Question

The exponential formula for the half-life of a radioactive isotope is y=y0ekt, where y is the amount of the isotope remaining after t years, y0 is the initial amount of the isotope, k is the decay constant, and e is the transcendental number approximately equal to 2.71828.

Answer 1

See explanation below.

Step-by-step explanation:

For this case we atart fom the proportional model given by the following differential equation:

`(dA)/(dt)= kA`

And if we rewrite this expression we got:

`(dA)/(A) = k dt`

If we integrate both sides we got:

`ln|A| = kt +C`

And using exponential on both sides we got:

`A(t) = e^(kt) e^C = A_o e^(kt)`

Where
`A_o` represent the initial amount for the isotope and t the time in years and A the amount remaining.

If we want to apply a model for the half life we know that after some time definfd
`t_(1/2)` the amount remaining is the hal, so if we apply this we got:

`(A_o)/(2) = A_o e^{kt_(1/2)}`

We can cancel
`A_o` and we got:

`(1)/(2)= e^{kt_(1/2)}`

If we solve for k we can apply natural log on both sides and we got:

`ln ((1)/(2)) = kt_(1/2)`

`k = (ln(1/2))/(t_(1/2))`

And that would be our proportional constant on this case.

If we replace this value for k int our model we will see that:

`A(t) =A_o e^{ (ln(1/2))/(t_(1/2)) t}`

And using properties of logs we can rewrite this like that:

`A(t) = A_o ((1)/(2))^{(t)/(t_(1/2))}`

And thats the common formula used for the helf life time.