Before You Begin

Question #43685

Difficulty: 8/10 | 🟢 Free

The Question

A parent nucleus ${}^{A}_{Z}X$ at rest undergoes alpha decay to form a daughter nucleus ${}^{A-4}_{Z-2}Y$ and an alpha particle ${}^{4}_{2}\text{He}$ . The total mass-energy released in the decay is $Q$ . The nuclear radius is given by $R = R_0 A^{1/3}$ , where $A$ is the mass number and $R_0$ is a constant. Let $M_Y$ and $M_\alpha$ be the exact masses of the daughter nucleus and the alpha particle, respectively. Assuming the daughter nucleus and alpha particle are spherical and just touching at the exact moment of creation, what is the kinetic energy of the alpha particle at this specific instant? (Let $k = 1/(4\pi\epsilon_0)$ and $e$ be the elementary charge. Assume no gamma emission occurs.)

Options

A.
$\frac{M_Y}{M_Y + M_\alpha} \left[ Q - \frac{2k(Z-2)e^2}{R_0((A-4)^{1/3} + 4^{1/3})} \right]$
B.
$\frac{M_Y}{M_Y + M_\alpha} Q$
C.
$\frac{M_Y}{M_Y + M_\alpha} \left[ Q - \frac{2k(Z-2)e^2}{R_0 A^{1/3}} \right]$
D.
$\frac{M_\alpha}{M_Y + M_\alpha} \left[ Q - \frac{2k(Z-2)e^2}{R_0((A-4)^{1/3} + 4^{1/3})} \right]$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Q-value Calculation|Energy Partitioning|Nuclear Density Invariance

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of the boundary conditions of energy conservation in nuclear reactions.

A common misconception is equating the $Q$ -value strictly to the kinetic energy of the products at all times.

Students often forget that at the exact moment of scission (when the products are just touching), a significant portion of the $Q$ -value is stored as Coulomb potential energy.

By requiring the application of Nuclear Density Invariance to find the correct separation distance ( $R_Y + R_\alpha$ ) rather than the parent nucleus radius, and then partitioning the remaining kinetic energy using momentum conservation, this problem effectively traps students who blindly apply the standard $K_\alpha = Q \cdot M_Y / (M_Y + M_\alpha)$ formula without considering the instantaneous spatial configuration.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Nuclei - Class 12
Concept	Q-value Calculation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free