Half-life

Radioactive decay proceeds according to a principal called the half-life. With the use of half life we can possibly predict what fraction of nuclei in a sample will decay within any given time. This rate of decay is constant for each isotope and different from any other. Thus, half life is the amount of time necessary for one-half of the nuclide of a sample to decay.

For example, the radioactive element bismuth (²¹⁰Bi) can undergo alpha decay to form the element thallium (²⁰⁶Tl) with a reaction half-life equal to five days. If we begin an experiment starting with 100 g of bismuth in a sealed lead container, after five days we will have 50 g of bismuth and 50 g of thallium in the jar. After another five days (ten from the starting point), one-half of the remaining bismuth will decay and we will be left with 25 g of bismuth and 75 g of thallium in the jar. As illustrated, the reaction proceeds in halfs, with half of whatever is left of the radioactive element decaying every half-life period.

If half-life of an isotope is represented by T, then the fraction of a sample that remain after time t is

Fraction Remaining = (1/2 ^t/T or (1/2)ⁿ

Where t/T or n is the number of half-life periods that have occurred. 

Example

               A 10.0 g sample of ²⁴Na is used to study the rate of blood flow in the circulatory system. If ²⁴Na has a half-life of 15 hours, how much is left after 60 hours.

Solution
 
       Method 1. Reduce the mass by one-half for each successive period until the total time reached.

After 15 hours, one-half of 10.0 g or 5.0 g will remain.

After 30 hours, one-half of 5.0 g or 2.5 g will remain.

After 45 hours, one-half of 2.5 g or 1.25 g will remain.

After 60 hours, one-half of 1.25 g or 0.625 g will remain.

Method 2. Divide the total time by the half-life to find the number of half-life periods, and then use the formula:

Fraction Remaining = (1/2)ⁿ where n is the number of half-life periods

In this case, n = 60/15 = 4

Fraction Remaining = (½)⁴= (1/16)

                                                   (1/16) x 10.0 g = 0.625 g


Click HERE for the list of Radioactive Elements and their Half-lives