Chapter 7, Problem 7.20QP

A particular form of electromagnetic radiation has a frequency of 8.11 × 10¹⁴ Hz. (a) What is its wavelength in nanometers? In meters? (b) To what region of the electromagnetic spectrum would you assign it? (c) What is the energy (in joules) of one quantum of this radiation?

Interpretation Introduction

Interpretation:

The wavelength (in nanometers and in meters) associated with a particular form of electromagnetic radiation having a frequency of $\text{9}{\text{.87 × 10}}^{\text{15}}\text{ Hz}$, the corresponding region of the electromagnetic spectrum and the energy (in joules) of one quantum of the given radiation should be calculated using the relation between speed, wavelength and frequency of a wave and Planck’s quantum theory.

Concept Introduction:

A wave is a disturbance or variation that travels through a medium transporting energy without transporting matter.  The wavelength is defined as the distance between the two similar points on consecutive waves.  The frequency is defined as the number of waves which move through any particular point in one second.

Figure.1

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 (${\text{s}}^{-1}$).

Planck’s quantum theory

1. 1. Different atoms or molecules emit or absorb energy in discreet quantities only.  The smallest amount of energy which is emitted or absorbed in the form of electromagnetic radiation is called quantum.
2. 2. The energy of the radiation absorbed or emitted is directly proportional to the frequency of the radiation.  The energy of radiation is expressed in terms of frequency as, $\text{E }=\text{ hν}$ where, $\text{E}$ = energy of the radiation; $\text{h}$ = Planck’s constant ($6.626\times {10}^{–34}\text{ Js}$) and $\text{ν}$ = frequency of radiation.

Electromagnetic spectrum is all forms of electromagnetic radiation where the only difference in the types of radiation is their wavelengths and frequencies.

Figure 2

The types of electromagnetic radiation which starts with the radiation having the longest wavelength and ends with the radiation having the shortest wavelength are given in the following order:

Radio waves, microwave, infrared, visible, ultraviolet, X rays, gamma rays

To find: Calculate the wavelength (in nanometers and in meters) associated with a particular form of electromagnetic radiation having a frequency of $\text{9}{\text{.87 × 10}}^{\text{15}}\text{ Hz}$

Answer to Problem 7.20QP

The wavelength associated with a particular form of electromagnetic radiation having a frequency of $\text{9}{\text{.87 × 10}}^{\text{15}}\text{ Hz}$ is $\text{30}\text{.4 nm}$ (in nanometers) and $\text{3}{\text{.04 × 10}}^{-\text{8}}\text{ m}$ (in meters)

Explanation of Solution

The speed, wavelength and frequency of a wave are related by $\text{c }=\text{ λν}$ where $\text{λ and ν}$ are expressed in meters ($\text{m}$) and reciprocal seconds (${\text{s}}^{-1}$) respectively.  Here, frequency is given; wavelength is to be calculated.  Hence, rearranging the equation for getting the wavelength is

  $\text{ν }=\frac{\text{c}}{\text{λ}}$

where frequency, $\text{ν}$ is $\text{9}{\text{.87 × 10}}^{\text{15}}\text{ Hz}$ and the speed of light, $\text{c}$ is $\text{3}{\text{.00 × 10}}^{\text{8}}\text{ m/s}$.  The speed of light is used in $\text{m/s}$.  Hence, frequency, $\text{ν}$ in $\text{Hz}$ is converted into that in $\text{s}$.  Here, $\text{9}{\text{.87 × 10}}^{\text{15}}\text{ Hz = 9}{\text{.87 × 10}}^{\text{15}}{\text{ s}}^{-1}$.  Substitute the given values in the formula:

  $\begin{array}{l}\text{λ = }\frac{\text{3}{\text{.00 × 10}}^{\text{8}}\text{ m/s}}{\text{9}{\text{.87 × 10}}^{\text{15}}\text{ /s}}\\ \text{λ = 3}{\text{.04 × 10}}^{-\text{8}}\text{ m}\\ \text{λ = 30}\text{.4 nm}\end{array}$

Here, $\text{1 nm }={\text{ 10}}^{-\text{9}}\text{ m}$

  $\begin{array}{l}\text{λ = 3}{\text{.04 × 10}}^{-\text{8}}\text{ m}\\ \text{λ = 3}{\text{.04 × 10}}^{-\text{8}}\text{ m }\times \left(\frac{{\text{10}}^{\text{9}}\text{ nm}}{\text{1 m}}\right)\\ \text{λ = 30}\text{.4 nm}\end{array}$

Therefore, the wavelength associated with a particular form of electromagnetic radiation having a frequency of $\text{9}{\text{.87 × 10}}^{\text{15}}\text{ Hz}$ is $\text{30}\text{.4 nm}$ (in nanometers) and $\text{3}{\text{.04 × 10}}^{-\text{8}}\text{ m}$ (in meters).

Interpretation Introduction

Interpretation:

The region of electromagnetic radiation having frequency of $\text{8}\text{.11 ×}{\text{10}}^{\text{14}}\text{Hz}$has to be identified.

Concept Introduction:

A wave is a disturbance or variation that travels through a medium transporting energy without transporting matter.  The wavelength is defined as the distance between the two similar points on consecutive waves.  The frequency is defined as the number of waves which move through any particular point in one second.

Figure.1

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 (${\text{s}}^{-1}$).

Planck’s quantum theory

1. 1. Different atoms or molecules emit or absorb energy in discreet quantities only.  The smallest amount of energy which is emitted or absorbed in the form of electromagnetic radiation is called quantum.
2. 2. The energy of the radiation absorbed or emitted is directly proportional to the frequency of the radiation.  The energy of radiation is expressed in terms of frequency as, $\text{E }=\text{ hν}$ where, $\text{E}$ = energy of the radiation; $\text{h}$ = Planck’s constant ($6.626\times {10}^{–34}\text{ Js}$) and $\text{ν}$ = frequency of radiation.

Electromagnetic spectrum is all forms of electromagnetic radiation where the only difference in the types of radiation is their wavelengths and frequencies.

Figure 2

The types of electromagnetic radiation which starts with the radiation having the longest wavelength and ends with the radiation having the shortest wavelength are given in the following order:

Radio waves, microwave, infrared, visible, ultraviolet, X rays, gamma rays

To find: Get the region of the electromagnetic spectrum associated with the wavelength

Answer to Problem 7.20QP

The region of the electromagnetic spectrum associated with the wavelength is ultraviolet region

Explanation of Solution

The wavelength associated with a particular form of electromagnetic radiation having a frequency of $\text{9}{\text{.87 × 10}}^{\text{15}}\text{ Hz}$ is $\text{30}\text{.4 nm}$.  This has a small value of wavelength.  The region associated with the wavelength of $\text{30}\text{.4 nm}$ is ultraviolet region.

Interpretation Introduction

Interpretation:

The energy in joules of one quantum of electromagnetic radiation with frequency of $\text{8}\text{.11 ×}{\text{10}}^{\text{14}}\text{Hz}$has to be determined.

Concept Introduction:

A wave is a disturbance or variation that travels through a medium transporting energy without transporting matter.  The wavelength is defined as the distance between the two similar points on consecutive waves.  The frequency is defined as the number of waves which move through any particular point in one second.

Figure.1

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 (${\text{s}}^{-1}$).

Planck’s quantum theory

1. 1. Different atoms or molecules emit or absorb energy in discreet quantities only.  The smallest amount of energy which is emitted or absorbed in the form of electromagnetic radiation is called quantum.
2. 2. The energy of the radiation absorbed or emitted is directly proportional to the frequency of the radiation.  The energy of radiation is expressed in terms of frequency as, $\text{E }=\text{ hν}$ where, $\text{E}$ = energy of the radiation; $\text{h}$ = Planck’s constant ($6.626\times {10}^{–34}\text{ Js}$) and $\text{ν}$ = frequency of radiation.

Electromagnetic spectrum is all forms of electromagnetic radiation where the only difference in the types of radiation is their wavelengths and frequencies.

Figure 2

The types of electromagnetic radiation which starts with the radiation having the longest wavelength and ends with the radiation having the shortest wavelength are given in the following order:

Radio waves, microwave, infrared, visible, ultraviolet, X rays, gamma rays

To find: Calculate the energy (in joules) of one quantum associated with a particular form of electromagnetic radiation having a frequency of $\text{9}{\text{.87 × 10}}^{\text{15}}\text{ Hz}$

Answer to Problem 7.20QP

The energy (in joules) of one quantum associated with a particular form of electromagnetic radiation having a frequency of $\text{9}{\text{.87 × 10}}^{\text{15}}\text{ Hz}$ is $\text{6}{\text{.54 × 10}}^{-\text{18}}\text{ J}$

Explanation of Solution

The energy of radiation is expressed in terms of frequency as,

$\text{E }=\text{ hν}$

where,

$\text{E}$ = energy of the radiation;

$\text{h}$ = Planck’s constant ($6.626\times {10}^{–34}\text{ Js}$);

$\text{ν}$ = frequency of radiation. 

Substitute the value of frequency in the formula,

  $\begin{array}{l}\text{E = hν}\\ \text{E = (6}{\text{.63 × 10}}^{-\text{34}}\text{ Js)(9}{\text{.87 × 10}}^{\text{15}}\text{ /s)}\\ \text{E = 6}{\text{.54 × 10}}^{-\text{18}}\text{ J}\end{array}$

Therefore, the energy of one quantum associated with a particular form of electromagnetic radiation having a frequency of $\text{9}{\text{.87 × 10}}^{\text{15}}\text{ Hz}$ is $\text{6}{\text{.54 × 10}}^{-\text{18}}\text{ J}$.