# If 1250 counts of an originally 10,000 count radioactive sample are being emitted after one (24 hour) day, what is the half-life of the element?

##### 1 Answer

#### Explanation:

**Formal Method:**

When an atom decays this is a random event which obeys the laws of chance.

The greater the number of undecayed atoms in a sample, the more chance there is of one of them decaying. We can, therefore, say that the rate of decay is proportional to the number of undecayed atoms:

Putting in the constant:

By doing some integration, which I won't go into here, we get the expression for radioactive decay:

The data given will enable us to calculate

Taking natural logs of

The count rate is proportional to the number of undecayed atoms so this becomes:

We can find the 1/2 life by setting the condition that when

Substituting these into

This is an example of 1st order exponential decay and is common in nature.

**Intuitive Method:**

In questions the numbers often drop out nicely and they can be solved intuitively.

The time taken for the count rate to fall by half its initial value is equal to 1 half - life.

So:

So a total of 3 half - lives has elapsed.