# How to Solve an Energy From Wavelength Problem

## Spectroscopy Example Problem

This example problem demonstrates how to find the energy of a photon from its wavelength.

### Energy from Wavelength Problem - Laser Beam Energy

The red light from a helium-neon laser has a wavelength of 633 nm. What is the energy of one photon?
You need to use two equations to solve this problem:

The first is Planck's equation, which was proposed by Max Planck to describe how energy is transferred in quanta or packets.
E = hν

where
E = energy
h = Planck's constant = 6.626 x 10-34 J·s
ν = frequency
The second equation is the wave equation, which describes the speed of light in terms of wavelength and frequency:
c = λν

where
c = speed of light = 3 x 108 m/sec
λ = wavelength
ν = frequency

Rearrange the equation to solve for frequency:
ν = c/λ

Next, replace frequency in the first equation with c/λ to get a formula you can use:
E = hν
E = hc/λ

All that remains is to plug in the values and get the answer:
E = 6.626 x 10-34 J·s x 3 x 108 m/sec/ (633 nm x 10-9 m/1 nm)
E = 1.988 x 10-25 J·m/6.33 x 10-7 m E = 3.14 x -19 J
The energy of a single photon of red light from a helium-neon laser is 3.14 x -19 J.

### Energy of One Mole of Photons

While the first example showed how to find the energy of a single photon, the same method may be used to find the energy of a mole of photons. Basically, what you do is find the energy of one photon and multiply it by Avogadro's number.

A light source emits radiation with a wavelength of 500.0 nm. Find the energy of one mole of photons of this radiation. Express the answer in units of kJ.

It's typical to need to perform a unit conversion on the wavelength value in order to get it to work in the equation. First, convert nm to m. Nano- is 10-9, so all you need to do is move the decimal place over 9 spots or divide by 109.

500.0 nm = 500.0 x 10-9 m = 5.000 x 10-7 m

The last value is the wavelength expressed using scientific notation and the correct number of significant figures.

Remember how Planck's equation and the wave equation were combined to give:

E = hc/λ

E = (6.626 x 10-34 J·s)(3.000 x 108 m/s) / (5.000 x 10-17 m)
E = 3.9756 x 10-19 J

However, this is the energy of a single photon. Multiply the value by Avogadro's number for the energy of a mole of photons:

energy of a mole of photons = (energy of a single photon) x (Avogadro's number)

energy of a mole of photons = (3.9756 x 10-19 J)(6.022 x 1023 mol-1) [hint: multiply the decimal numbers and then subtract the denominator exponent from the numerator exponent to get the power of 10)

energy = 2.394 x 105 J/mol

for one mole, the energy is 2.394 x 105 J

Note how the value retains the correct number of significant figures. It still needs to be converted from J to kJ for the final answer:

energy = (2.394 x 105 J)(1 kJ / 1000 J)
energy = 2.394 x 102 kJ or 239.4 kJ