Chapter 7, Problem 7.62P

(a)

Interpretation Introduction

Interpretation:

An expression in the form of $E=-\left(\text{constant}\right)\left(1{\text{/n}}^2\right)$ is to be written.

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.

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 equation to find the energy of a level in hydrogen-like atoms is,

$\Delta E=-2.18\times {10}^{-18}\text{ J}\left(\frac{1}{n^2}\right)$

(b)

Interpretation Introduction

Interpretation:

The $\Delta E$ between $n=2$ and $n=3$ 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.

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 equation to find the difference in the energy between the two levels in hydrogen-like atoms is,

$\Delta E=-2.18\times {10}^{-18}\text{ J}\left(\frac{1}{n_{\text{final}}^2}-\frac{1}{n_{\text{initial}}^2}\right)$ (2)

Here,

$\Delta E$ is the difference in the energy between two levels.

$n_{\text{final}}$ is the lower energy level.

$n_{\text{initial}}$ is the higher energy level.

(c)

Interpretation Introduction

Interpretation:

The wavelength of the photon that corresponds to $\Delta E=3.027\times {10}^{-19}\text{ J}$ is to be determined.

Concept introduction:

The difference between the energies of two energy levels is the amount of energy in a photon emitted or absorbed when an electron makes a transition. The energy of a photon emitted or absorbed by an electron is,

$\Delta E=h\nu$

Here,

$\Delta E$ is the energy.

$h$ is the Plank’s constant.

$\nu$ is the frequency

The equation to relate the frequency and wavelength of radiation is as follows:

$\nu =\frac{c}{\lambda }$

The above relation can be modified as follows:

$\Delta E=\frac{hc}{\lambda }$ (3)