Question #2e787

1 Answer
Oct 6, 2017

Answer:

#n_i = 8 #

Explanation:

Your tool of choice here will be the Rydberg equation, which looks like this

#1/(lamda) = R * (1/n_f^2 - 1/n_i^2)#

Here

  • #lamda# is the wavelength of the photon
  • #R# is the Rydberg constant, equal to #1.097 * 10^(7)# #"m"^(-1)#
  • #n_I# is the initial energy level of the transition
  • #n_f# is the final energy level of the transition

Now, you know that the electron starts on an initial energy level #n_i# and ends up on the third energy level, so

#n_f = 3#

Moreover, you know that this transition is accompanied by the emission of a photon of wavelength

#lamda = 9.54 * 10^(-7)# #"m"#

Rearrange the Rydberg equation to solve for #n_i#

#1/(lamda) = R * (n_i^2 - n_f^2)/(n_i^2 * n_f^2)#

This is equivalent to

#n_i^2 * n_f^2 = lamda * R * n_i^2 - lamda * R * n_f^2#

#lamda * R * n_i^2 - n_i^2 * n_f^2 = lamda * R * n_f^2#

#n_i^2 * (lamda * R - n_f^2) = lamda * R * n_f^2#

Finally, you should end up with

#n_i = sqrt( (lamda * R * n_f^2)/(lamda * R - n_f^2))#

Plug in your values to find

#n_i = sqrt( (9.54 * color(blue)(cancel(color(black)(10^(-7))))color(red)(cancel(color(black)("m"))) * 1.097 * color(blue)(cancel(color(black)(10^7)))color(red)(cancel(color(black)("m"^(-1)))) * 3^2)/(9.54 * color(blue)(cancel(color(black)(10^(-7))))color(red)(cancel(color(black)("m"))) * 1.097 * color(blue)(cancel(color(black)(10^7)))color(red)(cancel(color(black)("m"^(-1)))) - 3^2))#

#n_ i = 8.017 ~~ color(darkgreen)(ul(color(black)(8)))#

Therefore, you can say that this electron underwent a #n_i = 8 -> n_f = 3# transition.