Question #b058f
1 Answer
Explanation:
Start by calculating the energy of a single photon in joules.
You know that if you get
#90.5 color(red)(cancel(color(black)("kJ"))) * (10^3color(white)(.)"J")/(1color(red)(cancel(color(black)("kJ")))) = 9.05 * 10^4# #"J"#
of energy. As you know,
#1 color(red)(cancel(color(black)("photon"))) * (9.05 * 10^4color(white)(.)"J")/(6.022 * 10^(23)color(red)(cancel(color(black)("photons")))) = 1.503 * 10^(-19)# #"J"#
Now, the energy of a photon is directly proportional to its frequency, as described 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
Rearrange to solve for the frequency of the photon
#E = h * nu implies nu = E/h#
Plug in your values to find
#nu = (1.503 * 10^(-19) color(red)(cancel(color(black)("J"))))/(6.626 * 10^(-34)color(red)(cancel(color(black)("J"))) "s") = 2.268 * 10^14# #"s"^(-1)#
Finally, to find the wavelength of the photon, use the fact that frequency and wavelength have an inverse relationship that can be described 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 to solve for the wavelength of the photon
#nu * lamda = c implies lamda = c/nu#
Plug in your values to find
#lamda = (3 * 10^8 color(white)(.)"m" color(red)(cancel(color(black)("s"^(-1)))))/(2.268 * 10^(14)color(red)(cancel(color(black)("s"^(-1))))) = color(darkgreen)(ul(color(black)(1.32 * 10^(-6)color(white)(.)"m")))#
The answer is rounded to three sig figs, the number of sig figs you have for the energy of a mole of photons.