## Introduction

In this post, I will apply a detailed method to guide you understand how to calculate the wavelength of light

### How to Derive the Formula for Calculating Wavelength of Radiation

Since E = hf

Where E = Energy level, h = Planck’s constant, and f = frequency

If ΔE = E_{n} – E_{0}

Where E_{n} = Excitation energy of n^{th} level

E_{0} = Ground state energy level

We can now say that

E_{n} – E_{0} = hf

and if f = c/λ

where c = speed of light = constant = 3 x 10^{8}m/s, and λ = wavelength of the radiation

By λ subject of the formula from f = c/λ. We will have

λ = c/f

Now we put the equation (f = c/λ) into E_{n} – E_{0} = hf to substitute f. This expression will now become

E_{n} – E_{0} = h x c/λ

The above expression becomes

E_{n} – E_{0} = hc/λ

Now we have

λ = hc/(E_{n} – E_{0})

Therefore, the formula for calculating the wavelength of radiation is λ = hc/(E_{n} – E_{0})

## Examples of How to Calculate the Wavelength of Light

Here are the solved problems on how to calculate the wavelength of radiation

### Example 1

The ground state of hydrogen may be represented by the energy – 13.6eV, and the first excited state by – 3.4eV. The scale in which an electron is completely free of the atom is zero energy. Calculate the wavelength of the radiation.[ Take Plank’s constant = 6.6 x 10^{-34} Js, eV = 1.6 x 10^{-19}J ]

**Solution**

**Data:**

electron volt, eV = 1.6 x 10^{-19}J

The first excited state of the Hydrogen atom, E_{n} = E_{1} = -3.4eV = -3.4 x 1.6 x 10^{-19} = -54.4 x 10^{-20}J

Ground state, E_{0} = -13.6eV = -13.6 x x 1.6 x 10^{-19} = -217.6 x 10^{-20}J

planks constant, h = 6.6 x 10^{-34} Js

speed of light, c = 3 x 10^{8}m/s

λ = ?

Now we apply the formula for calculating the wavelength of radiation is λ = hc/(E_{n} – E_{0})

by substituting our data into the above equation, we now have

λ = (6.6 x 10^{-34} x 3 x 10^{8}) / (-3.4 -[13.6])eV

The above expression will now become

λ = (1980 x 10^{-28}) / (10.2eV)

which is also equal to

λ = (1980 x 10^{-28}) / (10.2 x 1.6 x 10^{-19})

Therefore, λ is equal to

λ = (1980 x 10^{-28}) / (163.2 x 10^{-20})

Thus,

λ = 12.132 x 10^{-8}m

Therefore, the wavelength λ, of the radiation is 12.132 x 10^{-8} meters

### Example 2

An electron jumps from one energy level to another in an atom radiating 4.5 x 10^{-19} joules of energy. If Planck’s constant is 6.6 x 10^{-34} Js. What is the wavelength of the radiation? [Take c = 3 x 10^{8} m/s]

**Solution:**

**Data**

Energy excitation, E = 4.5 x 10^{-19} J

Speed of light, c = 3 x 10^{8} m/s

Planck’s constant, h = 6.6 x 10^{-34} Js

The wavelength of the radiation, λ =?

and λ = hc/E = (6.6 x 10^{-34} x 3 x 10^{8})/(4.5 x 10^{-19}) = (1980 x 10^{-28}) / (4.5 x 10^{-19}) = 4.4 x 10^{-7}m

Therefore, the wavelength of the radiation λ is 4.4 x 10^{-7} meters

### Example 3

The work function of a metal is 8.6 x 10^{-19}J. Calculate the wavelength of its threshold frequency. [speed of light in vacuum = 3 x 10^{8} m/s and Planck’s constant = 6.6 x 10^{-34} Js]

**Solution**

**Data**

We apply the formula f = hc/W

The work function of a metal, W = 8.6 x 10^{-19}J

Threshold frequency, f =?

Planck’s constant, h = 6.6 x 10^{-34} Js

Speed of light, c = 3 x 10^{8} m/s

Thus,

f = hc/W = (6.6 x 10^{-34} x 3 x 10^{8}) / 8.6 x 10^{-19} = 2.3 x 10^{-7}

*You may also like to read:*

How to Calculate Power in a Circuit

**Reference**