Chapter 3.4, Problem 3.4.3SR

Interpretation Introduction

Interpretation:

The wavelength (in meters) of light emitted when an electron in a hydrogen atom goes from $\text{n }=\text{ 5 to n }=\text{ 3}$ should be calculated 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.

The speed, wavelength and frequency of a wave are interrelated by $\text{c }=\text{ λν}$ where $\text{λ and ν}$ are mentioned in meters ($\text{m}$) and reciprocal seconds (${\text{s}}^{-1}$).    Hence, rearrange the equation for getting the wavelength is,

$\text{ν }=\frac{\text{c}}{\text{λ}}$

Substitute the frequency formula and rearrange,

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

Therefore, this formula is used to find the wavelength of the given photon in the emission line process.  For the absorption line process in which an electron is removed from the nucleus, the sign is changed as,

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

To calculate: The wavelength (in meters) of light emitted when an electron in a hydrogen atom goes from $\text{n }=\text{ 5 to n }=\text{ 3}$