Solutions

KM#12: Age of the Earth (Level 4)
Mathematical Content: Basic algebra; exponential equations; logarithms; linear regression.

Part 1: Worked solutions are provided in modules at the “clicks” and following the answers for parts I and II below:

Answers Part I
Question 1. What is the value of the decay constant for rubidium-strontium?

λ = 1.42 x10^-5

Question 2. Suppose that in an isotope system, the parent decays into the daughter, and there was no daughter atoms present initially. What is the ratio of daughter atoms to parent atoms at time t?

Question 3. What is the age of the gabbro from Electra Lake?
The gabbro is, approximately, 1,429 million years old.

Solution for Part 2
Question. How old is the Earth?
In order to answer this question, you can repeat the procedure given in the dating of the gabbro; specifically:

Plot the data on an (x, y)-axis system to create the isochron diagram
Use linear regression to find the slope of the line of best fit
Set the slope equal to
Solve for t.

The slope is m = 0.065 (and ).
Solve

The age is approximately 4435 million years, that is, 4.435 billion years.

Worked Solutions for Part I
Solution to Question 1. Use the basic decay equation P(t) = P₀e^-λt. When
t = half-life for Rb-Sr = 4.88 x 10⁴ million years,
then the initial number of atoms is reduced by half (from to ) , so denoting half-life by h, set t = h and solve for l.
,

So now we solve for λ, with h = 4.88 x 10⁴.

Cancel P₀ from each side, then take the natural logarithm of both sides,

But

so
.

We have found the decay constant λ = 1.42 x10^-5 for the Rb-Sr system

.
Solution to Question 2. First, solve the basic decay equation for P₀,

Now let D(t) denote the number of daughter atoms at time t. The number of daughter atoms at time t is equal to the number of parent atoms present initially minus the number of parent atoms at time t, so