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Isotope abundance refers to the relative quantity of a particular isotope compared to others in a given sample. On planet X, the isotopes

• Lead-205 \(^ { 205 } \text{Pb}\), a radioactive isotope with a half-life of \(1.53 \times 10^{7}\) years.
• Lead-204 \(^ { 204 } \text{Pb}\), a stable isotope that does not undergo radioactive decay.

Initially, both isotopes were equally abundant, meaning their ratio \(\frac{n_{205}}{n_{204}}\) was 1:1. When discussing isotope abundance, it's important to note that stable isotopes, like \(^ { 204 } \text{Pb}\), remain constant over time. In contrast, the quantity of radioactive isotopes, like \(^ { 205 } \text{Pb}\), will decrease as they decay. Therefore, by understanding the current ratio of these isotopes \(\frac{n_{205}}{n_{204}} = 2 \times 10^{-7}\), one can trace back to determine how long ago the isotopes were in equal abundance. This decay process helps scientists determine the age of rocks or planets, such as planet X, by comparing current abundance ratios to what they were historically.

The concept of half-life is central to understanding radioactive decay. Half-life is the time required for half of the radioactive isotopes in a sample to decay. For \(^ { 205 } \text{Pb}\), this half-life is given as \(1.53 \times 10^{7}\) years.To employ half-life in calculations, it's useful to determine the decay constant, \(\lambda\), which is a measure of the rate of decay. It is calculated using the half-life \(\tau\) via the formula: \[ \lambda = \frac{\ln(2)}{\tau} = \frac{0.693}{1.53 \times 10^{7}} \approx 4.531 \times 10^{-8}\, \text{y}^{-1}\]The decay constant illustrates how quickly the isotope experiences decay. It is a foundational element in calculating how long it would take for significant changes in isotope abundance to occur. In this context, calculating the decay constant for \(^ { 205 } \text{Pb}\) is essential for understanding the age of planet X.Knowing the half-life and decay constant allows scientists to accurately track isotopic changes as well as the passage of time since the initial equal abundance of isotopes.

The exponential decay formula provides a way to describe how the number of radioactive isotopes decreases over time:\[ n(t) = n_0 \cdot e^{-\lambda t}\]where:

• \(n(t)\) is the number of isotopes remaining at time \(t\).
• \(n_0\) is the initial number of isotopes.
• \(\lambda\) is the decay constant.

The formula's exponential nature indicates that the change in isotope quantity doesn't happen at a constant rate but speeds up as the quantity of the parent isotope grows smaller.In our problem:- The initial abundance ratio \(\frac{n_{205}}{n_{204}}\) observed today is \(2 \times 10^{-7}\).- Substituting into the decay formula, you can express the ratio directly by setting \(n_{204}(t) = n_0\) (as it is stable):\[ \frac{n_{205}(t)}{n_{204}} = \frac{n_0 \cdot e^{-\lambda t}}{n_0} = e^{-\lambda t}\]This leads you to the equation:\[ e^{-\lambda t} = 2 \times 10^{-7}\]By solving this equation, we can calculate the age of planet X using the current observed isotope ratio. Taking the natural logarithm of both sides and solving for \(t\) gives the total time passed since initial equal abundance.