Question #cf90a

1 Answer
Nov 1, 2015

#"93.8 nm"#

Explanation:

That formula will not give you the wavelength of the emitted photon, it will give you its energy.

You would then use the Planck - Einestein equation to get the wavelength of the photon.

That value given to you , #2.18 * 10^(-18)"J"#, represents the Rydberg constant expressed in Joules, rather than in #"m"^(-1)#.

So, the energy of a photon emitted when an electron drops from the #n=6# to the #n=1# energy level is

#E = R_E * (1/n_1^2 - 1/n_2^2)" "#, where

#R_E# - the Rydberg constant* expressed in Joules
#n_1# - the final energy level of the electron
#n_2# - the initial energy level of the electron

In your case, the energy of the photon would be

#E = 2.18 * 10^(-18)"J" * (1/1^2 - 1/6^2)#

#E = 2.18 * 10^(-18)"J" * (1 - 1/36) = 2.12 * 10^(-18)"J"#

The Planck - Einstein equation looks like this

#E = h * nu" "#, where

#E# - the energy of the photon
#h# - Planck's constant, equal to #6.626 * 10^(-34)"J s"#
#nu# - the frequency of the photon

You know that frequency and wavelength have an inverse relationship

#nu * lamda = c" "#, where

#lamda# - wavelength
#c# - the speed of light in vacuum, approximately equal to #3.0 * 10^(8)"m s"^(-1)#

This means that the relationship between energy and wavelength looks like this

#E = h * c/(lamda)#

Plug in your values and rearrange to solve for #lamda#

#lamda = (h * c)/E = (6.626 * 10^(-34)color(red)(cancel(color(black)("J"))) color(red)(cancel(color(black)("s"))) * 3.0 * 10^8"m" color(red)(cancel(color(black)("s"^(-1)))))/(2.12 * 10^(-18)color(red)(cancel(color(black)("J"))))#

#lamda = 9.376 * 10^(-8)"m" = "93.8 nm"#

SIDE NOTE You can incorporate the Planck - Einstein equation into the Rydberg equation

#E = R_E * (1/n_1^2 - 1/n_2^2)#

#h * c/(lamda) = R_E * (1/n_1^2 - 1/n_2^2)#

Rearrange to get

#1/(lamda) = R_E/(h * c) * (1/n_1^2 - 1/n_2^2)#

The term #R_E/(h * c)# is another version of Rydberg's constant, this time expressed in #'m"^(-1)#, equal to

#R = (2.18 * 10^(-18)color(red)(cancel(color(black)("J"))))/(6.626 * 10^(-34)color(red)(cancel(color(black)("J"))) color(red)(cancel(color(black)("s"))) * 3.0 * 10^8 "m" color(red)(cancel(color(black)("s"^(-1))))) ~~ 1.9073... * 10^(7)"m"^(-1)#