Problem 114 - A Complete Resource Book in Chemistry for JEE Main

Question

A sample of \(\mathrm{U}^{238}\left(\mathrm{t}_{1 / 2}=4.5 \times 10^{9} \mathrm{yrs}\right)\) ore is found con- taining \(23.8 \mathrm{~g} \mathrm{U}^{238}\) and \(20.6 \mathrm{~g}\) of \(\mathrm{Pb}^{206} .\) Calculate the age of the ore. (a) \(4.9 \times 10^{9}\) year (b) \(9.0 \times 10^{11}\) year (c) \(9.4 \times 10^{9}\) year (d) \(4.5 \times 10^{9}\) year

Step-by-Step Solution

Answer

The age of the ore is approximately \(4.9 \times 10^9\) years, so the correct answer is (a).

Step 1: Understand the Decay Process

Uranium-238 (\(\mathrm{U}^{238}\)) decays into Lead-206 (\(\mathrm{Pb}^{206}\)) through a series of decay processes. The half-life (\(t_{1/2}\)) is the time it takes for half of a sample of the radioactive material to decay.

Step 2: Determine Initial and Current Amounts

Initially, all \(\mathrm{Pb}^{206}\) was \(\mathrm{U}^{238}\). The initial amount of \(\mathrm{U}^{238}\) is the current amount of \(\mathrm{U}^{238}\) plus \(\mathrm{Pb}^{206}\). So, initial \(\mathrm{U}^{238}\) = 23.8 g (present) + 20.6 g (converted to \(\mathrm{Pb}^{206}\)) = 44.4 g.

Step 3: Calculate Fraction Remaining

The fraction of \(\mathrm{U}^{238}\) remaining is the ratio of the current mass to the initial mass:\[\frac{\text{Current } \mathrm{U}^{238} \text{ mass}}{\text{Initial } \mathrm{U}^{238} \text{ mass}} = \frac{23.8}{44.4} = \frac{0.536}{\underline{\phantom{xx}}}\]

Step 4: Use the Decay Formula

The decay formula is:\[N = N_0 \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}}\]where \(N\) is the remaining amount, \(N_0\) is the initial amount, \(t\) is the time elapsed, and \(t_{1/2}\) is the half-life. We need to solve:\[\frac{1}{2}^{\frac{t}{t_{1/2}}} = 0.536\]

Step 5: Solve for Time Elapsed

Rearrange the formula and solve for \(t\):\[\frac{t}{t_{1/2}} = \log_{0.5}(0.536)\]Using the change of base formula,\[\frac{t}{t_{1/2}} = \frac{\log(0.536)}{\log(0.5)}\]Multiply by \(t_{1/2} = 4.5 \times 10^9\) years:\[t = \frac{\log(0.536)}{\log(0.5)} \times 4.5 \times 10^9 \approx 4.9 \times 10^9 \text{ years}\]

Step 6: Conclusion

The calculated age of the ore is closest to option (a)\(4.9 \times 10^9\) years, meaning that option (a) is the correct answer.

Key Concepts

Uranium-238 Decay

Uranium-238, abbreviated as \(\mathrm{U}^{238}\), is a naturally occurring radioactive element. It works through a series of radioactive decay processes, eventually becoming Lead-206, or \(\mathrm{Pb}^{206}\). This transformation occurs over an immensely long time period. Ul>\[\bullet\] **Radioactive decay** is a process where unstable isotopes lose energy by emitting radiation. \[\bullet\] **Decay series** describes the succession of decays that \(\mathrm{U}^{238}\) undergoes to become \(\mathrm{Pb}^{206}\). \[\bullet\] Each step in this series involves the emission of either an alpha particle or a beta particle. Understanding this process is crucial, as it is the backbone for calculating how old a sample of uranium ore is. The time it takes for \(\mathrm{U}^{238}\) to reduced by half (transform entirely into \(\mathrm{Pb}^{206}\)) is termed its half-life.

Half-Life Calculation

Half-life is how long it takes for half of a radioactive substance to decay. For \(\mathrm{U}^{238}\), the half-life is enormously long, about \(4.5 \times 10^{9}\) years. This measure is essential for determining the age of geological substances.To figure out how much \(\mathrm{U}^{238}\) remained undecayed over time, you start with what you have now and work backward. - **Initial mass** is the total amount you would have needed for your sample to end up having exactly what it does now. You lived this backwards math to find the starting and remaining amounts. - **Fraction remaining** is calculated using the formula \(\frac{\text{Current } \mathrm{U}^{238} \text{ mass}}{\text{Initial } \mathrm{U}^{238} \text{ mass}}\). This fraction helps you plug into decay formulas to find elapsed time.

Age of Ore Calculation

To calculate the age of a uranium ore, you use exponential decay formulas. In the case of \(\mathrm{U}^{238}\), calculations are based on current and initial quantities. - **Radioactive decay formula:** \[ N = N_0 \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}} \] This formula allows you to find the time \(t\), when you know \(N\), the remaining mass, and \(N_0\), the initial mass.- By rearranging the formula, you can solve for \(t\): \[ \frac{t}{t_{1/2}} = \log_{0.5}(\text{fraction remaining}) \]- Using this equation, you multiply by the known half-life to get absolute years: \(t = \frac{\log(0.536)}{\log(0.5)} \times 4.5 \times 10^9 \approx 4.9 \times 10^9\). This calculation confirms the age of the uranium ore sample and shows the powerful utility of radioactive decay principles in geology.