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RALPHK. BIRDWHISTELL University of West Fiortda Pensacala. FL 32504
The Concepts of Mass and Energy Hans Kolbenstvedt Department of Physics Reidar Stdevik Department of Chemistry. University of Trondheim, AVH, N-7055Dragvoll, Norway Energy and mass are fundamental concepts of chemistry and phvsics. Several recent textbwks on chemistri provide cont&ng evidence of the difficulty in explaking the mass+energy concept in a correct manner at an elementary ,~ to level. An article published in this J o ~ r n a l claiming answer the question "Can matter be converted to energy?", is frequently cited by textbook writers. In view of recent discussions in where the tendency is to characterize an object by a single mass parameter only, and abolish the redundant concept of relativistic mass completely, we wish to provide the readers of this Journal with a modern language version of the mass-energy problem. Einstein's Fundamental Equation In Newtonian mechanics a particle of mass M and velocity u has a kinetic energy One of the results of the relativity principle, demanding that the laws of mechanics must have the same form in all inertial frames of reference, is that the expression for the kinetic energy must be modified to where c = 3 x lo8d s , the velocity of light in vacuum. There is ample evidence for the correctness of eq 2 for velocities up to 0.999999...c, and modem particle accelerators would simply not work without this relation being fulfilled. The unfamiliar expression (eq 2) bears little resemblance to its classical counterpart (eq 1).However, for velocities much smaller than the velocity of light, eq2 reduces to eq 1,which is easily shown by a series expansion. Equation 2 can be rewritten as K=E-E,
(3)
E, = Mc2
(5)
and The quantity E can be interpreted as the total energy of the particle and E, as the rest energy of the particle. 'Bauman, R. P . J . Chem. ~duc.1966,43,366-367. '~dler,C. G. Am. J. Phys. 1987,55,739-743. 3 0 k u n , L. 8. Physics Today. 1989,31-36. %indler, W.; Vandyck, S.; Ruschin. C.; Sauter, C.: Okun, L. 6. Physics Today 1990,13-14 and 115-1 17.
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Contrary to common belief, eq 5, and not eq 4, is Einstein's fundamental equation. The rest energy is the energy residing within the particle and manifesting itself as mass. It was this very hypothesis put forward by Einstein that was confirmed by the detonation of the first atomic bomb. I t is also this equation that accounts for the process that the makes nuclear reactors work. Equation 4, however, tells us in essence merely how the kinetic energy varies with the velocity of the particle. The Mass of a Composite System Consider a system such as an atomic nucleus at rest in the laboratory-frurne. 'Thv nucleus consists of protons and neutrons rnovinc nbout within the b~wndarvofthenucleus. serve to The strong nuclear forces between the prevent them from flying apart. The energy residing within the nucleus, the internal energy, is made up of several parts: rest energy and kinetic energy of each nucleon and, in addition, mutual potential energy between the nucleons:
where V, is the potential energy between nucleons i and j. Einstein's fundamental equation (eq 5) simply states that the mass of the composite nucelus is found by dividing eq 6 by c2,that is
which means that all forms of internal energy contribute to mass in the same manner. Formally we could write
where the first term is the sum of the individual nucleon masses and Mkinand M,., are the contributions from internal kinetic and potential energies, respectively. Consequently,the mass of a composite systemis different from the sum of the masses of its constituents. As a matter of fact, the condition for the nucelus to be energetically stable, is that
implying that the part coming from the potential energies must he negatiue and dominate the contribution from the positiue kinetic energies. The discussion above holds for any composite system. The mass of, for example, a hydrogen atom is slightly smaller than the sum ofthe protonand the electronmasses because
of the negative contribution from the electromagnetic potential energy of the electron in the field of the proton, which outweighs the positive contribution from the kinetic electron energy. Analogously, in a molecule, kinetic and ~otentialelectron enerev. as well as rotational and vibrational mergy, will contribute to the total mi~is. The main difference between the cases of nuclei. atoms. and molecules, is the magnitude of the mass contribution$ from kinetic and potential energies. For a typical nucleus this contribution will be of the order of 10 MeV/cZ per nucleon or, equivalently,around 20 electron masses. For an atom the corresponding number will be of of the order of 1 eV/c2,that is, a factor of smaller and thus outside the limits of measurements. Change of Mass in a Reaction Consider a simple common reaction where a composite svstem(e.e.. a nuc1eus)ofrnassM. orieinallvat rest. breaks into k; parts of misses and vdozties ul) and (M2, u.). res~ectivelv.The law of conservation of enerw is valid inkelaiivistic cheory provided rest energy is incGded. For the reaction above we thus have
up
where Kl and K2 are given by equations of the same type as eqs 2 and 3 and are positive quantities. Clearly, mass is not conserved in the reaction since M > MI+M2.This is in fact a necessary condition for the reaction to take place. Part of the internal energy of the original system is transformed into kinetic energy of the reaction produds, the remaining part manifests itself as the masses MI and Mz. The nonconservation of mass simply means that energy can be transformed from one form into another; a phenomenon well known from thermodynamics. Mass Change and Photons The photon (quantum of light) has the unusual property that it is always moving with the velocity of light. Putting u = c in eq 4 yields an infinite value for the total energy unless M = 0. In this case E can have any value. A photon thus has zero mass and is always mouing with the speed of light in vacuum. The photon energy must consequently be determined by other parameters than mass and velocity The relevant parameter turns out to be the frequency v, and the energy of a photon is simply given by E = hv
(11)
where h is Planck's constant. When an atom is excited to a higher energy level by absorption of a photon, the photon energy is transformed into kinetic and potential energy of the atomic electrons. As a result of the absorption, the mass of the atom increases by an amount hvlc2. For the same reason as mentioned above, this mass increase is negligible in the case of atoms but can be considerable in the case of nuclei absorbing photons with energies of the order of megavolts (y-rays). Examples These ideas may be illustrated by numerical examples. In calculating a reaction energy from a mass change, the formula
e of mass releases about is used. Thus, the annihilation of 191000 billio;~. For the fusion reaction between a tritium (TI) and a deuterium'(Dt) nucleus to give a helium nucleus and a neutron (n)
In terms of energy, this equation can be written: where l g = u N,. The products have a mass 0.0191 u (u = 1.6606 x IFz4g) less than the reactants, corresponding to an energy release of about 430 billion J for every gram of helium produced. In addition to this reaction, two potentially useful fusion reactions for controlled energy production are Di+Di + T + + H + (14) with AM = 0.00429 u D ' + Di + 3 ~ e z+t n (15) with AM = 0.00344~.For both reactions the difference in mass between reactants and products is positive, corresponding to 80-90 billion J of energy released for every gram of deuterium converted. (Each heartbeat of a human being consumes an energy equivalent to about 1J.) A series of nuclear fusion reactions in the core of the sun are responsible for the production of solar energy. The overall reaction is as follows: 4H' j 4 ~ e 2++2e' + 2v + 2y (16) The total mass of the products is 0.0276 u less than that of the reactants, corresponding to a release of 6 x 10" J for every gram of hydro&n traiiformed. Each second, the sun converts about 4 million tons of hydrogen into helium. The hydrogen bombusesthe samemethodas thesun topn~ducc enerby, and the largest hydrogen homb detonated released about 2 x 10"d corresponding to a destruction of only 2.7 kg. It tnrnsout that uranium-235 isclose tothe point ofbeing able to undergo fiwion. A typical - spontaneous . ~. fission process is 23% + n + 2 3 6 ~+ 1 4 0 ~+a93Kr+ 3n (17) Also electrons, y-rays, and neutrinos are produced. Uranium-236 is an unstable nucleus. Although there are at least 30 different ways in which this nucleus can divide, the energy released is approximately 8 x 10" J for every gram of uranium destroyed. The kinetic energy of the fission fragments,including the neutrons, is about 85%ofthe total reaction energy. The energy liberated by the complete fission of 1kc of uranium-235 would be about 8 x 1013J. and this is roughly equivalent to the energy released in the exploeion of20,OOOtonsofthe ex~losiveTKT. We have seen that fission and fusion reactions broduce comparable quantitiesof energy, and that thev~roducefar more enerm .. -.than any other known sources. The apparent law of conservation of mass in connection with chemical reactions is only strictly true if there is no energy change in the reaction considered. For ordinary chemical reactions the difference in mass correspondingto the heat change involved is so small as to be outside the limits of measurement. Thus, 100 kJ is equivalent to about 1 n 4 a. 0 -"
High energetic y-rays(photons) can interact with matter in several ways. A ?ray photon can, in the vicinity of an and atomic nucleus. be transformed into an electron ~ a positron. The pc&ion is the antiparticle of the electron and has the same mass but o ~ ~ o s icharee. te The total enerw of the pair is equal to the G e r m hv or the incident ph&n. The kinetic energy of the pair is ~
K = ~ U - ~ M ~ ~
(18)
if the small recoil energy of the nucleus is neglected. For pair production to occur, hv must be greater than 2McZ. The pair production will not occur unless there is an atom nearby to absorb some of the momentum. Here we have an example of the creation of mass from photon energy. Volume 68 Number 10 October 1991
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~
Neither the electron nor the positron existed before the encounter of the photon with the atom; it was not an electron that was part of the atom. The reverse prosess electron + positron + photons
(19)
also occurs; this process is known as annihilation and can occur for free electrons and positrons a s long as (at least) two photons are created. The two photons have equal energies (Mc2)and move in exactly opposite directions. Afterword Practically all introductory chemistry and physics text books include the confusing concepts of rest mass, m , and relativisticvelocity dependent mass,m, connected through rn = rn,(l-u2/,2)-4
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Journal of Chemical Education
(201
(Here we deliberately use small letters m and m, formasses to avoid confusion with our discussions above.) The total energy of a particle is accordingly written (compare with eq (4))
This equation is frequently claimed to give the connection between mass and energy, but as stated before, it merely tells us how the energy varies with the particle velocity. The concepts ofrest mass and velocity-dependent mass are common sources of confusion among most students of science.' In our opinion, it would be better to abolish these concepts altogether and characterize a particle by its mass only.
