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.
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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.