Problem 42 - Master Resource Book in JEE Main Physics

Question

The half-life for the ?-decay of uranium \({ }_{92} \mathrm{U}^{238}\) is \(4.47 \times 10^{9} \mathrm{yr}\). If a rock contains sixty percent of its original \({ }_{92} \mathrm{U}^{238}\) atoms, its age is \([\log 6=0.778 ; \log 2=0.3]\) (a) \(3.3 \times 10^{9} \mathrm{yr}\) (b) \(6.6 \times 10^{9} \mathrm{yr}\) (c) \(1.2 \times 10^{8} \mathrm{yr}\) (d) \(5.4 \times 10^{7} \mathrm{yr}\)

Step-by-Step Solution

Answer

The rock is approximately \(3.3 \times 10^9\) years old.

Step 1: Understand the half-life formula

The formula to find the remaining quantity of a radioactive substance after a given time is \( N_t = N_0 \times (1/2)^{t/T} \), where \( N_t \) is the remaining amount, \( N_0 \) is the initial amount, \( T \) is the half-life, and \( t \) is the time elapsed.

Step 2: Determine the remaining quantity

Given that the rock contains sixty percent of its original \( { }_{92} \mathrm{U}^{238} \) atoms, we have \( N_t = 0.6N_0 \).

Step 3: Set up the equation

We substitute the given quantities into the formula: \( 0.6 = (1/2)^{t/T} \). Substitute \( T = 4.47 \times 10^9 \text{ yr} \) into the equation.

Step 4: Solve for the time \( t \)

Take the logarithm of both sides to solve for \( t \): \( \log(0.6) = t \cdot \log(1/2) / T \). Given \( \log(1/2) = -0.3 \), \( \log(0.6) = \log(6) - 1 = 0.778 - 1 = -0.222 \). Substitute these in: \( -0.222 = t \cdot (-0.3) / (4.47 \times 10^9) \).

Step 5: Calculate the value of \( t \)

Rearrange to solve for \( t \): \( t = (-0.222 / -0.3) \times 4.47 \times 10^9 \). Calculate: \( t \approx 0.74 \times 4.47 \times 10^9 \approx 3.3 \times 10^9 \text{ yr} \).

Key Concepts

Half-life calculation

Understanding the concept of half-life is crucial in studying radioactive substances. It describes the time it takes for half of a sample of a radioactive substance to decay. To calculate half-life, we use the formula:\[ N_t = N_0 \times \left( \frac{1}{2} \right)^{t/T} \]where:

• \( N_t \) is the amount remaining after time \( t \).
• \( N_0 \) is the initial amount.
• \( T \) is the half-life period.

This formula helps us determine how much of a substance will remain after a certain period of time has passed. For instance, when a rock is found to contain sixty percent of its original Uranium-238, the above formula helps us calculate its age. The half-life of Uranium-238 is \(4.47 \times 10^9\) years, which is vital for computing how long the decay process has been occurring.

Radioactive decay equations

Radioactive decay follows a predictable pattern, represented by mathematical equations. The decay of a radioactive substance is a random but predictable process.The key equation used is:\[ N_t = N_0 \times e^{-\lambda t} \]In this exponential decay formula:

• \( \lambda \) is the decay constant.
• \( t \) is the time elapsed.

The decay constant \( \lambda \) is related to the half-life (\( T \)) by the equation:\[ \lambda = \frac{\ln(2)}{T} \]This conversion is crucial when relating the half-life to the continuous decay process. The logarithmic approach in solving for \( t \), as seen in the exercise, simplifies the calculations by transforming the exponential equation into a linear form using logarithms. Knowing these equations allows scientists to determine past events, like when a particular rock formation started its decay process.

Radioisotope dating

Radioisotope dating is a vital technique that uses the known decay rates of radioactive isotopes to estimate the age of materials. This method is widely used in geology and archaeology. Scientists measure the ratio of the radioactive parent isotope to the stable daughter product. With this information, and the known half-life of the isotope, they can calculate the age of the sample using equations such as the half-life formula integrally discussed above. Examples include dating ancient rocks using Uranium-Lead dating. By measuring how much Uranium-238 has decayed into Lead-206, scientists determine the "clock" that started when the rock solidified. The accuracy of radioisotope dating is remarkable because it relies on the fundamental understanding of radioactive decay processes and mathematical consistency. This method not only assists in understanding the Earth's geological history but also in examining artifacts of historical significance.