Question

Question #83917

Answer 1

`4.35 * 10^(-7)"m"`

Explanation:

The idea here is that the energy of the photon is directly proportional to its frequency, as shown 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

This equation basically tells you that the energy of a photon increases as its frequency increases.

Now, the problem wants you to find the wavelength of the photon; as you know, 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 wave
• `c` is the speed of light in a vacuum, usually given as `3 * 10^8"m s"^(-1)`

Use the Planck - Einstein equation to find the frequency of the photon

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

Plug in your value to find

`nu = (4.57 * 10^(-19)color(red)(cancel(color(black)("J"))))/(6.626 * 10^(-34)color(red)(cancel(color(black)("J")))"s"^(-1)) = 6.897 * 10^(14)`

Now use this value to find the wavelength of the photon

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

Plug in your value to find

`lamda = (3 * 10^8 color(white)(.)"m" color(red)(cancel(color(black)("s"^(-1)))))/(6.897 * 10^(14)color(red)(cancel(color(black)("s"^(-1))))) = color(darkgreen)(ul(color(black)(4.35 * 10^(-7)"m")))`

The answer is rounded to three sig figs.