## Half-Lives

The half-life of a radioactive isotope is the time it takes for 50% of the particles in a sample (of the same isotope) to undergo radioactive decay. Note that this says nothing about the time it will take one individual particle to decay. Since radioactive isotopes are decaying at every moment, the half-life defines the time interval when the first 50% of a sample have decayed. It is important to notr that after two half-live, 25% of the original sample (one-half of one-half) remains. After three half-lives, the percentage is down to 12.5%, or one-eighth, equivalent to one-half of one-half of one-half. Given the half-life of an isotope, the amount of an isotope that remains after a certain interval of time can be calculated using the formula below:

## Half-Life Problems - How Much Remains?

If a 100. g sample of thorium-230 is analyzed, how much is left after 3.2×105 years?

First, find the half-life of the thorium-230. If this information is not given in the problem, it must be found in a table. Next, make sure the half-life's units of time agree with the length of time given in the problem. In this case, both are given in years, so there is no conversion necessary. If this is not the case, one value must be converted into the other's unit of time. For converting days to years, it is strongly recommended that 365.25 days be used as the equivalent for one year. Although it may seem negligible, the accumulation of many years may result in the loss of a multitude of days. Be sure to plug in the initial mass into A0, as the subscript 0 indicates it represents initial mass. The variable A represents the mass that remains.

## Half-Life Problems - How Much Has Decayed?

How much iron-55 will decay if a 112.4 g sample is left to decay for 225.3 days?

The trick to a problem that asks for hiow much has decayed is to realize that you are not done after you have solved for A. The difference between A0 (the initial mass) and A (the final mass) is equal to the mass that has decayed.

## Radioactivty - Graphical Perspective

Consider the data shown below, which examines the rate at which four random isotopes, W, X, Y, and Z undergo radioactive decay:

A graph of this data is shown below:

The blue data points represent isotope W, the red data points represent isotope X, the yellow data points represent isotope Y, and the green data points represent isotope Z. The half-life of each isotope can be estimated from the graph. Since the original amount of material was 120 g, after one half-life has passed 60 g will remain. A line can be drawn from the mass axis (y-axis) to intercept the curves. From there, the corresponding time coordinate (x-axis) can be estimated to give a rough idea of the value for the half-life. For isotope W, the half-life is approximately 17 days, for X it is about 24 days, for Y it is about 4 days, and finally for Z the half-life is approximately 45 days. Likewise, isotope Y with the shortest half-life can be assumed to be the least stable while isotope Z with the longest half-life can be inferred to be the most stable.