The Foundations of Chemistry

RATES OF DECAY AND HALF-LIFE

Radionuclides have different stabilities and decay at different rates. Some decay nearly
completely in a fraction of a second and others only after millions of years. The rates of
all radioactive decays are independent of temperature and obey first-order kinetics.In
Section 16-3 we saw that the rate of a first-order process is proportional only to the
concentration of one substance. The rate law and the integrated rate equation for a first-
order process (Section 16-4) are

rate of decayk[A] and ln

A

A

0

akt

Here A represents the amount of decaying radionuclide of interest remaining after some
time t,and A 0 is the amount present at the beginning of the observation. The kis the rate
constant, which is different for each radionuclide. Each atom decays independently of the
others, so the stoichiometric coefficient ais always1 for radioactive decay. We can there-
fore drop it from the calculations in this chapter and write the integrated rate equation as

ln 

A

A

0

kt

Because A 0 /A is a ratio, A 0 and A can represent either molar concentrations of a reactant
or masses of a reactant. The rate of radioactive disintegrations follows first-order kinetics,
so it is proportional to the amount of A present; we can write the integrated rate equa-
tion in terms of N,the number of disintegrations per unit time:

ln 

N

N

0

kt

26-10

26-10 Rates of Decay and Half-Life 1013

Gas molecules Path of a

single

radiation

Window

+

e– +

Figure 26-5 The principle of operation of a gas ionization counter. The center wire is
positively charged, and the shell of the tube is negatively charged. When radiation enters
through the window, it ionizes one or more gas atoms. The electrons are attracted to the
central wire, and the positive ions are drawn to the shell. This constitutes a pulse of electric
current, which is amplified and displayed on the meter or other readout.

A sample of carnotite, a uranium

ore, shown with a Geiger–Müller

counter.