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GATE 2023 CY – Question 44

Physical Chemistry · Hydrogen and Hydrogen-like Atoms: Atomic orbitals; radial and angular distribution functions and their probabilities; Atomic units, Multi-electron atoms: Orbital approximation; electron spin; Pauli exclusion principle; Slater determinants. Variational method and secular determinants; first order non-degenerate perturbation techniques. · 2 marks · Multiple choice

The wavefunction of $Be^{3+}$ in a state is $\psi=Ne^{-4r/a_0}$, with normalization constant N, electron–nucleus distance r, and Bohr radius $a_0$. The most probable electron distance is ______.

  1. $4a_0$
  2. $a_0/4$
  3. $8a_0$
  4. $a_0/8$

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Correct answer: (B) $a_0/4$

Explanation

The radial probability includes the volume factor: r²|ψ|²∝r²exp(−8r/a0). Set its logarithmic derivative 2/r−8/a0=0, to obtain **r=a0/4, B**. Maximizing |ψ|² alone would incorrectly locate the maximum at the nucleus.