Half life mass of sample

Question

Lead-210 has a half life of 20.4 years. What mass of a 2.00 g sample of Lead-210 will remain after 61.2 years?

Answer 1

For a 1st order decay, we can use FR = 0.5ⁿ (note: this will work even if the number of half lives isn't a whole number)

FR = fraction remaining = ?

n = number of half lives that have elapsed = 61.2 yrs x 1 half life / 20.4 yrs = 61.2 / 20.4 = 3 half lives

FR = 0.5³ = 0.125

Mass remaining = 2.00 g x 0.125 = 0.250 g (3 sig. figs.)

Answer 2

First let's define what "half life" means. A half life is the amount of time required for half the sample to remain. In other words if we have a 1g sample with a half life of 1 minute, then in 1 minute we will have 0.5g, then in another minute we will have 0.25g, then 0.125, and so on.

Applying the same principle, we have a sample of 2.00g that has a half life of 20.4 years. To figure out how many half-lives we have, we can divide the amount of time that has passed by the half life:

1. 61.2 years/20.4 years per half life = 3 half lives

Now that we know how many half lives there are, we can calculate how much of the sample remains. There is an equation you can look up for half lives, but with a nice round number like 3 we can calculate it ourselves. Remember each half live divides the sample by 2.

2. Start with 2.00g ---> 1.00g ---> 0.50g ---> 0.25g (each arrow is a half life)

Thus we end up with only 0.25g left of the sample.