Question

part ii: absolute dating
instructions: solve the radiometric decay problems below. show or explain your work.
information you will need:

• uranium - 238 to lead - 206 - half - life 4.5 billion years
• uranium - 235 to lead - 207 - half - life 713 million years
• carbon - 14 to nitrogen - 14 - half - life 5730 years

1. a zircon crystal from a lava flow was examined for geologic age. the scientists measured the 235u, and 207pb, and determined that 25% of the atoms were 235u, and 75% were 207pb.

a. how many half - lives have passed since the zircon formed?
b. what is the absolute age of the lava flow?
c. what is the age of the rocks below the lava flow?
d. what is the age of the rocks above the lava flow?

Explanation:

Step1: Determine the number of half - lives for part a

We use the formula \(N = N_0\times(\frac{1}{2})^n\), where \(N\) is the remaining amount of the parent isotope (\(N = 0.25N_0\) since \(25\%\) of \(^{235}U\) remains), and \(N_0\) is the initial amount of the parent isotope.
Substituting \(N = 0.25N_0\) into the formula \(0.25N_0=N_0\times(\frac{1}{2})^n\).
Dividing both sides by \(N_0\) (since \(N_0
eq0\)), we get \(0.25 = (\frac{1}{2})^n\).
We know that \((\frac{1}{2})^2=\frac{1}{4}=0.25\), so \(n = 2\) half - lives.

Step2: Calculate the absolute age for part b

The half - life of \(^{235}U\) to \(^{207}Pb\) is \(t_{1/2}=713\) million years.
Using the formula \(t=n\times t_{1/2}\), with \(n = 2\) (from part a) and \(t_{1/2}=713\) million years.
\(t=2\times713\) million years \( = 1426\) million years.

Step3: Determine the age of rocks below for part c

Rocks below the lava flow were formed before the lava flow. So they are older than the lava flow. Since the lava flow is \(1426\) million years old, the rocks below are older than \(1426\) million years.

Step4: Determine the age of rocks above for part d

Rocks above the lava flow were formed after the lava flow. So they are younger than the lava flow. Since the lava flow is \(1426\) million years old, the rocks above are younger than \(1426\) million years.

Answer:

a. \(2\) half - lives.
b. \(1426\) million years.
c. Older than \(1426\) million years.
d. Younger than \(1426\) million years.