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?

1 Answer
Nov 13, 2016

Answer:

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"#