Question

Question #8d66e

Answer 1

`2 * 10^16` `"photons"`

Explanation:

For starters, you should know that the energy of a photon is directly proportional to its frequency as given by the Planck - Einstein relation

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

Here

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

Now, frequency and wavelength have an inverse relationship as given by the equation

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

Here

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

This means that you can rewrite the Planck - Einstein relation using the wavelength of the photon

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

to get

`E = h * c/(lamda)`

Plug in your value to find the energy of a single photon of wavelength

`4000 color(red)(cancel(color(black)("pm"))) * "1 m"/(10^12color(red)(cancel(color(black)("m")))) = 4 * 10^(-9)` `"m"`

and you will get

`E = 6.626 * 10^(-34)color(white)(.)"J" color(red)(cancel(color(black)("s"))) * (3 * 10^8 color(red)(cancel(color(black)("m"))) color(red)(cancel(color(black)("s"^(-1)))))/(4 * 10^(-9)color(red)(cancel(color(black)("m"))))`

`E = 4.97 * 10^(-17)` `"J"`

This means that in order to get a total of `"1 J"` of energy, you need to have

`1 color(red)(cancel(color(black)("J"))) * "1 photon"/(4.97 * 10^(-17)color(red)(cancel(color(black)("J")))) = color(darkgreen)(ul(color(black)(2 * 10^(16)color(white)(.)"photons")))`

The answer is rounded to one significant figure, the number of sig figs you have for the wavelength of the photon.