Question

Question #3890e

Answer 1

`"1.7 g"`

Explanation:

Your go-to equation when it comes to calculating the amount of a radioactive sample that remains after the passing of a certain period of time looks like this

`color(blue)(A = A_0 * 1/2^n)" "`, where

`A` - the amount of the radioactive substance that remains after the passing of `n` half-lives
`A_0` - the initial mass of the sample
`n` - the number of half-lives that passed

As you know, the nuclear half-life of a radioactive substance is defined as the time needed for half of the atoms of that sample to undergo radioactive decay.

In other words, the nuclear half-life tells you how much time must before before a sample of a radioactive substance is reduced to half of its original mass.

In your case, the half-life of plutonium is known to `2.4 * 10^(4)` years. To determine the value of `n`, the number of half-lives that pass in a given period of time, divide the respective period of time by the half-life

`color(blue)(n = "total time"/"half-life")`

`n = (5.0 * 10^3color(red)(cancel(color(black)("years"))))/(2.4 * 10^4color(red)(cancel(color(black)("years")))) = 25/12 * 10^(-1) = 5/24`

This means that the amount of plutonium that remains undecayed after that much time will be

`A = "2.0 g" * 1/2^(5/24) = color(green)("1.7 g")`