Decay and half-life.

Step 1 of 5

A decaying sample

The nucleus of a radioactive atom is unstable: at some moment it gives off radiation and turns into a different nucleus. There is no knowing when a particular nucleus will decay, but a large sample follows a steady law: in each fixed stretch of time, half of the nuclei that are left decay. That stretch of time is called the Half-life.

This is exponential decay, as in the last Lesson: in each half-life the amount is multiplied by $\frac{1}{2}$⁠. The orange nuclei have not decayed yet, and the blue ones have. Here the half-life is 5 s. Watch the sample and predict how much of it will be left after two half-lives.

Step 2 of 5

Halving again

After $n$⁠ half-lives, $\left(\frac{1}{2}\right)^n = \frac{1}{2^n}$$\left(\frac{1}{2}\right)^n =$$\frac{1}{2^n}$⁠ of the sample is left. That is exactly the board from the first Lesson, turned round: instead of doubling, you halve. Choose On under Show the decay curve when you want to check.

Step 3 of 5

How long?

Here Half-life is locked at 4 s. To find how long it takes to reach a given fraction, first ask how many half-lives you need, and that is a question for a logarithm. Then multiply by the length of each one.

Step 4 of 5

A handful of nuclei

In a sample of $400$⁠ nuclei the orange line stays almost on the white curve, the ideal halving. Now the sample holds only $20$⁠ nuclei. Choose On under Show the decay curve and watch a few rounds: the sample starts afresh after five half-lives.

Step 5 of 5

Under one per cent

How many half-lives does it take until less than $1\%$⁠ of the sample is left? $1\%$⁠ is $\frac{1}{100}$⁠, so you are looking for the first power of $2$⁠ that passes $100$⁠. That is another logarithm: $\log_2 100$⁠. The graph here shows only five half-lives, and after them $\frac{1}{32}$⁠ is still left, more than $3\%$⁠.