Question

A sample initially contains only U-238 isotope of uranium. With time, some of the U-238 radioactively decays into Pb-206 while the rest of it remains undisintegrated.
When the age of the sample is P x10⁸ years, the ratio of mass of Pb-206 to that of U-238 in the sample is found to be 7. The value of P is _____.
[Given: Half-life of U-238 is 4.5 x10⁹ years; log_e2 = 0.693]

Answer 1

Correct Answer: 143

First-Order Reaction Time Calculation 

For a first-order reaction, the time \( t \) is given by:

\[ t = \frac{2.303}{k} \log \left( \frac{[A]_0}{[A]_t} \right) \]

Given:

\[ \frac{[m]_{\text{Pb}}}{[m]_{\text{U}}} = 7 \quad \Rightarrow \quad \frac{[A]_0}{[A]_t} = 1 + 7 \times \frac{1}{238 + 206} \approx 9 \]

Half-life of U-238:

The half-life \( t_{1/2} \) of U-238 is: \[ t_{1/2} = 4.5 \times 10^9 \quad \text{so} \quad k = \frac{0.693}{t_{1/2}} = \frac{0.693}{4.5 \times 10^9} \]

Substituting values into the equation:

\[ t = \frac{2.303}{k} \log 9 = \frac{2.303 \times 4.5 \times 10^9}{0.693} \log 9 \]

Simplifying:

\[ t \approx 14.27 \times 10^9 = 1.43 \times 10^{10} \quad \Rightarrow \quad P = 143 \]

Final Answer:

The value of P is 143.