Question

In a hypothetical scenario, the element Y has 4 stable isotopes \(^{197}{Y}\), \(^{198}{Y}\), \(^{199}{Y}\), and \(^{200}{Y}\). The isotope \(^{199}{Y}\) is radiogenic and is formed by \(\beta^-\)-decay of the isotope \(^{199}{X}\) of the element X with a half-life of 3.51 billion years. An igneous rock, which has behaved as a closed system, has three different minerals P, Q, and R, which crystallized from the same magma 2 billion years ago, with initial X/Y ratios of 1.75, 2.05, and 0.75, respectively. Which one of the following statements is true for the ratio \(^{199}{Y}/^{200}{Y}\) in this igneous rock in the present day?

• A. \(P>Q>R\)
• B. \(R>P>Q\)
• C. \(Q>P>R\)
• D. \(Q>R>P\)

Hint:

In isotopic systems with radiogenic decay, minerals with a higher initial parent/daughter ratio accumulate more radiogenic daughter isotopes over time, resulting in higher current daughter/parent ratios.

Answer 1

The Correct Option is C

Step 1: Understanding the scenario.
The isotopic ratio \( \frac{199Y}{200Y} \) in the rock will depend on the initial X/Y ratio and the radiogenic decay of the isotope 199X to 199Y. Over time, the radiogenic 199Y will accumulate in minerals based on the initial X/Y ratio and the time that has passed since the rock crystallized.

Step 2: Evaluating the initial X/Y ratios.
Mineral P has the highest initial X/Y ratio (1.75), meaning it initially contained more of element X relative to Y, so it would have accumulated more of the radiogenic 199Y.
Mineral Q has a moderate initial X/Y ratio (2.05), which is slightly higher than R but lower than P, meaning it will accumulate less of the radiogenic 199Y compared to P.
Mineral R has the lowest initial X/Y ratio (0.75), meaning it had less of element X initially, and therefore it would accumulate the least amount of radiogenic 199Y.

Step 3: Considering radiogenic decay and the time elapsed.
Given that all the minerals crystallized 2 billion years ago, the rate of accumulation of 199Y in each mineral is proportional to the initial X/Y ratio, but since the time is the same for all, the mineral with the highest initial X/Y ratio (P) would have the highest 199Y/200Y ratio, followed by Q, and R with the lowest.

Step 4: Correct interpretation of the ratio.
After 2 billion years, the correct order of 199Y/200Y ratio in the minerals would be: Q > P > R, based on the initial X/Y ratios and the corresponding amount of 199Y formed.