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Question #43797

Difficulty: 8/10 | 🟢 Free

The Question

A perfectly reflecting spherical cavity of initial volume $V_0$ contains a photon gas (thermal radiation) in equilibrium at temperature $T_0$ . According to Wien's displacement law, the peak emission wavelength $\lambda_{\text{max}}$ of this radiation initially matches the Lyman series limit of a $\text{Li}^{2+}$ ion. The cavity undergoes a reversible adiabatic expansion until its volume becomes $64 V_0$ . Assuming the radiation remains in thermal equilibrium, the new peak emission wavelength of the cavity matches the series limit of which of the following? (Let $R_H$ be the Rydberg constant)

Options

A.
Balmer series limit of $\text{Li}^{2+}$
B.
Brackett series limit of $\text{Li}^{2+}$
C.
Paschen series limit of $\text{He}^{+}$
D.
Balmer series limit of $\text{He}^{+}$

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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|Wien’s Displacement Law|Thermal Properties of Matter

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of quantum atomic transitions, blackbody radiation, and thermodynamics.

The primary trap lies in the adiabatic expansion of the cavity.

Students often reflexively treat any 'gas' undergoing adiabatic expansion as an ideal monatomic gas with $\gamma = 5/3$ .

Doing so would yield $T_2 = T_0/16$ and incorrectly point to the Brackett series limit of $\text{Li}^{2+}$ (Option B).

However, a photon gas (thermal radiation) has a radiation pressure $P = u/3$ , resulting in an effective adiabatic index of $\gamma = 4/3$ .

Recognizing this distinction is crucial for correctly coupling the evolving boundary conditions of the cavity to the spectral series limits.

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