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Question #43680

Difficulty: 8/10 | 🟢 Free

The Question

A hypothetical superheavy nucleus of initial mass number $A_0$ undergoes a continuous decay process where it loses nucleons. The decay rate is governed by $\frac{dA}{dt} = -\lambda A$ , where the decay parameter $\lambda$ is not a constant. Instead, $\lambda$ depends on the instantaneous surface area $S$ and volume $V$ of the nucleus as $\lambda = c \frac{S}{V}$ , where $c$ is a positive constant with dimensions of $[L T^{-1}]$ . Assuming the strong nuclear force maintains a strictly invariant nuclear density $\rho_n$ (nucleons per unit volume) throughout the decay process, what is the time $T$ required for the nucleus to completely decay ( $A \to 0$ )?

Options

A.
$\frac{1}{c} \left( \frac{3A_0}{4\pi \rho_n} \right)^{1/3}$
B.
$\frac{3}{c} \left( \frac{3A_0}{4\pi \rho_n} \right)^{1/3}$
C.
$\frac{1}{3c} \left( \frac{3A_0}{4\pi \rho_n} \right)^{1/3}$
D.
$\infty$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Nuclear Density Invariance|Statistical Nature of Decay|Derived Units via Fundamental Constants

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to reconcile statistical decay laws with the physical constraint of nuclear density invariance.

The primary trap is the assumption that the decay constant $\lambda$ is a universal, time-independent constant, which would lead to the standard exponential decay law $A(t) = A_0 e^{-\lambda t}$ and an infinite time for complete decay (Option D).

By making $\lambda$ dependent on the surface-to-volume ratio, the problem couples the statistical decay rate to the geometric shrinkage of the nucleus.

Students must recognize that invariant density forces the nucleus to shrink in radius ( $R \propto A^{1/3}$ ) as it loses mass.

Failing to properly apply the chain rule to the coupled differential equation often results in missing the integration factor of 3, leading to Option C.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Nuclei - Class 12
Concept	Nuclear Density Invariance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free