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
Dec 2, 2015

Answer:

#"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")#