Chapter 6, Problem 1QAP

A photon of violet light has a wavelength of 423 nm.

Calculate

(a) the frequency.

(b) the energy in joules per photon.

(c) the energy in kilojoules per mole.

Interpretation Introduction

(a)

Interpretation:

The frequency of a photon of violet light should be calculated for 423 nm wavelength.

Concept introduction:

Frequency is defined as the number of cycles of wave (successive troughs and crests) which passes from a given point at unit time. The unit of frequency is hertz which represents one cycle per second.

An expression which gives the relation between frequency, wavelength and speed of light is given as:

$c=\lambda \nu$

Where c= speed of light

$\lambda$ = wavelength of light

$\nu$ = frequency of light

Answer to Problem 1QAP

Frequency of a photon of violet light = $\text{7}\text{.0}\times {\text{10}}^{14}{\text{ s}}^{-1}$

Explanation of Solution

The mathematical expression for calculating frequency is given by:

$c=\lambda \nu$

Convert the given value of wavelength in nm into m.

1 nm = ${10}^{-9}$ m

Conversion factor is $\frac{1\text{ nm}}{{10}^{-9}\text{m}}$

Therefore, 423 nm is equal to $423\text{ nm }\times \frac{1\text{ m}}{{10}^9\text{nm}}=423\times {\text{ 10}}^{-9}\text{m}$

Put the given value of wavelength and speed of light that is $423\times {\text{ 10}}^{-9}\text{m}$ and $3\times {10}^8\text{ m/s}$

${\text{3×10}}^{\text{8}}{\text{m/s=423×10}}^{\text{9 }}\text{m ×ν}$

$\text{ν}=\frac{{\text{3×10}}^{\text{8}}\text{m/s}}{{\text{423×10}}^{\text{-9 }}\text{m }}\text{=7}\text{.0}\times {\text{10}}^{14}{\text{ s}}^{-1}$

Interpretation Introduction

(b)

Interpretation:

The energy in joules per photon should be calculated for 423 nm wavelength.

Concept introduction:

An expression which gives the relation between energy, wavelength and speed of light is given as:

$E=h\frac{c}{\lambda }$ (1)

Where c= speed of light

$\lambda$ = wavelength of light

Equation (1) is also written as:

$E=h\nu ,\nu =\frac{c}{\lambda }$ (2)

$\nu$ = frequency of light

$h$ = Planck’s constant

Answer to Problem 1QAP

Energy in joules per photon = $46.382\times {10}^{-20}\text{J}$

Explanation of Solution

The mathematical expression for calculating energy for one photon is given by:

$E=h\frac{c}{\lambda }$

Convert the given value of wavelength in nm into m.

1 nm = ${10}^{-9}$ m

Conversion factor is $\frac{1\text{ nm}}{{10}^{-9}\text{m}}$

Therefore, 423 nm is equal to $423\text{ nm }\times \frac{1\text{ m}}{{10}^9\text{nm}}=423\times {\text{ 10}}^{-9}\text{m}$

Put the value of Planck’s constant and speed of light that is $6.626\times {10}^{-34}\text{ J}\text{.s}$ and $3\times {10}^8\text{ m/s}$

$E=6.626\times {10}^{-34}\text{ J}\text{.s}\times \frac{3\times {10}^8\text{ m/s}}{423\times {\text{ 10}}^{-9}\text{m}}$

$E=6.626\times {10}^{-34}\text{ J}\text{.s}\times \text{7}\text{.0}\times {\text{10}}^{14}{\text{ s}}^{-1}=46.382\times {10}^{-20}\text{J}$

Interpretation Introduction

(c)

Interpretation:

The energy in kilojoules per mole should be calculated for 423 nm wavelength.

Concept introduction:

A conversion factor is a numerical ratio which is used to show a measurement in one unit as another unit. It is always equal to 1.

Answer to Problem 1QAP

Energy in kilojoules per mole = $279.3\text{ kJ}$

Explanation of Solution

1 kJ = 1000 J

1 mole photons = $6.022\times {10}^{23}\text{ photons}$

Energy in joules per photon = $46.382\times {10}^{-20}\text{J}$

Energy per mole of photons is calculated by:

Energy per mole of photons

= $\text{1 mol of photon}\times \frac{46.382\times {10}^{-20}\text{J}}{1\text{ photon}}\times \frac{6.022\times {\text{10}}^{23}\text{ photons}}{1\text{ mol photon}}\times \frac{1\text{ kJ}}{1000\text{ J}}$

= $279.3\text{ kJ}$