Deutrium was discovered in 1932 by Harold Urey by measuring the small change in wavelength for a particular transition in `.^(1)H` and `.^(2)H`. 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 `mu`, revolving around the nucleus at a distance equal to the electron -nucleus separation. Here `mu = m_(e) M//(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 `.^(1)H` and `.^(2)H`. (mass of `.^(1)H` nucleus is `1.6725 xx 10^(-27)` kg, mass of `.^(2)H` nucleus is `3.3374 xx 10^(-27)` kg, Mass of electron `= 9.109 xx 10^(-31) kg`.)

Explanation

If we take into account the nuclear motion, the stationary state energies will be given by
`E_(n)=-(mu Z^2e^4)/(8in_(0)^(2)h^2) (1/(n^2))`
Let `mu_(H)` be reduced mass of hydrogen `(H^1)` and `mu_(D)` be reduced mass of deuterium `(H^2)`.
The frequency of 1st of Lyman series in hydrogen is given by
`hv_(H)=(mu_(H)e^4)/(8pi in_0^2h^2)(1/(1^2)-1/(2^2))= (3//4 mu_(H)e^4) /(8in_0^2h^2)`
The wave number of transition is `barlambda_(H)=(3//4 mu_(H)e^4) /(8in_0^2h^3 c)`
Similarly, for deuterium, `barlambda_(D)=(3//4 mu_(D)e^4) /(8in_0^2h^3 c)`
`Deltabar lambda=barlambda_(D)-barlambda_(H)`
`:.` The percentage difference in wave number is `(Deltabar(lambda))/(bar(lambda)_(H)) xx 100 = ((bar(lambda)_(D)-bar(lambda)_(H)) xx100)/(bar(lambda)_(H)) = (mu_(D)-mu_(H))/(mu_(H)) xx 100`
Using `mu_(H)=(m_(e)M_(H))/(m_(e)+M_(H))` and `mu_(D)=(m_(e)M_(D))/(m_(e)+M_(D))`, we g et `(bar(Delta lambda))/(barlambda_(H))xx100=((m_(e)M_(D))/(m_(e)+M_(D))-(m_(e)M_(H))/(m_(e)+M_(H)))/(m_(e)M_(H)//(m_(e)+M_(H)))xx100`
As `m_(e)lt lt M_(H)lt ltM_(D)`
`:. (barDeltalambda)/(barlambda_(H))=[(M_(D))/(M_(H))xx(M_(H))/(M_(D))((1+m_(e)//M_(H))/(1+m_(e)//M_(D)))-1]xx100=[(1+(m_(e))/(M_(H)))(1+(m_(e))/(M_(D)))^(-1)-1]xx100`
`=[1+(m_(e))/(M_(H))-(m_(e))/(M_(D))-1]xx100=m_(e)[1/(M_(H))-1/(M_(D))]xx100`
`(bar(Deltalambda))/(bar(lambda)_(H))xx100=9.1xx10^(-31)[1/(1.6725xx10^(-27))-1/(3.3374xx10^(-27))]xx100`
`=9.1xx10^(-4)(0.5979-0.2996)xx100=2.714xx10^(-2)%`