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 = 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
