Question

Question #39370

Answer 1

`1/256`

Explanation:

You know that the nuclear half-life of a radioactive nuclide, `t_"1/2"`, tells you the time that must pass in order for half of an initial sample of said to nuclide to undergo radioactive decay.

This means that with every passing half-life, the mass of the sample gets reduced by half.

If you take `A_0` to be the initial mass of the sample, you can say that you will be left with

`A_t = A_0 * (1/2)^color(red)(n)`

Here

• `A_t` is the mass of the sample that remains undecayed after a period of time `t`
• `color(red)(n)` represents the number of half-lives that pass in a given time period `t`

In your case, you know that

`t_"1.2" = "3 hours"`

You also know that the total time that passes is equal to

`"1 day = 24 hours"`

You can thus say that you have

`color(red)(n) = (24 color(red)(cancel(color(black)("hours"))))/(3color(red)(cancel(color(black)("hours")))) = color(red)(8) ->`this means that eight half-lives pass in a `"24-hour"` period.

Therefore, the mass of the sample that remains undecayed is equal to

`A_t = A_0 * (1/2)^color(red)(8)`

`A_t = A_0 * 1/256`

To find the fraction that remains undecayed, simply divide the amount that remains undecayed by the initial amount

`A_t/A_0 = (color(red)(cancel(color(black)(A_0))) * 1/256)/color(red)(cancel(color(black)(A_0))) = 1/256`