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Question #44791

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The Question

A pure sample of a radioactive parent nuclide $X$ (initial number of nuclei $N_0 = 2.0 \times 10^{10}$ ) decays into a radioactive daughter nuclide $Y$ , which further decays into a stable nuclide $Z$ . The decay constants are $\lambda_X = 1.0 \times 10^{-2} \text{ s}^{-1}$ and $\lambda_Y = 2.0 \times 10^{-2} \text{ s}^{-1}$ , respectively. A radiation detector continuously measures the total activity ( $A_{total}$ ) of the sample, defined as the sum of the activities of $X$ and $Y$ . Assuming 100% detection efficiency, what is the instantaneous rate of change of the total activity ( $\frac{dA_{total}}{dt}$ ) exactly at the moment when the activity of $Y$ reaches its maximum value?

Options

A.
$-1.0 \times 10^6 \text{ Bq s}^{-1}$
B.
$0 \text{ Bq s}^{-1}$
C.
$-2.0 \times 10^6 \text{ Bq s}^{-1}$
D.
$-4.0 \times 10^6 \text{ Bq s}^{-1}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Numerical_Precision

Improve calculation stamina and avoid rounding errors in high-stakes competitive examinations.

Concepts Behind This Question

Instantaneous vs. Average Rate|Radioactive Decay as Kinetics

Why This Question Is Tricky

Common Mistakes

The trap lies in the calculus of radioactive decay chains (Bateman equations).

Students often assume that when the daughter nuclide reaches maximum activity (transient equilibrium), the total activity is also at a maximum, incorrectly concluding the derivative is 0 (Option B).

Alternatively, they attempt to differentiate the entire Bateman equation, getting lost in complex algebra and making sign errors.

The elegant 'hidden variable' is realizing that at $t_{max}$ for $Y$ , $\frac{dA_Y}{dt} = 0$ , meaning the instantaneous rate of change of the entire system's activity is dictated solely by the second derivative of the parent nuclide's decay profile ( $-\lambda_X^2 N_X$ ).

Option C is a trap where students calculate $-\lambda_X A_{total}$ instead of $-\lambda_X A_X$ .

Quick Facts About This Question

Question Information

Subject	Chemistry
Chapter	Chemical Kinetics - Class 12
Concept	Instantaneous vs. Average Rate
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free