Question

Deutrium 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 depend 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 = m_eM/(m_e +M) where M is the nuclear mass and m_e 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.)

Answer 1

Taking into account the nuclear motion, the stationary state energies shall be,

Let μ_H be the reduced mass of Hydrogen and μ_D that of Deutrium. Then the frequency of the 1^st Lyman line in Hydrogen is

Thus the wavelength of the transition is

 

The wavelength of the transition for the same line in Deutrium is