Question

What is the rock's age? A rock contains one-fourth of its original amount of potassium-40 The half-life of potassium-40 is 1.3 billion years?

Answer 1

The rock's age is approximately `"2.6 billion"` years.

There are essentially two ways of solving nuclear half-life problems. One way is by applying the half-life formula, which is

`A(t) = A_0(t) * (1/2)^(t/t_(1/2))` , where

`A(t)` - the quantity that remains and has not yet decayed after a time t;
`A_0(t)` - the initial quantity of the substance that will decay;
`t_(1/2)` - the half-life of the decaying quantity;

In this case, the rock contains `"1/4th"` of the orignal amount of potassium-40, which means `A(t)` will be equal to `(A_0(t))/4`. Plug this into the equation above and you'll get

`(A_0(t))/4 = A_0(t) * (1/2)^(t/t_(1/2))`, or `1/4 = (1/2)^(t/t_(1/2))`

This means that `t/t_(1/2) = 2`, since `1/4 = (1/2)^2`.

Therefore,

`t = 2 * t_(1/2) = 2 * "1.3 = 2.6 billion years"`

A quicker way to solve this problem is by recognizing that the initial amount of the substance you have is halved with the passing of each half-life, or `t_(1/2)`.

This means that you'll get

`A = (A_0)/2` after the first 1.3 billion years

`A = (A_0)/4` after another 1.3 billion years, or `2 * "1.3 billion"`

`A = (A_0)/8` after another 1.3 billion years, or `2 * (2 * "1.3 billion")`

and so on...