Rate of Radioactive Decay

During natural radioactive decay, not all atoms of an element are instantaneously changed to atoms of another element. The decay process takes time and there is value in being able to express the rate at which a process occurs. A useful concept is half-life, which is the time required for half of the starting material to change or decay. Half-lives can be calculated from measurements on the change in mass of a nuclide and the time it takes to occur. The only thing we know is that in the time of that substance's half-life, half of the original nuclei will disintegrate. Although chemical changes were sped up or slowed down by changing factors such as temperature, concentration, etc, these factors have no effect on half-life. Each radioactive isotope will have its own unique half-life that is independent of any of these factors.

The half-lives of many radioactive isotopes have been determined and they have been found to range from extremely long half-lives of 10 billion years to extremely short half-lives of fractions of a second.

The quantity of radioactive nuclei at any given time will decrease to half as much in one half-life. For example, if there were 100g of Cf-251 in a sample at some time, after 800 years, there would be 50g of Cf-251 remaining and after another 800 years (1600 years total), there would only be 25g

remaining.

Remember, the half-life is the time it takes for half of your sample, no matter how much you have, to remain. Each half-life will follow the same general pattern as Cf

-251. The only difference is the length of time it takes for half of a sample to decay.

Using the graph, what is the half-life of an isotope that produces the following graph of decay over time:

Solution

We know that the half-life is the time it takes for half of a sample to change. How long did it take for half of our isotope to change? It took approximately 200 years for 100% of our sample to leave only 50% (half of the original amount) remaining. The half-life is 200 years.

*Notice that after another 200 years (400 years total), 25% remains (half of 50%)

Look carefully at the graph in the previous example. All types of radioactive decay make a graph of the same general shape. The only difference is the scale and units of the x -axis, as the half-life time will be different.

If there are 60 grams of Np -240 present, how much Np-240 will remain after 4 hours? (Np -240 has a half-life of 1 hour)

Solution

Np -240 with a half life of only 1 hour.

After 4 hours, only 3.75g of our original 60g sample would remain the radioactive isotope Np -240.

A sample of Ac -225 originally contained 80 grams and after 50 days only 2.55 grams of the original Ac-225 remain. What is the half life of Ac -225?

Solution

We are going to tackle this problem similar to the last problem. The difference is that we are looking for the half-life time. Let's set up a similar table, though:

We know that 50 days is the same as 5 half-lives. Therefore, 1 half-life is 10 days. The half-life of Ac -225 is 10 days.