Question

Question #1249f

Answer 1

Here;s how you can do that.

Explanation:

The idea here is that you must prove that the energy of a photon and its wavelength have an inverse relationship, i.e. when the wavelength increases, the energy decreases and vice versa.

To do that, you can start from the Planck - Einstein relation, which shows that 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

Now, you should also know that 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)`

Rearrange this equation to find an expression for the frequency of the wave in terms of its wavelength

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

Plug this into the Planck - Einstein relation to find

`E = h * c/(lamda)`

`color(darkgreen)(ul(color(black)(E = h * c * 1/(lamda))))`

Since the product between `h` and `c` is always constant, you can say that

`E = "constant" xx 1/(lamda)`

which is equivalent to

`E prop 1/(lamda)`

The energy of a photon, `E`, is inversely proportional to its wavelength, `lamda`, which is equivalent to saying that the energy of a photon is directly proportional to the inverse of its wavelength, `1/(lamda)`

You can even find a numeral value for the product between `h` and `c`

`h * c = 6.626 * 10^(-34)"J" color(red)(cancel(color(black)("s"))) * 3 * 10^(8)"m" color(red)(cancel(color(black)("s"^(-1))))`

`h * c = 1.9878 * 10^(-25)"J m"`

This means that you have

`color(darkgreen)(ul(color(black)(E = 1.9878 * 10^(-25)"J m" * 1/(lamda))))`