Chapter 7, Problem 7.43QP

Interpretation Introduction

Interpretation:

The inadequacies of Bohr’s theory should be explained using the concept of Bohr’s theory.

Concept Introduction:

The emission of radiation given by an energized hydrogen atom to the electron falling from a higher-energy orbit to a lower orbit give a quantum of energy in the form of light.  Based on electrostatic interaction and law of motion, Bohr derived the following equation.

${\text{E}}_{\text{n}}=-\text{2}{\text{.18 × 10}}^{-\text{18}}\text{ J }\left(\frac{\text{1}}{{\text{n}}^{\text{2}}}\right)$

where $\text{n}$ gets an integer values such as $\text{n }=\text{ 1, 2, 3}$ and so on.  This is the energy of electron in ${\text{n}}^{\text{th}}$ orbital.

The electrons are excited thermally when the light is used by an object.  As a result, an emission spectrum comes.  Line spectra consist of light only at specific, discrete wavelengths.  In emission, the electron returns to a lower energy state from ${\text{n}}_{\text{f}}$ (the $\text{i and f}$ subscripts denote the initial and final energy states).  In most cases, the lower energy state corresponds to the ground state but it may be any energy state which is lower than the initial excited state.  The difference in the energies between the initial and final states is

$\text{ΔE }={\text{ E}}_{\text{f}}-{\text{ E}}_{\text{i}}$

This transition results in the photon’s emission with frequency $\text{v}$ and energy $\text{hv}$.  The following equation is resulted.

$\text{ΔE }=\text{ hν }=-\text{2}{\text{.18 × 10}}^{-\text{18}}\text{ J }\left(\frac{\text{1}}{{\text{n}}_{\text{f}}{}^{\text{2}}}-\frac{\text{1}}{{\text{n}}_{\text{i}}{}^{\text{2}}}\right)$

When ${\text{n}}_{\text{i}}{\text{ > n}}_{\text{f}}$, a photon is emitted.  The term in parentheses is positive, making $\text{ΔE}$ negative.  As a result, energy is lost to the surroundings.  When ${\text{n}}_{\text{i}}{\text{ < n}}_{\text{f}}$, a photon is absorbed.  The term in parentheses is negative, so $\text{ΔE}$ is positive.  As a result, energy is absorbed from the surroundings.

Bohr’s theory of the hydrogen atom describes an atom as a small, positively charged nucleus with electrons travelling around it in circular orbits.  It explains the appearance of an emission spectrum because the “orbits” available for the electrons to occupy in their excited states are quantized.  The resulting transition emits a specific wavelength of light because the energy of the “orbit” is quantized.

Figure.1

Visible line emission spectrum of hydrogen

The emission spectrum of hydrogen shows a wide range of wavelengths from the infrared to the ultraviolet region.  The given table lists the series of transitions in the hydrogen spectrum, each with a different value of ${\text{n}}_{\text{f}}$.  The series are named after their discoverers (Lyman, Balmer, Paschen and Brackett).  The Balmer series was the first to be studied because some of its lines occur in the visible region.

Series	${\text{n}}_f$	${\text{n}}_{\text{1}}$	Spectrum region
Lyman	1	2, 3, 4, ...	Ultraviolet
Balmer	2	3, 4, 5, ...	Visible and Ultraviolet
Paschen	3	4, 5, 6, ...	Infrared
Brackett	4	5, 6, 7, ...	Infrared

The given figure shows transitions associated with spectral lines in each of the emission series.  Each horizontal line represents one of the allowed energy levels for the electron in a hydrogen atom.  The energy levels are labeled with their $\text{n}$ values.

Figure.2

 Energy levels in the hydrogen atom and the various emission series.

Each series ends at a different value of $\text{n}$.