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?
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.
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 a total of 3 half - lives has elapsed.