What must be the velocity of electrons if their associated wavelength is equal to the longest wavelength line in the Lyman series?

Express your answer using four significant figures.

Should I be using the Balmer equation?
ν=(3.2881×1015s−1)(122−1n2)

Or should I apply the Rydberg one?
1λ=R∞(1n21−1n22)

1 Answer
Michael · Stefan V.
Nov 14, 2016

v=5.987×103xm/s

Explanation:

The longest wavelength in the Lyman series corresponds to the

n=2→n=1

transition and can be calculated using the Rydberg equation

1λ=R∞⋅(1n2f−1n2i)

with

  • R∞≈1.097373⋅107m−1
  • nf=1
  • ni=2

Rearrange to solve for λ

λ=1R∞⋅(1−1n2i)

Plug in your values to find

λ=11.097373⋅107m−1⋅(1−122)=1.215⋅10−7m

The wavelength of a moving electron is given by the de Broglie expression:

λ=hmv

m is the mass which is 9.1094×10−31xkg

h is the Planck Constant which has the value 6.626×10−34xJ.s

Rearranging:

v=hmλ

Putting in the numbers:

v=6.626×10−319.1094×10−31×1.215×10−7xm/s

v=5.987×103xm/s

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