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Question #44700

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The Question

A student models the third ionization energy ( $IE_3$ ) of Thallium, corresponding to the process \ce{Tl^{2+}(g) -> Tl^{3+}(g) + e^-} at $298 \text{ K}$ , using the modified Rydberg equation: $IE = 13.6 \times \frac{Z_{eff}^2}{n^2} \text{ eV}$ Using standard non-relativistic Slater's rules, the student calculates a predicted value ( $IE_{pred} \approx 12.3 \text{ eV}$ ). However, the experimentally observed value is significantly different ( $IE_{obs} = 29.8 \text{ eV}$ ). Which of the following correctly identifies the fundamental limitation of the Bohr-Rydberg model in this scenario and the physical origin of the discrepancy?

Options

A.
$IE_{obs} > IE_{pred}$ ; The Bohr model assumes a constant electron rest mass. In heavy atoms, relativistic velocities of penetrating electrons increase their effective mass ( $m = \gamma m_e$ ). Since the Bohr radius $r_n \propto 1/m$ and binding energy $IE \propto m$ , the $6s$ orbital undergoes severe relativistic contraction. This, combined with poor $4f/5d$ shielding, anomalously stabilizes the $6s$ electron.
B.
$IE_{obs} < IE_{pred}$ ; The Rydberg equation fails to account for the Lanthanide contraction, which increases core electron density and enhances the shielding of the $6s^1$ electron, thereby expanding its radial distribution and lowering its binding energy.
C.
$IE_{obs} > IE_{pred}$ ; The $6s$ orbital experiences strong spin-orbit coupling which splits the $n=6$ shell, causing the $6s$ electron to transition into a highly stable $4f$ vacancy, a mechanism ignored by the principal quantum number scaling.
D.
$IE_{obs} < IE_{pred}$ ; The Bohr model assumes a point-charge nucleus. The finite nuclear size of Thallium reduces the effective nuclear charge felt by the $6s$ electron, destabilizing the orbital and decreasing the ionization energy.

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Explanation

How the Examiner Thinks

Exception_Exposure

Learn to identify edge cases and boundary exceptions that JEE examiners hide in math problems.

Concepts Behind This Question

Rydberg Equation Constants|Inert Pair Effect|Bohr’s Model Limitations

Why This Question Is Tricky

Common Mistakes

This question targets the pedagogical blind spot where students apply single-electron models (Bohr/Rydberg) and empirical shielding rules (Slater's) to heavy post-transition metals (like Tl, Pb, Bi).

The trap lies in the 'Inert Pair Effect', which is often memorized merely as a periodic trend.

The hidden variables are Relativistic Orbital Contraction and the Lanthanide Contraction.

By forcing a quantitative comparison ( $IE_{obs}$ vs $IE_{pred}$ ), the student must recognize that the Rydberg equation's assumption of a constant electron rest mass ( $m_e$ ) collapses for high-Z elements, where relativistic mass increase ( $m = \gamma m_e$ ) directly contracts the Bohr radius ( $a_0 \propto 1/m$ ) and increases binding energy.

Quick Facts About This Question

Question Information

Subject	Chemistry
Chapter	Structure of Atom - Class 11 Part 1
Concept	Rydberg Equation Constants
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free