Chemistry, Third edition

MATHEMATICS OF RADIOACTIVE DECAY

Gamma radiation

Gamma radiation is very short wavelength electromagnetic radiation, very similar to

X-rays. Their wavelengths are usually 10^10 –10^13 m, corresponding to about

106 –10^9 kJ per mol of photons.

Gamma rays are usually emitted by radionuclides following the emission of alpha

or beta particles. Just as the brightness of a torch falls as the observer moves away

from the torch, an observer armed with a Geiger counter will detect a lower intensity

of gamma rays as he or she moves away from a radioactive source. In principle, the

intensity of gamma rays never falls to zero, although in practice the observed intens-

ity does fall to an undetectable level. This ‘zero level’ may also be reached by placing

concrete or lead between the source and the detector.

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Alpha and beta decay

An element P, radioactively decays to an element Q with the emission of an alpha particle. Q

decays to element R by beta emission and R decays to element S by beta emission. What

can you state about P and S?

Exercise 21C

Energy of gamma rays

Confirm that a gamma ray wavelength of 10^10 m corresponds to an energy of about 10^6 kJ

per mole of photons

Exercise 21D

Mathematics of radioactive decay

The number of atoms of a radionuclide Ntremaining after an interval of tseconds

may be calculated from the formula

Nt=N 0 ekt

whereN 0 is the number of atoms at t0 and kis the first-order rate constant (often

called the decay constant) with the units of s^1.

Since the number of atoms of a radionuclide is proportional to its mass, Ntand N 0

may be replaced by the mass of radionuclide at time t(symbolizedmt) and t 0

(symbolizedm 0 ) respectively:

mt=m 0 ekt

These equations show that the fall in the number of atoms of radionuclide with time

is exponential (Fig. 21.3(a)). In fact, any quantity which is proportional to the

number of atoms (such as the number of counts per minute registered on a geiger

counter) may also be substituted in these equations.

The half-life (t (^1) ⁄ 2 ) of a radionuclide is the time which elapses before half the nuclei
21.4