# 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?

##### 1 Answer

#### 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 **every** **half** of the atoms of thorium you have in your sample will decay.

This is of course equivalent to saying that **every** *left with* **half** of the atoms of thorium you have in your sample.

So, if you start with

#A * 1/2 = A/2 -># after the passing ofonehalf-life#A/2 * 1/2 = A/4 -># after the passing oftwohalf-lives#A/4 * 1/2 = A/8 -># after the passing ofthreehalf-lives

#vdots#

and so on.

So, if you start 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

*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")#