Chapter 7, Problem 7.22P

Interpretation Introduction

Interpretation:

Whether the Bohr model could predict the spectra for the $\text{H}$ atom and ${\text{Be}}^{3+}$ ion accurately, is to be determined. Also, whether their spectra would be identical is to be determined.

Concept introduction:

Bohr’s atomic model had the following postulates:

1) The electrons in atoms revolve around the nucleus in certain circular orbits. The orbits are the energy levels. The orbits closer to the nucleus are of lesser energy than the ones further from the nucleus.

2) The energy of an electron changes sharply upon transition from one energy level to another. The electrons absorb energy when the transition occurs from lower to higher orbits, whereas the release of energy in the form of radiation occurs when electrons move from higher to lower orbits. In a stationary state, an electron in an atom would not radiate energy.

3) The angular momentum of an electron moving around the nucleus is quantized. Thus the angular momentum of an electron is a whole number multiple of $\text{h/}2\pi$.

The Bohr’s equation for the energy of an electron in hydrogen-like atoms is,

$E_n=\frac{\left(-2.18\times {10}^{-18}\text{ J}\right)\left(Z^2\right)}{n^2}$

Here,

$E_n$ is the energy for the electron.

$-2.18\times {10}^{-18}\text{ J}$ is the value for the Rydberg’s constant.

$Z$ is the atomic number for the atom or ion.

$n$ is the principal quantum number.