Deuterium was discovered in 1932 by Harold Urey by measuring the small change in wavelength for a particular transition in 1H and 2H. This is because the
wavelength of transition depends to a certain extent on the nuclear mass. If nuclear motion is taken into account then the electrons and nucleus revolve
around their common centre of mass. Such a system is equivalent to a single particle with a reduced mass µ, revolving around the nucleus at a distance equal
to the electron-nucleus separation. Here µ = me M/(me +M) where M is the nuclear mass and me is the electronic mass. Estimate the percentage difference in
wavelength for the 1st line of the Lyman series in 1H and 2H. (Mass of 1H nucleus is 1.6725 × 10–27 kg, Mass of 2H nucleus is 3.3374 × 10–27 kg, Mass of
electron = 9.109 × 10–31 kg.)
The energy of an electron in the nth state is
En = -μZ2e4/8ε02h2(1/n2)
For hydrogen atom, μH = meMh/(me + Mh)
For deuterium atom, μD = meMd/(me + Md)
The wavelength of the transition for hydrogen is = λH = 3/4 μHe4/8ε02h2c
The wavelength of the transition for deuterium is = λD = 3/4 μDe4/8ε02h2c
The difference between the wavelength is = λ = λH – λD = 2.714 × 10-2%