TEACHING MASS-ENERGY EQUIVALENCE1 LAURENCE S. FOSTER Ordnance Materials Research Oftice, Watertown Arsenal, Watertown, Massachusetts
THE
problem of introducing nuclear physics iutoelementary courses in chemistry has arisen because of the impossibility of avoiding such topics as radioactivity, fission of uranium, nuclear reactors, atomic power, or hydrogen bombs. It is necessary to integrate these modern topics into the elementary course without disturbing too drastically the traditional chemical content because the course still has to serve as preparation for later courses in chemistry. I t is the purpose of this review to show how the concept of mass-energy eqnivalence may he introduced while retaining a chemical tinge to the subject. The meaning of the Einstein equation, E = me2, might seem t o be beyond the scope of the elementary chemistry course, particularly in the secondary school, but one cannot dodge its implications. Fortunately, with only a slight extension of the knowledge the student already possesses, it is not difficult t o demonstrate the applicability of this important principle. It is not necessary to go far afield into nuclear physics. It can be done by referring only to atomic weights. When it comes time to talk about atomic power, nuclear reactors, fission bombs, and the origin of solar energy, the student hasalready learnedquitealot about atoms. He will have some i d e ~ sabout the electronic configuration of atoms and the origin of the chemical bond. The nucleus has been described to him as an extraordinarily small, phenomenally dense, highly charged, mysterious speck which contains protons and neutrons. He knows that the atomic number is the number of protons and that the mass number is the sum of the number of protons and neutrons. He is familiar with the concept of isotopes, particularly those of hydrogen and uranium. He does not know enough, however, about the masses of atomic nuclei. Up to this point, he has been permitted to use atomic weights
having only three significant figures a t m o s h x a c t arithmetic has properly been sacrificed to emphasize principles. My suggestion now is that by increasing his awareness of the accuracy to which the masses of atoms are one can introduce concepts of atomic energy without using any new ideas or difficult words. Masr (weight) is something the student understands intuitively. Atomic weights are a part of the chemistry course. The idea of loss in mass and the equivalent appearance of energy is a simple extension of earlier ideas. The approachis one that can be made historically as well, which has certain pedagogic advantages. Before delving into the mass-energy equivalence conversion, however, it is necessary for the teacher t o clarify in his own mind the units in which the changes in mass and in energy will be expressed. Unit of Mass. It is unfortunate that two scales for atomic masses are now in common use. Most of the tables of isotopic masses present the data in terms of the physical atomic mass unit (AMU) or mass unit (MU), that is based on a scale for which the weight of the is 16.00000. Chemists, on the oxygen isotope, 016, other hand, still use atomic weights based on a scale for which the weighted average of the three oxygen isotopes, OL6,01',and OL8,is given as the whole number 16 (printed in bold face type in thelatest International Atomic Weight Table). I n 1950, the American Chemical Society officially adopted the word avogram for the unit of mass on the chemical atomic-weight scale. It has a value of one gram divided by the value of Avogadro's number on the chemical scale, 6.020402 X loza. (Note that on the physical scale, the value of Avogadro's number is 6.02566 X 1023).4 The conversion factor between the two atomic-weight scales, based on the relative abundances of the three oxygen isotopes, is 1 Abstract of a t a k presented at the Seventeenth Summer ConNIER,A. O., S e i m e , 121, 737 (1955).
ference of the NEACT, Tufts University, Medford, Massachusetts, August 18, 1955.
300
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MATTAUCR, J., Sciaee, 121, 745 (1955).
BEA~DEN, J. A,, AND H. M. WATPS, Phys. Rev.,81,73 (1951)
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VOLUME 33, NO. 6, JUNE, 1956
such that one atomic mass unit is equal to 1.000272 avograms. Unit of Energy. To simplify the mathematics still further it is helpful to substitute the energy unit million electron uolts (Mev.) or ergs. Since: m = 1 avogram = 1.66 X c = 3 X loL0 om./sec.
g.
for one avogram of mass the Einstein equation becomes: E = mXc" = 1.66 X = 14.94 X
--
,~i' 7.016255 3vograms
+
,HI
1.007873 avograms
8.024128
=
2 , ~ ~ ' 2(4.002760) or 8.005520 avograms
+
E 17.32 Mev.
r nE- - ~ -i l Y. O-PoP-R -
8.024128 avograms before collision 8.005520 avogmma after collision 0.018608 avograms loss in mass
~p
lo-'
X (3 X 10'o)2ergs ergs
The electron-volt (ev.) is the unit of energy corresponding to the kinetic energy gained by oneelectron, having a charge of 16 X 10-20coulombs,falling through a potential differenceof one volt. ev. = 1 volt X 16 X = 16 X joules = 16 X lO-Iz ergs
coulombs
If we use ev. units instead of ergs in the Einstein equation, for a mass change of one auogram, it becomes: E
to show the masses of the colliding particles (nuclei) with the accuracy of seven significant figures to which they are known from mass spectroscopic data.2ta
(14.94 X lo-') ergs (16 X 10-IS) ergs/ev. = 931 X lo8 ev. = 931 Mev.
=
From the energy gain we may compute the equivalent mass loss: E (Mev.) = n (avograms) X 931 17.32 Mev. = 0.0186 avogram
The calculated loss in mass agrees well with observed difference. Additional examples are available in the literature, but invariably expressed in the physical mass unit.' It is necessary t o convert the values t o avograms before presenting them to beginners in chemistry if confusion is to he avoided. Changes in. Mass of Nuclems. We may use similar calculations to explain the energy release in the fission of uranium-235 and the fusion of hydrogen to form helium. Again it is necessary to use isotopic masses known to six or seven significant figures. The protons and neutrons in atomic nuclei are called nucleons. The mass of an average nucleon in an atomic nucleus can easily be calculated by dividing the mass of the nucleus in avograms by the number of nucleons. For example, the actual mass of the U-235 atom, as determined in a
By expressing chemical atomic weights with a snfficient number of significant figures and using the unit auogram for the actual isotopic masses (and never mentioning physical atomic weights), it is possible to present to the beginning student in a simple but accurate manner the experimental evidence that confirms the validity of the Einstein equation. What became the first laboratory proof of Einstein's ' HUMPHRIES, R. F., AND ROBERT BERINGER, "Fimt Principles law started as an experiment of Cockcroft and Walton of Atomic Physics," Harper & Bros., New Yark, 1950, Chap. 27, in 1932.5 Thev constructed a linear accelerator and especially Pars. 27.227.5. bombarded Gthium by means of protons having energies up t o 0.7 Mev. The most abundant isotopes of lithium has mass number 7. Its nucleus is dismpted by the fast moving proton in a nuclear reaction and two alpha particles emerge with high velocities, 180" apart. These particles have velocities corresponding t o kinetic energies of 8.66 Mev. for each, or 17.32 6.4 Mev per Nucleon Mev. for the two together.= Where did this large increase in energy come from? The answer becomes apparent when the reaction is written COCKCROFT, J. D.,
AND
E.
T. S. WALMN, PTW.Rag. Soe. (Londonl A 137. 229 (1933). ' 6 K&N, I., ; ~ ~ u c l e aPhysr ics," Addison-Wesley Publishing Co., Cambridge, Massachusetts, 1955, pp. 224-25.
ATOMIC
NUMBER
JOURNAL OF CHEMICAL EDUCATION
302
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4,H' ~He4 mass spectrograph, is 235.0601 avograms. Each electron IH' nucleon = 1.007305 avograms weighs 0.00055 avogram. Subtracting the mass of the %He'nucleon = 1.000417 rwogrrtms 92 electrons, the U-235 nucleus has a mass of 235.0096 Loss in mass 0.006888 avogram avograms. Thus, each of the 235 nucleons has on the 0.006888 X 931 = 6.413 Mev. per nucleon, or 25.6 average a mass of 235.0096/235 = l.M)004 avograms. Mev. per He atom formed. The figure8 presents the results o f similar calculations The energy released ,by several other possible fusion that show the variations in average mass of the nucleons for all the elements from 1 to 92. The mass of reactions are? the nucleon is less than 1.00000 in the middle of the Enevgv per He alom. Me". plot, but is greater than 1.00000 a t both ends. When fission of U-235 occurs to form elements of about half the atomic number, the nucleons each lose mass; this slight loss in mass appears as tremendous amounts of energy, in accordance with the Einstein equation. Typical ultimate products of fission are the stable atoms &og4 and ,La1a9. The masses of their nucleons are 0.998926 and 0.999168 respectively, or an average ACKNOWLEDGMENT The idea of comparing masses of nucleons instead of of 0.999047. Subtracting this from the mass of the U-235 nucleon (1.000040 - 0.999047), the net loss using the packing fraction t o compute the energy rein mass is 0.000993 avograms per nucleon. Multiply- leased in fission was first suggested t o the author by ing by 931 t o convert to Mev. units, the result is 0.924 Capt. W. B. Murray, Sandia Base, Albuquerque, New Mexico. The figure is similar t o one used by him in a Mev. per nucleon, or 217 Mev. per atom of U-235. The average amount of energy released on fission of lecture he gave as a member of the AFSWP Technical U-235 corresponds to 0.85 Mev. per nucleon or 200 Training Group in 1952. The data on which the presMev. per atom. This value can be read approximately ent figure is based were computed by George A. Consofrom the figure because in this plot the mass of the lazio, formerly of Watertown Arsenal Laboratory, and nucleon can be read t o five significant figures. Simi- now of Atlantic Gelatin Division of General Foods, larly, it is possible to compute the energy released in Inc. The author appreciates his kindness in making the data available to him. the fusion of hydrogen t o form helium: WEAVER.E. C.. AND L. S. FOSTER."Chemistrv for Our Times," ~ n d e d . ,~d~raw- ill Book Co.,'lnc., New 'iork, 1954, p. 593.
SNEED,M. C., J. I,. MAYNARD, AND R. C. BRASTED, "Cornprohenaive Inorganic Chemistry," Val. 1, D. Van Nostrand Co., Inc., N e w York, 1953, p. 129. Q
