Answer:
Age of the rock is 3.02x10^9 years
Explanation:
First, we need to get clear in which is the expression we need to use here:
N = No * e(-kt) (1)
Where:
N: mass, moles, concentration of one element after t time has passed (Depends of the units)
No: innitial mass, mol, concentration of one element
t: time that has passed, it could also be time of half life.
k: constant of decay.
Now, the problem already gave us the ratio between the Ar and K, which is 4.2, this means the following
rAr / rK = 4.2
or simply 4.2 : 1
This means that the innitial concentration of both of them, is just sum the value of both of them so:
No = 4.2 + 1 = 5.2
and the final N, after half life has passed, we can assume it's one (Because Argon is decaying to potassium. If the ratio is 4.2 : 1, means that Argon decays from 4.2 to 1).
With No, N and t, we can calculate the constant of decay, and then, the age of the rock. First, we'll find the constant with the expression:
k = ln(2) / t1/2 (2)
k = ln(2) / 1.27x10^9
k = 5.46x10^-10 yr^-1
With this value, we'll use equation (1) to get the age of the rock
If:
N = No * e(-kt)
then:
N/No = e(-kt)
ln(N/No) = -kt
t = -ln(N/No) / k (2)
Solving for t:
t = -ln(1/5.2) / 5.46x10^-10
t = 3.02x10^9 yr
