How much energy does a mole of yellow photons of wavelength 527 nm have?
1 Answer
Explanation:
The first thing to do here is to calculate the energy of a single photon of wavelength equal to
So, according to the Planck - Einstein relation, 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
As you know, the frequency of a wave is inversely proportional to its wavelength as described by the equation
#color(blue)(ul(color(black)(lamda * nu = 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 the above equation to solve for
#lamda * nu = c implies nu = c/(lamda)#
Convert the wavelength of the photon from nanometers to meters and plug in your values to find
#nu = (3 * 10^8color(red)(cancel(color(black)("m")))"s"^(-1))/(527 * 10^(-9)color(red)(cancel(color(black)("m")))) = 5.693 * 10^(14)"s"^(-1)#
This photon will have an energy of
#E = 6.626 * 10^(-34)"J" color(red)(cancel(color(black)("s"))) * 5.693 * 10^(14)color(red)(cancel(color(black)("s"^(-1))))#
#E = 3.772 * 10^(-19)"J"#
Finally, use Avogadro's constant
#color(blue)(ul(color(black)("1 mole" = 6.022 * 10^(23)"photons")))#
to calculate the energy of one mole of photons
#3.772 * 10^(-19)"J"/color(red)(cancel(color(black)("photon"))) * (6.022 * 10^(23)color(red)(cancel(color(black)("photons"))))/"1 mole photons" = color(darkgreen)(ul(color(black)("227 kJ")))#
The answer is expressed in kilojoules--keep in mind that