Discussion
In the 87Rb -87Sr 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.
Answer: C
The approximation holds because \(\lambda\) is very small for \(^{87}\)Rb.
The exponential can be expanded as a series, and when \(\lambda t\) is small every term after the second is negligible:
\[ \begin{aligned} e^{\lambda t} &= 1 + \lambda t + \tfrac{(\lambda t)^2}{2} + \cdots \\ e^{\lambda t} - 1 &\approx \lambda t \end{aligned} \]
For \(^{87}\)Rb, \(\lambda \approx 1.42 \times 10^{-11}\) per year, so even for the age of the Earth, about \(4.5 \times 10^{9}\) years, \(\lambda t\) is only about 0.064 and the squared term is about 0.002.
The decay itself is exponential, not linear, so the first option is false. Alteration of the Rb/Sr ratio affects whether an isochron is valid, not this algebra. The last option is the wrong way round: the half-life of \(^{87}\)Rb, about 49 billion years, is far longer than the age of the Earth, which is exactly why \(\lambda t\) stays small.
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