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 10^{8} 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 10^{8} 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**Answer:**

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 10^{9}.

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 10^{8} 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 10^{23} 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 10^{5} J/mol

for one mole, the energy is 2.394 x 10^{5} 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 10^{5} J)(1 kJ / 1000 J)

energy = 2.394 x 10^{2} kJ or 239.4 kJ