Chapter 28, Problem 53A

To determine

To calculate:The differences in the energy levels for each of the three visible lines.

Answer to Problem 53A

The difference in energies of blue and green photon is $0.58\text{ eV}$ , that of blue and yellow photon is $0.70\text{ eV}$ , and the difference in energies of green and yellow photon is $0.12\text{ eV}$ .

Explanation of Solution

Given info:

Wavelength of blue line, ${\lambda }_{blue}=436\text{ nm}$

Wavelength of green line, ${\lambda }_{green}=546\text{ nm}$

Wavelength of yellow line, ${\lambda }_{yellow}=579\text{ nm}$

Formula used:

Energy-wavelength relation:

  $E=\frac{hc}{\lambda }$

Wherethe energy of the visible photon is $E$ , $h$ is the Planck’s constant, $c$ is the speed of light and $\lambda$ is the wavelength of the photon.

Calculation:

As per problem,

  $c=3\times {10}^8\text{ m/s}$

  $h=6.63\times {10}^{-34}\text{ Js}$

Calculate the energy of the blue photon

  ${\lambda }_{blue}=436\text{ nm}$

  $\begin{array}{l}{E}_{blue}=\frac{hc}{{\lambda }_{blue}}\\ E_{blue}=\frac{(6.63\times {10}^{-34}\text{ Js)(3}\times {\text{10}}^8\text{ m/s)}}{436\times {10}^{-9}\text{ m}}\\ E_{blue}=4.56\times {10}^{-19}\text{ J}\\ E_{blue}=2.85\text{ eV}\end{array}$

Calculate the energy of green photon

  ${\lambda }_{green}=546\text{ nm}$

  $\begin{array}{l}{E}_{green}=\frac{hc}{{\lambda }_{green}}\\ E_{green}=\frac{(6.63\times {10}^{-34}\text{ Js)(3}\times {\text{10}}^8\text{ m/s)}}{546\times {10}^{-9}\text{ m}}\\ E_{green}=3.64\times {10}^{-19}\text{ J}\\ E_{green}=2.27\text{ eV}\end{array}$

Calculate the energy of yellow photon

  ${\lambda }_{yellow}=579\text{ nm}$

  $\begin{array}{l}{E}_{yellow}=\frac{hc}{{\lambda }_{yellow}}\\ E_{yellow}=\frac{(6.63\times {10}^{-34}\text{ Js)(3}\times {\text{10}}^8\text{ m/s)}}{579\times {10}^{-9}\text{ m}}\\ E_{yellow}=3.43\times {10}^{-19}\text{ J}\\ E_{yellow}=2.15\text{ eV}\end{array}$

The difference between energy of blue and green photon

  $\begin{array}{l}\Delta E=E_{blue}-E_{green}\\ \Delta E=(2.85\text{ eV})-(2.27\text{ eV)}\\ \Delta E=0.58\text{ eV}\end{array}$

The difference between energy of blue and yellow photon

  $\begin{array}{l}\Delta E=E_{blue}-E_{yellow}\\ \Delta E=(2.85\text{ eV})-(2.15\text{ eV)}\\ \Delta E=0.70\text{ eV}\end{array}$

The difference between energy of green and yellow photon

  $\begin{array}{l}\Delta E=E_{green}-E_{yellow}\\ \Delta E=(2.27\text{ eV})-(2.15\text{ eV)}\\ \Delta E=0.12\text{ eV}\end{array}$

Conclusion:

Thus, the difference in energies of blue and green photonis $0.58\text{ eV}$ , that of blue and yellow photon is $0.70\text{ eV}$ , and the difference in energies of green and yellow photon is $0.12\text{ eV}$ .