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Question #44696

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The Question

Calculate the change in molar internal energy ( $\Delta U_m$ ) for the first ionization of Helium gas at $300\text{ K}$ :

$\ce{He(g) -> He+(g) + e-(g)}$

Assume the initial Helium gas behaves as a van der Waals real gas with an internal pressure correction factor such that $\frac{a}{V_m} = 4.905\text{ kJ/mol}$ . The resulting $\ce{He+}$ ions and electrons behave as ideal monoatomic gases. Use Slater's rules to estimate the effective nuclear charge ( $Z_{eff}$ ) for the electronic energy calculations.

[Given: Ground state energy of a Hydrogen atom $E_H = -1312\text{ kJ/mol}$ . Universal gas constant $R = 8.30\text{ J K}^{-1}\text{mol}^{-1}$ . Shielding constant for a $1s$ electron by another $1s$ electron is $0.30$ .]

Options

A.
2344.00 kJ/mol
B.
5256.64 kJ/mol
C.
2335.36 kJ/mol
D.
2340.27 kJ/mol

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

Bohr’s Model Limitations|Internal Energy of Real Gases|Atomic vs Ionic Radii

Why This Question Is Tricky

Common Mistakes

This problem tests the 'Bohr Model Limitation' and 'Real Gas Deviations'.

Students often fall into the trap of using the standard Bohr energy formula ( $E \propto Z^2/n^2$ ) for multi-electron atoms, ignoring the Pauli Exclusion Principle and shielding effects (Slater's rules), which leads to a massive overestimation of the ionization energy (Option B).

Furthermore, it tests the conflation of idealized electronic transitions with macroscopic thermodynamic state functions.

Students must recognize that the internal energy of a real gas is lowered by intermolecular attractions ( $-a/V_m$ ), which must be overcome during ionization, and they must account for the translational kinetic energy of the newly liberated electron (Options C and D represent failures to include these thermodynamic corrections).

Quick Facts About This Question

Question Information

Subject	Chemistry
Chapter	Structure of Atom - Class 11 Part 1
Concept	Bohr’s Model Limitations
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free