# 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?

Dec 2, 2015

$\text{25 g}$

#### 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

• $A \cdot \frac{1}{2} = \frac{A}{2} \to$ after the passing of one half-life
• $\frac{A}{2} \cdot \frac{1}{2} = \frac{A}{4} \to$ after the passing of two half-lives
• $\frac{A}{4} \cdot \frac{1}{2} = \frac{A}{8} \to$ after the passing of three half-lives
$\vdots$

and so on.

So, if you start with $\text{100 g}$ of thorium-234, you can say that you'll be left with

• $\text{100 g" * 1/2 = "50 g} \to$ after $24$ days
• $\text{50 g" * 1/2 = "25 g} \to$ after $48$ days

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

$\textcolor{b l u e}{A = {A}_{0} \cdot \frac{1}{2} ^ n} \text{ }$, where

${A}_{0}$ - the initial mass of the sample
$n$ - the number of half-lives that pass in a given period of time.

$n = \left(48 \textcolor{red}{\cancel{\textcolor{b l a c k}{\text{days"))))/(24color(red)(cancel(color(black)("days}}}}\right) = 2$
A = "100 g" * 1/2^2 = "100 g" * 1/4 = color(green)("25 g")