Problem 6 - Chemistry: The Central Science

Question

A rock contains \(0.257 \mathrm{mg}\) of lead-206 for every milligram of uranium-238. The half-life for the decay of uranium-238 to lead-206 is \(4.5 \times 10^{5} \mathrm{yr}\). How old is the rock? SOLUTION Analyze We are told that a rock sample has a certain amount of lead206 for every unit mass of uranium-238 and asked to estimate the age of the rock. Plan Lead-206 is the product of the radioactive decay of uranium-238. We will assume that the only source of lead-206 in the rock is from the decay of uranium-238, with a known half-life. To apply firstorder kinetics expressions (Equations \(21.19\) and 21.20) to calculate the time elapsed since the rock was formed, we first need to calculate how much initial uranium-238 there was for every \(1 \mathrm{mg}\) that remains today. Solve Let's assume that the rock currently contains \(1.000 \mathrm{mg}\) of uranium-238 and therefore \(0.257 \mathrm{mg}\) of lead-206. The amount of uranium-238 in the rock when it was first formed therefore equals \(1.000 \mathrm{mg}\) plus the quantity that has decayed to lead-206. Because the mass of lead atoms is not the same as the mass of uranium atoms, we cannot just add \(1.000 \mathrm{mg}\) and \(0.257 \mathrm{mg}\). We have to multiply the present mass of lead-206 \((0.257 \mathrm{mg})\) by the ratio of the mass number of uranium to that of lead, into which it has decayed. Therefore, the original mass of \({ }_{92}^{239} \mathrm{U}\) was $$ \text { Original } \begin{aligned} { }_{98}^{238} \mathrm{U} &=1.000 \mathrm{mg}+\frac{238}{206}(0.257 \mathrm{mg}) \\ &=1.297 \mathrm{mg} \end{aligned} $$ Using Equation 21.20, we can calculate the decay constant for the process from its half-life: $$ k=\frac{0.693}{4.5 \times 10^{9} \mathrm{yr}}=1.5 \times 10^{-10} \mathrm{yr}^{-1} $$ Rearranging Equation \(21.19\) to solve for time, \(t\), and substituting known quantities gives $$ t=-\frac{1}{k} \ln \frac{N_{t}}{N_{0}}=-\frac{1}{1.5 \times 10^{-10} \mathrm{yr}^{-1}} \ln \frac{1.000}{1.297}=1.7 \times 10^{9} \mathrm{yr} $$ Comment To check this result, you could use the fact that the decay of uranium-235 to lead-207 has a half-life of \(7 \times 10^{8} \mathrm{yr}\) and measure the relative amounts of uranium-235 and lead-207 in the rock.

Step-by-Step Solution

Answer

The age of the rock is approximately \(1.7 \times 10^{9}\) years, or 1.7 billion years.

Step 1: Calculate the Original Mass of Uranium-238

: First, we will assume that the rock currently contains 1.000 mg of uranium-238 and 0.257 mg of lead-206. To find the original mass of uranium-238, we need to add the given 1.000 mg to the mass of lead-206 multiplied by the ratio of mass numbers of uranium and lead. The calculation is as follows: Original mass of Uranium-238 = \(1.000\,mg + \frac{238}{206}(0.257\,mg) = 1.297\,mg\)

Step 2: Calculate the Decay Constant

: Next, we will compute the decay constant (k) using the given half-life of uranium-238 decay: \(k = \frac{0.693}{4.5 \times 10^{9}\,yr} = 1.5 \times 10^{-10} yr^{-1}\)

Step 3: Calculate the Age of the Rock

: Now we have all the necessary information to calculate the time elapsed or the age of the rock using the first-order decay equation: \(t = -\frac{1}{k} \ln \frac{N_t}{N_0} = -\frac{1}{1.5 \times 10^{-10}\,yr^{-1}} \ln \frac{1.000}{1.297} = 1.7 \times 10^{9}\,yr\) So, the age of the rock is approximately 1.7 billion years.

Key Concepts

Uranium-Lead Dating

Uranium-lead dating is a technique used by geologists to determine the age of rocks. Naturally occurring uranium-238 undergoes radioactive decay to form lead-206, which is a stable isotope. Here's how it works in simple terms:

• As the uranium in a rock decays over time, it turns into lead.
• By comparing the ratio of uranium-238 to lead-206, scientists can estimate how long this process has been occurring.
• This method is especially useful for dating very old rocks, since the half-life of uranium-238 is about 4.5 billion years.

In this particular exercise, the rock sample has a specific amount of lead-206 for every milligram of uranium-238, which helps calculate the rock's age. The assumption is that the lead-206 present is solely from the uranium-238's decay, making this method highly reliable and accurate for geological dating.

Half-Life Calculation

The concept of half-life is crucial in radioactive dating. It's the time required for half of the radioactive nuclei in a sample to decay. For uranium-238, this is 4.5 billion years. This long half-life makes it a perfect candidate for dating ancient geological formations.

Understanding half-life helps in knowing:

• How quickly or slowly a radioactive isotope decays over time.
• How to utilize this decay process to determine the age of a sample.

In the provided solution, we apply the half-life to calculate the decay constant. The formula used is:
\( k = \frac{0.693}{\text{half-life}} \)
For uranium-238, it helps us find the decay constant, which then plays a role in calculating the time that has elapsed since the rock was formed.

First-Order Kinetics

Radioactive decay, including uranium-238 decaying to lead-206, follows first-order kinetics, which means the rate of decay is proportional to the amount of substance present.

Here's why it's important:

• The expression for first-order kinetics allows us to calculate the time elapsed based on the ratio of remaining uranium to the initial amount.
• In this case, we rearrange the first-order kinetics equation to solve for time (\( t \)), using: \[ t = -\frac{1}{k} \ln \frac{N_t}{N_0} \]
• \( N_t \) is the current amount of uranium-238, \( N_0 \) is the original amount, and \( k \) is the decay constant calculated from the half-life.

This mathematics of this process tells us the age of the rock in question, which turns out to be approximately 1.7 billion years.