Question

Thorium-234 has a half-life of 24 days. if you started with 100 gram sample of thorium-234, how much would remain after 48 days?

Answer 1

`"25 g"`

Explanation:

Think about what a nuclear half-life represents, i.e. the time needed for an initial sample of a radioactive substance to be halved.

In your case, you know that thorium-234 has a half-life of `24` days. That means that every `24` days, half of the atoms of thorium you have in your sample will decay.

This is of course equivalent to saying that every `24` days, you'll be left with half of the atoms of thorium you have in your sample.

So, if you start with `A` grams of thorium-234, you can say that you'll be left with

• `A * 1/2 = A/2 ->` after the passing of one half-life
• `A/2 * 1/2 = A/4 ->` after the passing of two half-lives
• `A/4 * 1/2 = A/8 ->` after the passing of three half-lives
  `vdots`

and so on.

So, if you start with `"100 g"` of thorium-234, you can say that you'll be left with

• `"100 g" * 1/2 = "50 g" ->` after `24` days
• `"50 g" * 1/2 = "25 g" ->` after `48` days

As you can see, you can calculate the amount of a sample that remains undecayed by using the equation

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

`A_0` - the initial mass of the sample
`n` - the number of half-lives that pass in a given period of time.

In your case, you'd have

`n = (48 color(red)(cancel(color(black)("days"))))/(24color(red)(cancel(color(black)("days")))) = 2`

Therefore,

`A = "100 g" * 1/2^2 = "100 g" * 1/4 = color(green)("25 g")`