Question

If the half-life of uranium-232 is 70 years, how many half-lives will it take for 10 g of it to be reduced to 1.25 g?

Answer 1

It will take 210 years.

Explanation:

The formula for radioactive decay is

`color(blue)(bar(ul(|color(white)(a/a) N/N_0 = (1/2)^n color(white)(a/a)|)))" "`

where

`N_0 = "original amount of isotope"`
`N = "amount of isotope remaining"`
`n = "number of half-lives"`

and

`n = t/t_½`

where

`t = "the time for the decay"`
`t_½ = "the half-life"`

In your problem,

`N_0 = "10 g"`
`N = "1.25 g"`
`t_½ = "70 years"`

`N/N_0 = (1/2)^n`

`(1.25 color(red)(cancel(color(black)("g"))))/(10 color(red)(cancel(color(black)("g")))) = (1/2)^n`

`1/8 = 1/2^n`

`n = 3`

So, the uranium has decayed for 3 half-lives.

`n = t/t_½`

`t = nt_½ = "3 × 70 years" = "210 years"`