GATE 2022 CY – Question 60

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 · Numerical answer

A hydrogen-like atom has wavefunction $\Psi(r,\theta,\phi)=\dfrac{2^{1/2}}{81\pi^{1/2}}(Z/a_0)^{5/2}(6-Zr/a_0)r e^{-Zr/(3a_0)}\cos\theta$. Its number of nodes is ______.

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Correct answer: 2 to 2

Explanation

The radial polynomial(6−Zr/a0) vanishes at positive radius6a0/Z, giving one radial node. The angular factor cosθ vanishes in the equatorial plane, giving one angular node. The origin is not counted as a nodal surface. Total **2 nodes**, consistent with a 3p state.