How Old Is The Rock?

Question

A certain element has a half life of 4.5 billion years.

a. You find a rock containing a mixture of the element and lead. You determine that 35​% of the original element​ remains; the other 65​% decayed into lead. How old is the​ rock?

b. Analysis of another rock shows that it contains 15​% of its original​ element; the other 85​% decayed into lead. How old is the​ rock?

Answer 1

For half life:

Let N₀= initial amount = 100% = 100

Equation (1) for amount at any given time:

N(t) = N₀ x e^-kt , where k is a constant and t is the given time.

What do we know?

N(4.5 by) = 50 = 100e^{-k (4.5 by)} ---> We have 50% of the original amount @ t = 4,500,000,000 yrs

Solve for k:

50/100 = e^{-k (4.5 by)}

---> take the natural log

ln (1/2) = - 4.5by (k) x ln e , but ln e = 1

so,

k = (ln 0.5) / -4500000000

k = 1.54 x 10^-10

Now substitute on Equation (1):

N(t) = 35 = N₀ x e^-kt ---> We have 35% of the original amount @ t = unknown

N(t) = 35 = 100e^-kt

35/100 = e^-kt

---> take the natural log

ln (0.35) = - 1.54x10^-10t x ln e , but ln e = 1

so,

ln 0.35 = -1.54x10^-10 x t

therefore,

t = (ln 0.35) / -1.54 x 10^-10

t = 6.8 x 10⁶ which is 6.8 by old.

Answer for (a): The rock is 6.8 billion years old.

***This makes senses because it only has 35% left of the original material

For letter (b), you need to do the same thing as we did for (a). You already solved the constant k. You just need to plug in the values.