Question

How much energy does a mole of yellow photons of wavelength 527 nm have?

Answer 1

`"227 kJ"`

Explanation:

The first thing to do here is to calculate the energy of a single photon of wavelength equal to `"527 nm"`, then use Avogadro's number to scale this up to the energy of a mole of such photons.

So, according to the Planck - Einstein relation, the energy of a photon is directly proportional to its frequency

`color(blue)(ul(color(black)(E = h * nu )))`

Here

• `E` is the energy of the photon
• `h` is Planck's constant, equal to `6.626 * 10^(-34)"J s"`
• `nu` is the frequency of the photon

As you know, the frequency of a wave is inversely proportional to its wavelength as described by the equation

`color(blue)(ul(color(black)(lamda * nu = c )))`

Here

• `lamda` is the wavelength of the wave
• `c` is the speed of light in a vacuum, usually given as `3 * 10^8"m s"^(-1)`

Rearrange the above equation to solve for `nu`

`lamda * nu = c implies nu = c/(lamda)`

Convert the wavelength of the photon from nanometers to meters and plug in your values to find

`nu = (3 * 10^8color(red)(cancel(color(black)("m")))"s"^(-1))/(527 * 10^(-9)color(red)(cancel(color(black)("m")))) = 5.693 * 10^(14)"s"^(-1)`

This photon will have an energy of

`E = 6.626 * 10^(-34)"J" color(red)(cancel(color(black)("s"))) * 5.693 * 10^(14)color(red)(cancel(color(black)("s"^(-1))))`

`E = 3.772 * 10^(-19)"J"`

Finally, use Avogadro's constant

`color(blue)(ul(color(black)("1 mole" = 6.022 * 10^(23)"photons")))`

to calculate the energy of one mole of photons

`3.772 * 10^(-19)"J"/color(red)(cancel(color(black)("photon"))) * (6.022 * 10^(23)color(red)(cancel(color(black)("photons"))))/"1 mole photons" = color(darkgreen)(ul(color(black)("227 kJ")))`

The answer is expressed in kilojoules--keep in mind that `"1 kJ" = 10^3` `"J"`--and is rounded to three sig figs, the number of sig figs you have for the wavelength of the photon.