Question

Regolith is sampled on mars and he concentration of potassium-40 is 120 ppm. In a crater, potassium-40 is only 12 ppm. If the half life of potassium-40 is 1.2 billion years, how long ago was this crater formed?

I do not understand how to get the answer.

Answer 1

`4.98` billion years

Explanation:

The half-life decay of some elements can be used to calculate how long a particular sample has been around, if the initial or baseline value is known. The "half-life" is the amount of time it takes for `1/2` of an element to decay (radioactive) into a different element.

Sometimes we can just do a serial division by two to get to the desired number. More accurately we have a specific equation that can do this for any value. I'll show the approximation first, then the equation.
Half-life decay approximation:
120ppm Start -> 1 half-life results in 60 ppm left (1.2B yr)
60ppm -> 1 half-life results in 30 ppm left (1.2B yr)
30ppm -> 1 half-life results in 15 ppm left (1.2B yr)
15ppm -> 1 half-life results in 7.5 ppm left (1.2B yr)
From this we can see that the sample is between 3.6 and 4.8 billion years, because 3-4 half-lifes were required.

The equation is: `[A]_t = [A_0]e^(-0.693t/t_(1/2))`
`12 = 120(e^(-0.693t/1.5)) ; 0.1 = e^(-0.693t/1.5)`
`ln 0.1 = -0.693t/1.5 ; -2.303 = -0.693t/1.5`
`-3.45 = -0.693t`; `t = 4.98` billion years.

A short-cut equation is :
Original Amount/Final Amount `= 2^k`
k * Half-life = years to final amount

`120/12 = 10` ; `10 = 2^x ; log10 = xlog2`
`1 = x xx 0.301`; `x = 3.322`

`3.322 * 1.5 = 4.98` billion years