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: The exact decay term.
The isochron slope carries the term $e^{\lambda t} - 1$. Here $\lambda$ is the decay constant of $^{87}Rb$ and $t$ is the age.

Step 2: A handy expansion.
For any small number $x$ we can write \[ e^{x} \approx 1 + x. \] So $e^{\lambda t} - 1 \approx \lambda t$, but only when $\lambda t$ is small.

Step 3: Why $\lambda t$ is small here.
$^{87}Rb$ has a very tiny decay constant because its half life is enormous, about fifty billion years. Even over billions of years the product $\lambda t$ stays small.

Step 4: Test the other options.
Decay is not linear, so that option is wrong. Whether the Rb to Sr ratio can be altered, or how the Earth age compares to the half life, has nothing to do with this math step.

Step 5: Final choice.
The approximation holds because $\lambda$ is very small for $^{87}Rb$.
\[ \boxed{\lambda \text{ is very small for } ^{87}Rb} \]