A rock sample contains 75 atoms of a parent isotope and 25 atoms of a daughter isotope. The half-life of the parent isotope is 100 years. How old is this rock?

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Answer 1 · 2016-10-01T14:15:39

The rock is 41.5 years old.

For each half-life, you divide the total amount of the isotope by 2, so

#"Amount remaining" = "original amount"/2^n#, where #n = "the number of half-lives"#.

#A = A_0/2^n#

You can rearrange this to

#A_0/A = 2^n#

If there were originally 100 atoms, and 75 of the atoms remain, we have

#100/75 = 2^n#

#4/3 = 2^n#

#log(4/3) = nlog2#

#n = log(4/3)/log2 = 0.1249/0.3010 = 0.415#

#t = 0.415t_(1/2) = "0.415 × 100 yr" = "41.5 yr""#

The rock is 41.5 years old.