in mass.) On the other hand, the equation that is usually quoted, eqn. (I), requires a relativistic interpretation and shows that the energy and mass of any system must increase together or decrease together. We commonly add to this a separate assumption, the first law of thermodynamics or law of conservation of energy, to obtain the statement: The total energy of any isolated system is constant and is directly proportional to the total mass of the system. I t seems reasonable to expect that, more than sixty years after the publication of the special theory of relativity, we should recognize, and teach, that this theory clearly prohibits any conversion, in the strict sense, of mass to energy or energy to mass. The tern1 "conversion of matter into energy" must be interpreted as a conventional phrase that does not really say what it may seem to say. Volume 43, Number 7, July 1966
/
367
