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Question #43800

Difficulty: 8/10 | 🟢 Free

The Question

A stationary parent nucleus of mass $M_P$ undergoes alpha decay, producing an alpha particle of mass $m_\alpha$ and a hydrogen-like daughter ion of mass $M_D$ . The Q-value of the decay is $Q$ . The emitted alpha particle is observed to have a de Broglie wavelength $\lambda_\alpha$ . Assume that the portion of the Q-value not converted into the kinetic energy of the product nuclei is entirely used to excite the single electron of the daughter ion from its ground state exactly to the threshold of ionization. When an electron from infinity recombines with this ion to the ground state, it emits a photon at the spectral series limit. What is the wavelength $\lambda_L$ of this emitted photon? (Here $h$ is Planck's constant and $c$ is the speed of light).

Options

A.
$\frac{hc}{Q - \frac{h^2}{2 \lambda_\alpha^2} \left( \frac{m_\alpha + M_D}{m_\alpha M_D} \right)}$
B.
$\frac{hc}{Q - \frac{h^2}{2 m_\alpha \lambda_\alpha^2}}$
C.
$\frac{hc}{Q - \frac{h^2}{2 \lambda_\alpha^2} \left( \frac{M_D - m_\alpha}{m_\alpha M_D} \right)}$
D.
$\frac{hc}{Q - \frac{h^2}{2 \lambda_\alpha^2 \sqrt{m_\alpha M_D}}}$

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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

Spectral Series Limits|Q-value Calculation|Nuclear Binding Energy

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of nuclear energetics, quantum mechanics (de Broglie), and the Bohr model.

The primary trap lies in the misapplication of energy conservation during nuclear decay.

Students frequently assume that the entire Q-value minus the alpha particle's kinetic energy is available for excitation, completely ignoring the recoil kinetic energy of the massive daughter nucleus (Option B).

By forcing the calculation through the de Broglie wavelength, the student must explicitly reconstruct the momentum vectors and recognize that $p_D = p_\alpha$ , leading to the necessary inclusion of the $p^2/2M_D$ term in the energy balance before determining the spectral limit.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Spectral Series Limits
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free