## How to calculate the half-life of an element

Half-life is defined as the time it takes for one half of the atoms of a particular element to decay.

Suppose you had 100 atoms of element X.

The half life for element X is 10 minutes.

After 10 minutes ½ of the atoms would decay. You would now have 50 atoms of element X.

After 10 more minutes (or 20 minutes total) ½ of the remaining 50 atoms of element X would decay. You would now have 25 atoms of element X.

After 10 more minutes (or 30 minutes total) ½ of the remaining 25 atoms of element X would decay. You would now have 12.5 atoms of element X.

After 10 more minutes (or 40 minutes total) ½ of the remaining 12.5 atoms of element X would decay. You would now have 6.25 atoms of element X.

The formula that represents the process that we just uncovered is:

starting amount x ½^{number of half-lives} = ending amount

Using the formula for the problem above, the starting amount is 100 atoms. The number of half-lives this element underwent was 4.

100 atoms x ½^{4} = 6.25 atoms

(Note: To use your calculator you enter 100 x .5^4)

**Problem 1**

The half-life of Zn-71 is 2.4 minutes. Suppose that you had 100.0 g at the beginning, how many grams would be left after 7.2 minutes has elapsed?

In this problem you are given the half-life = 2.4 minutes.

You are given the starting amount of 100.0 g.

You are not told how many half-lives have have passed, but this is easy to obtain from the total amount of time passed. Divide 7.2 minutes by 2.4 minutes to see that 3 half-lives have passed.

100.0 g x ½^{3} = 12.5 g

**Problem 2**

After 24.0 days, 2.00 g of an original 128.0 g sample remain. What is the half-life of the sample?

This problem can be worked through logic, or through algebra.

Logically working through this problem:

After 1 half-life: 128.0 g becomes 64.0 g. (divide 128.0 by 2)

After 2 half-lives: 64.0 g becomes 32.0 g. (divide 64.0 by 2)

After 3 half-lives: 32.0 g becomes 16.0 g. (divide 32.0 by 2)

After 4 half-lives: 16.0 g becomes 8.0 g. (divide 16.0 by 2)

After 5 half-lives: 8.0 g becomes 4.0 g (divide 8.0 by 2)

After 6 half-lives: 4.0 g becomes 2.0 g (divide 4.0 by 2)

Therefore, it took 6 half-lives for the original 128.0 g sample to decay to 2.0 g. 24 days divided by 6 half-lives gives us a half-life of 4 days.

Working this problem with algebra:

128.0 g x ½^{n} = 2.0 g

To solve for n, we must first determine the decimal fraction of the original sample that remains. We do this through division: 2.0 g / 128.0 g = 0.015625

Next, we determine how many half-lives must have elapsed to get to 0.015625 remaining?

(1/2)^{n} = 0.015625

To solve for n, we must take the log of each side of this equation.

n log 0.5 = log 0.015625

Isolate n by dividing both sides by log 0.5

n = log 0.5 / log 0.015625

n = 6

24 days / 6 half-lives = 4 days (the length of the half-life)

**You try this one:**

Pd-100 has a half-life of 3.6 days. If you had 6.02 x 10^{23} atoms at the start, how many atoms would be present after 20.0 days? (Answer: 20.0 days elapsed / 3.6 days per half-life = 5.56 half-lives have elapsed 6.02 x 10^{23} atoms x ½^{5.56} = 1.28 x 10^{22} atoms remain)

(source-page 7)