Calculate the wavelength of the n= 4 -> 3 transition in He

  • Thread starter Flucky
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  • #1
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Homework Statement


Calculate the wavelength of the n = 4 → 3 transition in 4He+ to an accuracy of 4 significant figures. (R∞=109 737 cm-1.) (Fine structure effects can be neglected.)

Homework Equations


[itex] \frac{1}{λ} = \frac{m}{m_e} R_∞ (\frac{1}{n_1^2} - \frac{1}{n_2^2})[/itex]

where λ is wavelength, m is the reduced mass, and [itex]R_∞[/itex] is Rydberg constant (1.1x105 cm-1)

[itex] m = m_e \frac{m_N}{m_e + m_N} [/itex]

where m is the reduced mass, [itex]m_e[/itex] is mass of an electron, and mN is the mass of the nucleus.

The Attempt at a Solution


In order to come out with a positive wavelength I set n1 to 3 and n2 to 4.

Now the problem I have is with the reduced mass, m. To try and make sense of it I did everything in SI units:

me = 9.11x10-31
mN = 2 protons + 2 neutrons = 2x(1.673x10-27 kg) + 2x(1.675x10-27 kg) = 6.696x10-27 kg

So that gives:

m = me x [itex]\frac{6.696x10^-27}{9.11x10-31 + 6.696x10^-27}[/itex] = 0.9999me

Plugging all the numbers in gives:

[itex]\frac{1}{λ}[/itex] = 5347 cm-1

∴ λ = 1.870x10-4 cm = 1.870x10-6 m = 1870 nm


However the answer is 468.7 nm which means the reduced mass m should equal 3.99 me.

It's only a 1 mark question so I am obviously understanding it all wrong somewhere.

Any help would be very much appreciated.
 

Answers and Replies

  • #2
954
117
You are dealing with helium and not hydrogen. Helium has a nuclear charge of 2. You need to take that into account in modifying your Rydberg formula.
 
  • #3
95
1
You are dealing with helium and not hydrogen. Helium has a nuclear charge of 2. You need to take that into account in modifying your Rydberg formula.
I had no idea the equation would differ. Just looked up 'Rydberg formula' and a Z2 appears. Big thanks, I was beginning to pull my hair out.
 

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