CHEMISTRY · Multi-part · CBSE · Chemistry · Structure of Atom · Structure of Atom
Problem
Hydrogen-like ions contain only a single electron. For such ions, the average distance of the electron from the nucleus in the first shell (n = 1) is one-half the Bohr radius divided by the nuclear charge Z, and the average distance in the second shell (n = 2) is four times larger than that value.
Consider the He⁺ ion (Z = 2).
(a) Calculate the ratio of the average electron–nucleus distances for the n = 2 and n = 1 shells in He⁺.
(b) Assuming that ionization energy is directly proportional to the electrostatic attraction between the nucleus and the electron, derive an expression for the ratio of the first excited-state (n = 2) ionization energy to the ground-state (n = 1) ionization energy in a hydrogen-like ion in terms of the distances found in part (a).
(c) Evaluate the ratio obtained in part (b) for He⁺ and compare it to the numerical ratio that would be predicted from the energy-level spacings of the Bohr model (ground-state ionization energy = 54.4 eV, first excited-state ionization energy = 13.6 eV). Reconcile any difference between the two ratios.