BioPHYSICAL chemistry

260 PART 2 QUANTUM MECHANICS AND SPECTROSCOPY

the energy of each particle corresponding to their masses (technically the rest mass as the

mass is dependent upon the motion in relativity). Electrons and positrons have the same

mass, me, corresponding to energies of mec^2. Thus, a total energy of at least 2mec^2 is needed

to create an electron–positron pair at a synchrotron. Once created, the positron will rapidly

recombine with an electron, releasing an energy of 2mec^2 , which is equal to 1022 keV, and

it is this release of energy that is often discussed in literature. To conserve momentum, the

energy will be released as two photons emitted at a relative angle of 180° (assuming that

the particles are at rest). For example, in the book Angels and Demonsby Dan Brown, scientists

make use of the Conseil Europeen pour la Recherche Nucleaire, denoted by the acronym

CERN, a center for particle physics that straddles the French–Swiss border. In that story,

scientists at CERN generated 0.25 g of positrons that are trapped in a special container

(a plot device that has never been accomplished). This mass corresponds to 0.25 gme−^1 , or

2.7 × 1026 positrons, and hence each recombination of every positron with an equivalent

amount of electrons will release an energy of 1.4 × 1029 keV or equivalently 5.3 kt, an amount

comparable to that of the first atomic bombs.

me=9.1 × 10 −^31 kg

mec^2 =(9.1 × 10 −^31 kg)(3 × 108 ms−^1 )^2 =511 keV

1 eV =3.8 × 10 −^32 kt (kilotonnes)

Multi-electron atoms

The solution of Schrödinger’s equation when more than one

electron is present is usually solved by use of approxima-

tions to incorporate the interactions between the electrons.

There are two basic approaches that can be used. Both rely

on the idea that the wavefunctions derived for the hydrogen

atom are basically correct and simply need minor corrections

in order to be used for many electron atoms.

Empirical constants

One approach is to modify the constants used in the expres-

sions to include the new interactions. This can easily be done

by modifying the nuclear-charge parameter, Z. The presence

of the additional negatively charged electrons effectively

decreases the positive charge of the nucleus (Figure 12.13).

i

i

2

(^12)

1

=− =− 1

Limited net effect

of these electrons

Net effect roughly

equivalent to a point

charge at the center

r

Figure 12.13For atoms with more
than one electron, the nuclear
charge, Zatome, that any given
electron experiences is effectively
reduced by the presence of the
other electrons, thus reducing the
charge to the effective nuclear
charge Zeffatom.