NUCLEAR CHEMISTRY

Rate of Radioactive Decay

Radioactivity is the spontaneous disintegration of atoms of certain elements into atoms of other elements with the emission of smaller particles.
Elements, such as uranium, which spontaneously emit energy without the absorption of energy are said to be naturally radioactive.
Imagine having a very small sample of ²³⁸U, say 1 mg. This small mass contains many trillions of atoms. Now consider just one of these atoms. When will this particular atom undergo radioactive decay? The answer is that it is impossible to predict when, if ever, an atom will decay. Fortunately, the chemist is rarely concerned with the behaviour of a single particle. 1 mg of ²³⁸U contains 2.5 x 10¹⁸ atoms. With such large numbers of atoms it is possible to make precise predictions about rates of decay.
Experiments show that radioactive decay rates are directly proportional to the number of atoms present:

rate = k x N

where N = the number of atoms.
Different radioisotopes decay at different rates, so how can we compare their decay rates?

• The above expression tells us that for a radioactive isotope, the time taken for a fixed proportion of the original number of atoms to decay is constant.
• The proportion scientists have taken is 'one-half' of the original number of atoms, and the time taken for these to decay is called the half-life, denoted as t_½.
• In this way, the decay rates of different radioisotopes can be compared. The half-life is a direct measure of the stability of an isotope: the shorter the half-life, the faster is the rate of decay.
• The half-life is a characteristic of each radioisotope, but it is independent of the number of atoms.

⁹⁰Sr undergoes beta decay and has a half-life of 28 years. Given 8 g of this isotope, 4 g would remain after 28 years, 2 g would remain after 56 years, and 1 g after 84 years.

The table below gives the half-lives of some radioisotopes, all of which undergo beta decay:

Isotope Half-Life 3H 12.3 years 14C 5.73 x 103 years 32P 14.3 days 59Fe 45.1 days 60Co 5.27 years 90Sr 28 years 131I 8.04 days

the time required for one-half of a given quantity of a reactant to react. Half-life is commonly used as a measure of the rate of first-order reactions. It indicates the kinetic stability of a reactant: the longer the half-life, the greater the stability. The decomposition of dinitrogen pentoxide in tetrachloromethane is a typical first-order reaction: N2O5(sol) ® 2NO2(sol) + ½O2(g) Its rate equation is: rate = k[N2O5(sol)] where k is a constant called the rate constant. It can be deduced that: k = 0.693/t½ The half-life may be evaluated from the rate constant or vice versa. The shorter the half-life, the faster the reaction. The half-life for the decomposition of N2O5 in tetrachloromethane at 30 ºC is 2.4 hours. If we start with 10.0 g of N2O5 at t = 0 (q0), then, after a period of 2.4 hours, 5.0 g will remain. After a second period of 2.4 hours (4.8 hours in total), 2.5 g remain. After a total of 7.2 hours, 1.25 g remain, and so on. For 3 half-lives, the amount of N2O5 remaining at time t (qt) is: qt = 10 g x (0.5)3 = 1.25 g Generalising, we have qt = q0(0.5)t/t½ in which t/t½ gives the number of half-lives. XD