Chapter 6.7, Problem 3.3ACP

Interpretation Introduction

Interpretation: The energy of a photon of line in the yellow region with wavelength $5.89\times {10}^{-7}\text{m}$ and $5.8959\times {10}^{-7}\text{m}$ has to be calculated and the energy difference between the two photons has to be determined.

Concept introduction:

• Planck’s equation,

    $\begin{array}{c}\text{E}\text{=}\text{hν}\text{=}\frac{\text{hc}}{\text{λ}}\\ \text{where, ν}=\text{frequency}\\ \text{λ}=\text{wavelength}\end{array}$

The energy increases as the wavelength of the light decreases.

Answer to Problem 3.3ACP

• The energy of a photon of line in the yellow region with wavelength $5.89\times {10}^{-7}\text{m}$ is $3.3729\times {10}^{-19}\text{J}$
• The energy of a photon of line in the yellow region with wavelength $5.8959\times {10}^{-7}\text{m}$ is $3.3692\times {10}^{-19}\text{J}$
• The energy difference between the two photons is $3.4\times {10}^{-22}\text{J}$.

Explanation of Solution

The energy of a photon of line in the yellow region with wavelength $5.89\times {10}^{-7}\text{m}$ and $5.8959\times {10}^{-7}\text{m}$ is calculated and the energy difference between the two photons is also determined.

Given,

The wavelength of line in the yellow region is $589\text{nm}\text{=}5.89\times {10}^{-7}\text{m}$

The wavelength of line in the yellow region is $589.59\text{nm}\text{=}5.8959\times {10}^{-7}\text{m}$

• The energy of a photon of line in the yellow region with wavelength $\underset{\_}{5.89\times {10}^{-7}\text{m}}$ is calculated,

  $\text{E}\text{=}\frac{\text{hc}}{\text{λ}}$

Substituting the values

  $\begin{array}{c}\text{E}\text{=}=\frac{\text{6}\text{.626}\text{×}{\text{10}}^{-\text{34}}\text{J}\text{.s}/\text{photon}\times \text{2}\text{.998}\text{×}\text{10}{}^{\text{8}}\text{m/s}}{5.89\times {10}^{-7}\text{m}}\\ =3.3726\times {10}^{-19}\text{J}\end{array}$

The energy of a photon of line in the yellow region with wavelength $5.89\times {10}^{-7}\text{m}$ is $3.3726\times 10^{-19}J$

• The energy of a photon of line in the yellow region with wavelength $\underset{\_}{5.8959\times {10}^{-7}\text{m}}$ is calculated,

  $\text{E}\text{=}\frac{\text{hc}}{\text{λ}}$

Substituting the values

    $\begin{array}{c}\text{E}=\frac{\text{6}\text{.626}\text{×}{\text{10}}^{-\text{34}}\text{J}\text{.s}/\text{photon}\times \text{2}\text{.998}\text{×}\text{10}{}^{\text{8}}\text{m/s}}{5.8959\times {10}^{-7}\text{m}}\\ =3.3692\times {10}^{-19}\text{J}\end{array}$

The energy of a photon of line in the yellow region with wavelength $5.8959\times {10}^{-7}\text{m}$ is $3.3692\times 10^{-19}J$

• The energy difference between the pair of lines in the yellow region with wavelength $\underset{\_}{5.89\times {10}^{-7}\text{m}}$ and $\underset{\_}{5.8959\times {10}^{-7}\text{m}}$ is calculated

  $\begin{array}{c}\text{ΔE}\text{=}{\text{E}}_{\text{λ=589}\text{nm}}-{\text{E}}_{\text{λ=589}\text{.59}\text{nm}}\\ =3.3726\times {10}^{-19}\text{J}-3.3692\times {10}^{-19}\text{J}\\ =3.4\times {10}^{-22}\text{J}\end{array}$

The energy difference between the pair of lines in the yellow region with wavelength $5.89\times {10}^{-7}\text{m}$ and $5.8959\times {10}^{-7}\text{m}$ is $3.4\times 10^{-22}J$