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How to Calculate the Wavelength of Light

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Introduction

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

How to Calculate the Wavelength of Radiation
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 = En – E0

Where En = Excitation energy of nth level

E0 = Ground state energy level

We can now say that

En – E0 = hf

and if f = c/λ

where c = speed of light = constant = 3 x 108m/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 En – E0 = hf to substitute f. This expression will now become

En – E0 = h x c/λ

The above expression becomes

En – E0 = hc/λ

Now we have

λ = hc/(En – E0)

Therefore, the formula for calculating the wavelength of radiation is λ = hc/(En – E0)

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 10-19J ]

Solution

Data:

electron volt, eV = 1.6 x 10-19J

The first excited state of the Hydrogen atom, En = E1 = -3.4eV = -3.4 x 1.6 x 10-19 = -54.4 x 10-20J

Ground state, E0 = -13.6eV = -13.6 x x 1.6 x 10-19 = -217.6 x 10-20J

planks constant, h = 6.6 x 10-34 Js

speed of light, c = 3 x 108m/s

λ = ?

Now we apply the formula for calculating the wavelength of radiation is λ = hc/(En – E0)

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

λ = (6.6 x 10-34 x 3 x 108) / (-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 10-8m

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 108 m/s]

Solution:

Data

Energy excitation, E = 4.5 x 10-19 J

Speed of light, c = 3 x 108 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 108)/(4.5 x 10-19) = (1980 x 10-28) / (4.5 x 10-19) = 4.4 x 10-7m

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 10-19J. Calculate the wavelength of its threshold frequency. [speed of light in vacuum = 3 x 108 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 10-19J

Threshold frequency, f =?

Planck’s constant, h = 6.6 x 10-34 Js

Speed of light, c = 3 x 108 m/s

Thus,

f = hc/W = (6.6 x 10-34 x 3 x 108) / 8.6 x 10-19 = 2.3 x 10-7

You may also like to read:

How to Calculate Power in a Circuit

Reference

wikipedia

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