Slide 1

Rates of Radioactive Decay

Radioactive Decay

Different nuclei undergo radioactive decay at different rates.

Some radioisotopes, such as uranium-238, decay at a barely perceptible rate over billions of years; others, such as carbon-17, decay within milliseconds.

All radioactive decays obey first-order kinetics.Therefore the rate of radioactive decay at any time t for No Nt is given by Rate of decay at time t, - d [N]/dt = k [N] k is the decay constant N is the number of radioactive nuclei present at time t.

A first order rate can be converted into an integrated rate law:-d[N]/[N] = k dt-d[N]/[N] = k dt

N0 is the number of radioactive nuclei initially present in a sample.Nt is the number of radioactive nuclei at time t.

NoNt

ln= kt

HALF-LIFELike all first-order processes, radioactive decay is characterized by a half-life.HALF-LIFE is the time required for the number of radioactive nuclei in a sample to drop to half its initial value.

ln [No / Nt ] = ktln [ No /0.5No ] = ktln 2 = kt1/2 t1/2 = ln 2 /k

ExampleFor example: Ra-234 has a half-life of 3.6 daysIf we start with 50 grams of Ra-234, calculate mass of this sample after 10.8 days?

For each half-life, one half of the substance decomposes.After 3.6 days only 25 g of Ra-234 will remainAfter 7.2 days only 12.5 g of Ra-234 remainAfter 10.8 days only 6.25 g of Ra-234 remain

Radiocarbon DatingBecause the half-life of any particular nuclide is constant, the half-life can serve as a nuclear clock to determine the ages of different objects.

Carbon-14 has been used to determine the age of organic materials.

The procedure is based on the formation of carbon-14 by capture of solar neutrons in the upper atmosphere:

14N + 1n 14C + 1H

This reaction provides a small reasonably constant source of carbon-14. The carbon-14 is radioactive, undergoing beta decay with a half-life of 5730 years.

In using radiocarbon dating, we generally assume that the ratio of carbon-14 to carbon-12 in the atmosphere has been constant for the past 50,000 year.

14C 14N + 0b

67-1t = 5730 years

The carbon-14 is incorporated into carbon dioxide, which is in turn incorporated, through photosynthesis, into more complex carbon-containing molecules within plants.

When the plants are eaten by animals, the carbon-14 becomes incorporated within them.

Because a living plant or animal has a constant intake of carbon compounds, it is able to maintain a ratio of carbon-14 to carbon-12 that is nearly identical with that of the atmosphere.

Once the organism dies, it no longer ingests carbon compounds to replenish the carbon-14 that is lost through radioactive decay.

The ratio of carbon-14 to carbon-12 therefore decreases.

By measuring this ratio and contrasting it to that of the atmosphere, we can estimate the age of an object.