Chapter 1, Problem 1A.9E

Interpretation Introduction

Interpretation:

The given table has to be completed.

Frequency	Wavelength	Energy of photon	Event
$8.7\times {10}^{14}Hz$
		$3.3\times {10}^{-19}J$
$300MHz$
	$2.5nm$

Concept Introduction:

The relationship between wavelength, frequency and speed of light is given below.

  $\lambda \times \nu =c$

Where, $\lambda$ is the wavelength, $\nu$ is the frequency of radiation and $c$ is the speed of light.

Energy of a photon can be expressed mathematically as given below.

  $E=h\nu =\frac{hc}{\lambda }$

The wavelengths of electromagnetic radiation and their corresponding frequencies are given below in the table.

Radiation type	Frequency / $\left({10}^{14}Hz\right)$	Wavelength / $\left(nm\right)$	Energy of photon / $\left({10}^{-19}J\right)$
x-rays and $\gamma$- rays	$\ge {10}^3$	$\le 3$	$\ge {10}^3$
Ultraviolet	$8.6$	$350$	$5.7$
Visible light
Violet	$7.1$	$420$	$4.7$
Blue	$6.4$	$470$	$4.2$
Green	$5.7$	$530$	$3.8$
Yellow	$5.2$	$580$	$3.4$
Orange	$4.8$	$620$	$3.2$
Red	$4.3$	$700$	$2.8$
Infrared	$3.0$	$1000$	$2.0$
Microwaves and radio waves	$\le {10}^{-3}$	$\ge 3\times {10}^6$	$\le {10}^3$

Explanation of Solution

First row:

The frequency of the radiation is $8.7\times {10}^{14}Hz.$

The calculation of wavelength and energy of the radiation can be done as shown below.

  $\begin{array}{l}\lambda =\frac{c}{\nu }=\frac{3\times {10}^{8}m/s}{8.7\times {10}^{14}{s}^{-1}}=3.44\times {10}^{-7}m=344nm\\ \\ E=\frac{hc}{\lambda }=\frac{\left(6.6\times {10}^{-34}J.s\right)\times \left(3\times {10}^{8}m/s\right)}{3.44\times {10}^{-7}m}=5.7\times 10^{-19}J\end{array}$

Second row:

The energy of the radiation is $3.3\times {10}^{-19}J.$

The calculation of wavelength and frequency of the radiation can be done as shown below.

  $\begin{array}{l}E=\frac{hc}{\lambda }\\ \\ \lambda =\frac{hc}{E}=\frac{\left(6.6\times {10}^{-34}J.s\right)\times \left(3\times {10}^{8}m/s\right)}{3.3\times {10}^{-19}J}=6\times {\text{10}}^{\text{-7}}m=600nm\\ \\ \nu =\frac{c}{\lambda }=\frac{3\times {10}^{8}m/s}{600\times {10}^{-9}m}=5.0\times 10^{14}Hz\end{array}$

Third row:

The frequency of the radiation is $300MHz.$

The calculation of wavelength and energy of the radiation can be done as shown below.

  $\begin{array}{l}\lambda =\frac{c}{\nu }=\frac{3\times {10}^{8}m/s}{300\times {10}^6{s}^{-1}}=1m\\ \\ E=\frac{hc}{\lambda }=\frac{\left(6.6\times {10}^{-34}J.s\right)\times \left(3\times {10}^{8}m/s\right)}{1m}=2\times 10^{-25}J\end{array}$

Fourth row:

The wavelength of the radiation is $2.5nm.$

The calculation of frequencyand energy of the radiation can be done as shown below.

  $\begin{array}{l}\nu =\frac{c}{\lambda }=\frac{3\times {10}^{8}m/s}{2.5\times {10}^{-9}m}=1.2\times 10^{17}Hz\\ \\ E=h\nu =\left(6.6\times {10}^{-34}J.s\right)\times \left(\text{1}{\text{.2×10}}^{\text{17}}{s}^{-1}\right)=7.92\times 10^{-17}J\end{array}$

Now, the given table can be filled up as given below.

Frequency	Wavelength	Energy of photon	Event
$8.7\times {10}^{14}Hz$	$344nm$	$5.7\times 10^{-19}J$	Sun tan
$5.0\times 10^{14}Hz$	$600nm$	$3.3\times {10}^{-19}J$	Reading
$300MHz$	$1m$	$2\times 10^{-25}J$	Microwave popcorn
$1.2\times 10^{17}Hz$	$2.5nm$	$7.92\times 10^{-17}J$	Dental x-ray