Problem 76P

A 68-mg sample of a radioactive nuclide is administered to a patient to obtain an image of her thyroid. If the nuclide has a half-life of 12 hours, how much of the nuclide remains in the patient after 4.0 days?

Given:

Mass of the sample= 68mg

Half life of nuclide = 12 hours

Know:

How much of the nuclide remains in the patient after 4.0 days.

Explanation:

Half-life is the amount of time it takes for 1/2 of a radioactive substance to decay to something else. For example, if the half-life of this particular isotope is 12 hours, that means every 12 hours, the amount remaining will be cut in half.

So if you start with 68 mg, after 12 hours there will be 34 mg left, since 34 is half of 68. The other 34 grams will have decayed into something else.

After another 12 hours (total of 24 hours), there will be 17 mg of the original sample left...

After a total of 36 hours, there will be 8.5 mg left...

...and so on.

To put it another way: after 1 half-life there is 1/2 of the sample remaining.

After 2 half-lives there is 1/2 1/2, or (1/2)2, or 1/4 of the sample remaining.

After 3 half-lives there is 1/2 1/2 1/2, or (1/2)3, or 1/8 of the sample remaining.

After 4 half-lives there is 1/2 1/2 1/2 1/2, or (1/2)4, or 1/16 of the sample remaining.

So if you know how many half-lives have passed, you can raise 1/2 to that power, and that's what fraction of the original amount will remain.

For example, 4.0 days is 96 hours. That's 8.0 half-lives, so the fraction remaining would be (1/2)8.0, or 0.003906

Take that fraction of 58 mg, and you'll know how many milligrams remain after 4.4 days.

68 mg 0.003906 = 0.265 mg

Thus 0.265mg of the nuclide remains in the patient after 4.0 days.