Use appropriate relationships from chapter 8 of the textbook to determine the wa
ID: 552124 • Letter: U
Question
Use appropriate relationships from chapter 8 of the textbook to determine the wavelength of the line in the emission spectrum of He+ produced by an electron transition from ni = 7 to nf = 3.
I got 1005nm before but it is apparently incorrect:
This is the correct wavelength for the transition in a hydrogen atom, but this species is the He+ cation which has two protons (Z=2) in its nucleus, and will have an entirely different emission spectrum compared to that of a single electron atom with only one proton. Use the equation
En=Z2RHn2
where En is the energy of the electron in the nth level, Z is the atomic number corresponding to the number of protons in the nucleus (or total nuclear charge), RH is the Rydberg constant, and n is the energy level, to find the energy of the transition in a single electron species with more than one proton. Then, the wavelength can be calculated once the energy of the transition is known.
Explanation / Answer
As you correctly said, we are dealing with He+ ion and not hydrogen atom. He+ ion is hydrogen LIKE in that they both have one electron; but the atomic number of helium is 2 and that must be taken into consideration. To do that we simply need to multiply the usual energy level formula by Z2. Make sure to use RH value = 2.18 X 10-18 J as energy is in Joules. Substitute all the values in the formula:
Delta E = Z2RH [(1/nf2) - (1/ni2)]
E3 - E7 = Z2RH [(1/32) - (1/72)]
E3 - E7 = 4 X 2.18 X 10-18 J X [(1/9) - (1/49)]
E3 - E7 = 8.72 X 10-18 J (0.0907)
E3 - E7 = 7.909 X 10-19 J
We know the relation -
E = h X frequency
It can also be written in terms of the wavelength.
E = (h X c) / wavelength
wavelength = (h X c) / E
h = Planck's constant in Joules.second and c = speed of light in m/s
wavelength = (6.626 X 10-34 J.s X 3 X 108 m/s) / 7.909 X 10-19 J
wavelength = 2.513 X 10-7 m
Convert the wavelength from metres to nanometres.
We know that 1 m = 1 X 109 nm
wavelength = 251.3 nm
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