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Question #44702

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The Question

A stationary radioactive nucleus $\ce{^{A}_{Z}X}$ undergoes $\alpha$ -decay to form $\ce{^{A-4}_{Z-2}Y}$ , releasing a total energy $Q$ . The rest mass of the $\alpha$ -particle is $m_\alpha$ . A student calculates the de Broglie wavelength of the emitted $\alpha$ -particle using classical mechanics ( $\lambda_{class}$ ). However, due to the high energy of the decay, relativistic effects must be considered to find the true wavelength ( $\lambda_{rel}$ ). Assuming the daughter nucleus is non-relativistic, which of the following expressions correctly represents the ratio $\frac{\lambda_{rel}}{\lambda_{class}}$ ? (Assume $c$ is the speed of light in vacuum).

Options

A.
$\left( 1 + \frac{(A-4)Q}{2 A m_\alpha c^2} \right)^{-1/2}$
B.
$\left( 1 + \frac{(A-4)Q}{A m_\alpha c^2} \right)^{-1/2}$
C.
$\left( 1 + \frac{A Q}{2 (A-4) m_\alpha c^2} \right)^{-1/2}$
D.
$\left( 1 - \frac{(A-4)Q}{2 A m_\alpha c^2} \right)^{1/2}$

Check Your Answer

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

De Broglie Wavelength Logic|Radioactive Decay as Kinetics

Why This Question Is Tricky

Common Mistakes

This problem tests a critical 'Hidden Variable' trap: Students habitually use the classical momentum equation $p = \sqrt{2mK}$ for all particles unless explicitly told it is a photon or a high-energy electron.

For $\alpha$ -particles emitted in radioactive decay (energies ~4-9 MeV), the velocity is a significant fraction of $c$ (~0.05c).

While classical mechanics provides a close approximation, exact analytical precision requires the relativistic mass-energy equivalence.

The trap lies in failing to use the relativistic invariant $E^2 = (pc)^2 + (m_0 c^2)^2$ and incorrectly assuming $\lambda_{rel} = \lambda_{class}$ , or misapplying the Q-value distribution $K_\alpha = \frac{A-4}{A}Q$ .

Quick Facts About This Question

Question Information

Subject	Chemistry
Chapter	Structure of Atom - Class 11 Part 1
Concept	De Broglie Wavelength Logic
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free