The GATE Grind

GATE 2020 CY – Question 51

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 hydrogen-like 3s radial function is $$R_{3,0}=N\left(6-2\rho+\frac{\rho^2}{9}\right)e^{-\rho/6},$$ where $$N=\frac1{9\sqrt3}\left(\frac Z{a_0}\right)^{3/2},\qquad \rho=2Zr/a_0.$$ Here Z is atomic number, r is distance from nucleus and a₀ is Bohr radius. Find radial-node positions in units of a₀.

  1. $(3+\sqrt3)/(2Z),\ (3-\sqrt3)/(2Z)$
  2. $(6+3\sqrt3)/(2Z),\ (6-3\sqrt3)/(2Z)$
  3. $(9+3\sqrt3)/(2Z),\ (9-3\sqrt3)/(2Z)$
  4. $(3+3\sqrt3)/(2Z),\ (3-3\sqrt3)/(2Z)$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (C) $(9+3\sqrt3)/(2Z),\ (9-3\sqrt3)/(2Z)$

Explanation

The prefactor and exponential do not vanish at finite positive r, so solve $6-2\rho+\rho^2/9=0$. Multiplying by nine gives ρ²−18ρ+54 = 0, with roots ρ = 9±3√3. Since r/a₀ = ρ/(2Z), the two positions are (9±3√3)/(2Z), C.

More 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. questions

All 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. questions →