Question

In the \(^{87}Rb-^{87}Sr\) isotopic systematics applied to terrestrial rocks, the slope of the isochron \( (e^{\lambda t} - 1) \) can be approximated as \( \lambda t \), where \( \lambda \) is the decay constant and \( t \) is time. Choose the correct option that justifies this approximation.

• A. \( ^{87}Rb \) decays to \( ^{87}Sr \) following a linear law
• B. The Rb/Sr ratio is susceptible to alteration in terrestrial rocks
• C. \( \lambda \) is very small for \( ^{87}Rb \)
• D. Age of the Earth > half-life of \( ^{87}Rb \)

Hint:

For small decay constants, the exponential decay equation can be approximated as a linear function for short timescales.

Answer 1

The Correct Option is C

Step 1: Understanding the formula for the isochron.
In radiometric dating, the formula \( (e^{\lambda t} - 1) \) describes the decay of isotopes over time, where \( \lambda \) is the decay constant and \( t \) is time. For small values of \( \lambda t \), the term \( e^{\lambda t} \) can be approximated as \( 1 + \lambda t \), which simplifies to \( \lambda t \).

Step 2: Analyzing the decay constant for \( ^{87}Rb \).
The decay constant \( \lambda \) for \( ^{87}Rb \) is very small, meaning that over geologically significant timescales, the decay is gradual. This allows for the approximation \( e^{\lambda t} - 1 \approx \lambda t \), as the term \( \lambda t \) is small enough for this linear approximation to hold true.

Step 3: Analyzing Option (A).
Option (A) suggests that \( ^{87}Rb \) decays to \( ^{87}Sr \) following a linear law, but this is not true for radioactive decay. The decay follows an exponential law, not a linear one.

Step 4: Analyzing Option (B).
Option (B) discusses the susceptibility of the Rb/Sr ratio to alteration in terrestrial rocks, but this does not justify the approximation \( \lambda t \). It is unrelated to the mathematical simplification of the exponential decay formula.

Step 5: Analyzing Option (D).
Option (D) suggests that the age of the Earth is greater than the half-life of \( ^{87}Rb \), which is true in a general sense, but it is not directly relevant to the approximation of the isochron formula. The approximation depends on the smallness of \( \lambda \), not on the age of the Earth.

Step 6: Conclusion.
The correct answer is (C), as the small decay constant \( \lambda \) for \( ^{87}Rb \) justifies the linear approximation \( e^{\lambda t} - 1 \approx \lambda t \).
\[ \boxed{\lambda \text{ is very small for } ^{87}Rb} \]