Question #1249f

1 Answer
Feb 16, 2017

Answer:

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))))#