Question

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?

Answer 1

`sf(8color(white)(x)"hr")`

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:

`sf(-N_t/tpropN_t)`

Putting in the constant:

`sf(-N_t/t=lambdat)`

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

`sf(N_t=N_0e^(-lambdat)" "color(red)((1)))`

`sf(N_t)` is the number of undecayed atoms at time `sf(t)`

`sf(N_0)` is the initial number of undecayed atoms

`sf(lambda)` is the decay constant

`sf(t)` is the time.

The data given will enable us to calculate `sf(lambda)` and, from that, the half - life.

Taking natural logs of `sf(color(red)((1))rArr)`

`sf(lnN_t=lnN_0-lambdat)`

`:.` `sf(ln(N_t/N_0)=-lambdat)`

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

`sf(ln(1250/10000)=-lambdaxx24)`

`:.` `sf(ln(1/8)=-lambdaxx24)`

`sf(-lambda=-2.079/24)`

`sf(lambda=0.0866color(white)(x)"hr"^(-1))`

We can find the 1/2 life by setting the condition that when `sf(N_t=N_0/2)` then `sf(t=t_(1/2))`

Substituting these into `sf(color(red)((1))rArr)`

`sf(cancel(N_0)/2=cancel(N_0)e^(-lambdat_(1/2)))`

`:.` `sf(ln2=lambdat_(1/2))`

`sf(t_(1/2)=0.693/lambda=0.693/0.0866=8color(white)(x)"hr")`

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:

`sf(10,000rarr5000)` is 1 half life

`sf(5000rarr2500)` is another half life

`sf(2500rarr1250)` is another half - life.

So a total of 3 half - lives has elapsed.

`:.` 3 half - lives = 24 hrs

`:.` 1 half - life =24/3 = 8 hrs.