Question

Question #b058f

Answer 1

`1.32 * 10^(-6)"m"`

Explanation:

Start by calculating the energy of a single photon in joules.

You know that if you get `1` mole of photons at this wavelength, you get

`90.5 color(red)(cancel(color(black)("kJ"))) * (10^3color(white)(.)"J")/(1color(red)(cancel(color(black)("kJ")))) = 9.05 * 10^4` `"J"`

of energy. As you know, `1` mole of photons will contain `6.022 * 10^(23)` photons, as given by Avogadro's constant. This implies that the energy of a single photon will be equal to

`1 color(red)(cancel(color(black)("photon"))) * (9.05 * 10^4color(white)(.)"J")/(6.022 * 10^(23)color(red)(cancel(color(black)("photons")))) = 1.503 * 10^(-19)` `"J"`

Now, the energy of a photon is directly proportional to its frequency, as described by the Planck - Einstein equation

`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

Rearrange to solve for the frequency of the photon

`E = h * nu implies nu = E/h`

Plug in your values to find

`nu = (1.503 * 10^(-19) color(red)(cancel(color(black)("J"))))/(6.626 * 10^(-34)color(red)(cancel(color(black)("J"))) "s") = 2.268 * 10^14` `"s"^(-1)`

Finally, to find the wavelength of the photon, use the fact that frequency and wavelength have an inverse relationship that can be described by the equation

`color(blue)(ul(color(black)(nu * lamda = 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 to solve for the wavelength of the photon

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

Plug in your values to find

`lamda = (3 * 10^8 color(white)(.)"m" color(red)(cancel(color(black)("s"^(-1)))))/(2.268 * 10^(14)color(red)(cancel(color(black)("s"^(-1))))) = color(darkgreen)(ul(color(black)(1.32 * 10^(-6)color(white)(.)"m")))`

The answer is rounded to three sig figs, the number of sig figs you have for the energy of a mole of photons.