Question

A few questions about half-lives? If anybody can look at the questions at least, please help.

Answer 1

Q1. After three half-lives of an isotope, 1 billion of the original isotope's atoms still remain in a certain amount of this element. How many atoms of the daughter product would you expect to be present?

Q2. By measuring the amounts of parent isotope and daughter product in the minerals contained in a rock, and by knowing the half-life of the parent isotope, a geologist can calculate the absolute age of the rock. A rock contains 125 g of a radioisotope with a half-life of 150 000 years and 875 g of its daughter product. How old is the rock according to the radiometric dating method?

I did everything else I am just stuck on these.

Answer 2

After each half-life, you have 1/2 the previous amount of the parent. After 3 half-lives, what is the remaining fraction of the parent?

Answer 3

You can write this in a formula as \[P(t) = P_0(2)^{-t/t_{1/2}}\]where \(P_0\) is the original amount, and \(t_{1/2}\) is the half life.

After 3 half lives,
\[P(3*t_{1/2}) = P_0(2)^{-3} = \frac{P_0}{8}\]

That means (if we assume that the daughter product hasn't subsequently decayed into something else, not always a good assumption!) that we should have \[P_0 - \frac{P_0}{8} = \]as the daughter product quantity

Answer 4

We could write another equation for the population of the daughter product:

\[D(t) = P_0(1-P(t)) = P_0(1-2^{-t/t_{1/2}}\]

Here's a graph that shows the relative size of parent and daughter as a function of how many half-lives have passed. You can clearly see that they are equal at 1 half-life (as we would expect).