Question

Question #72974

Answer 1

`2.41 * 10^5" J/mol"`

Explanation:

The idea here is that you can use the Planck - Einstein equation to write a relationship between wavelength and energy per photon.

Once you know how much energy you get per photon, you can use Avogadro's number to find how much energy you'd get per mole of photons.

So, the Planck - Einstein equation loooks like this

`color(blue)(E = h * nu)" "`, where

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

Notice that energy is proportional to frequency.

You know that frequency and wavelength have an inverse relationship

`color(blue)(nu * lamda = c)" "`, where

`lamda` - the wavelength of the photon
`c` - the speed of light in vacuum, approximately equal to `3.0 * 10^(8)"m s"^(-1)`

This implies that energy and wavelength will have an inverse relationship as well, since you can write

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

Now you have everything you need to calculate the energy of a single photon. Plug in your value for wavelength - do not forget to convert it from nanometers to meters

`E = (6.626 * 10^(-34)"J" * color(red)(cancel(color(black)("s"))) * 3.0 * 10^(8)color(red)(cancel(color(black)("m"))) color(red)(cancel(color(black)("s"^(-1)))))/(497 * 10^(-9)color(red)(cancel(color(black)("m")))) = 3.9996 * 10^(-19)"J"`

Now, one mole of photons will contain exactly `6.022 * 10^(23)` photons - this is known as Avogadro's number.

In your case, the energy of one mole of photons of wavalength `"497 nm"` will be

`3.9996 * 10^(-19)"J"/color(red)(cancel(color(black)("photon"))) * (6.022 * 10^(23)color(red)(cancel(color(black)("photons"))))/"1.00 mole" = color(green)(2.41 * 10^5" J/mol")`