Chapter 7.2, Problem 7.3AFP

(a)

Interpretation Introduction

Interpretation:

The change in energy of the hydrogen atom upon emission of a photon of IR radiation 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,

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

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.

(b)

Interpretation Introduction

Interpretation:

The wavelength of the photon in $\stackrel{\text{o}}{\text{A}}$ 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 $\text{h/}2\pi$.

According to Plank-Einstein equation the energy of a photon is directly proportional to the frequency and is expressed as follows:

$E=h\nu$ (1)

Here,

$E$ is the energy.

$h$ is the Plank’s constant.

$\nu$ is the frequency.

The equation that relates to the frequency and wavelength of electromagnetic radiation is as follows:

$\nu =\frac{c}{\lambda }$ (2)

Here,

$c$ is the speed of light.

$\lambda$ is the wavelength.

Equation (1) can be modified as follows:

$E=h\left(\frac{c}{\lambda }\right)$

Rearrange the above equation for $\lambda$.

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