Problem 42 - Physics

Question

The half-life for the \(\alpha\) decay of uranium \(\frac{238}{92} \mathrm{U}\) is \(4.47 \times 10^{9} \mathrm{yr} .\) Determine the age (in years) of a rock specimen that contains sixty percent of its original number of \(\frac{238}{92} \mathrm{U}\) atoms.

Step-by-Step Solution

Answer

The age of the rock specimen is approximately \(3.29 \times 10^9\) years.

Step 1: Understanding Half-Life

The half-life is the time required for half of a radioactive substance to decay. Here, the half-life of uranium-238 is given as \(4.47 \times 10^9\) years.

Step 2: Using the Decay Formula

The decay formula is \( N(t) = N_0 \cdot \left( \frac{1}{2} \right)^{t/T} \), where \(N(t)\) is the remaining quantity, \(N_0\) is the initial quantity, \(t\) is the time, and \(T\) is the half-life. Since the specimen contains 60% of its original amount, \( N(t) = 0.6N_0 \).

Step 3: Set Up the Equation

Substitute the given values into the decay formula: \( 0.6N_0 = N_0 \cdot \left( \frac{1}{2} \right)^{t/(4.47 \times 10^9)} \). Cancel \(N_0\) from both sides to simplify: \( 0.6 = \left( \frac{1}{2} \right)^{t/(4.47 \times 10^9)} \).

Step 4: Solve for Time \(t\)

Take the natural logarithm of both sides to solve for \(t\): \( \ln(0.6) = \frac{t}{4.47 \times 10^9} \cdot \ln\left(\frac{1}{2}\right) \). Now solve for \(t\): \[ t = \frac{\ln(0.6)}{\ln(0.5)} \cdot 4.47 \times 10^9. \]

Step 5: Calculate \(t\)

Plug the values into a calculator: \( \ln(0.6) \approx -0.5108 \) and \( \ln(0.5) \approx -0.6931 \). Thus, \[ t \approx \frac{-0.5108}{-0.6931} \cdot 4.47 \times 10^9 \approx 0.737 \cdot 4.47 \times 10^9 \approx 3.29 \times 10^9. \] The age of the rock specimen is approximately \(3.29 \times 10^9\) years.

Key Concepts

Uranium-238 Decay

Uranium-238 is a naturally occurring isotope found in the Earth's crust. It's known for its radioactive properties, decaying over time by emitting alpha particles. This type of decay results in the formation of a new element, thorium-234. This process is called "alpha decay" and takes a very long time.

One key measure of Uranium-238's decay is its half-life, the time it takes for half of the uranium atoms in a sample to decay. The half-life of Uranium-238 is incredibly long, about 4.47 billion years. This makes it suitable for dating very old geological formations.

In the context of rocks and fossils, scientists use the decay measurement to estimate their age. When Uranium-238 decays, it gradually diminishes, allowing researchers to determine how long it's been decaying and thus the age of the rock.

Radioactive Decay Formula

The radioactive decay formula is essential for understanding how substances like Uranium-238 change over time. This formula models the decrease in the number of radioactive atoms in a sample.

The general decay formula is given by:

• \( N(t) = N_0 \cdot \left( \frac{1}{2} \right)^{t/T} \)

In this formula:

• \(N(t)\) is the remaining quantity of the substance at time \(t\).
• \(N_0\) is the initial quantity of the substance.
• \(T\) represents the half-life of the substance.
• The exponent \(t/T\) represents the number of half-lives that have passed.

This formula allows you to plug in the percentage of the remaining substance to find the time that has passed. For example, if 60% of the original Uranium-238 remains, you can determine the age of the specimen.

Natural Logarithm Calculation

Natural logarithms are used in radioactive decay calculations to solve exponential equations. When you have an equation involving exponential decay, taking the natural logarithm of both sides can simplify finding the time \(t\).

The natural logarithm, denoted as \(\ln\), is the logarithm to the base \(e\), where \(e\) is approximately 2.718. In our specific example of finding the age of a rock, you set up the equation:

• \(0.6 = \left( \frac{1}{2} \right)^{t/(4.47 \times 10^9)}\)

Take the natural logarithm of both sides:

• \(\ln(0.6) = \frac{t}{4.47 \times 10^9} \cdot \ln\left(\frac{1}{2}\right)\)

You can then solve for \(t\) by rearranging the equation:

• \(t = \frac{\ln(0.6)}{\ln(0.5)} \cdot 4.47 \times 10^9\)

This approach helps find how many years have passed since the original cost of the uranium atoms has decreased, providing an estimate for the age of the rock.