(a) (i) Find the wavelength of the radiation required to excite thye elctron in `Li(++)` from the first to the third Bohr orbit.
(ii) How many spectal lines are observed in the emission spectrum of the above excited system ?
(b) The energy needed to detach the electron of a hydrogen-like ion in ground state in 4 rydberg.
(i) What is the wavelength of the radiation emitted when the electron jumps from the first excited state ot the ground state ?
(ii) What is the radius of first orbit ?
(c ) A hydrogen sample is prepared in a particular excited state A. Photons of energy 2.55 eV get absorbed into the sample to take some electrons to a further excited state B. Find the quantum number of the A and B.
(d) A hydrogen atom in a state having a binding energy of 0.85 eV makes transition to a state with excitation energy 10.2 eV. (i) Identify the quantum number n of the upper and the lower energy states.
Find `lambda`.

Explanation

(a) (i)` (1)/(lambda)=Z^(2)R ((1)/(n_(1)^(2))-(1)/(n_(2)^(2)))=(3)^(2) R((1)/(1)^(2))-(1)/(3^(3)))=8 R`
`lambda=(1)/(8R)=(1)/(8xx1.097xx10^(7))=0.114xx10^(-7)=11.4 nm`
(ii) `(n(n-1))/(2)=(3(3-1))/(2)=3` (b) 1 rydberg=13.6 eV
`DeltaE=13.6 Z^(2)((1)/(n_(1)^(2)-(1)/(n_(2)^(2))))`
`4xx13.6=13.6 Z^(2)((1)/(1^(2))-(1)/(oo)) implies Z^(2)=4`
Z=2
(i) `(1)/(lambda)=Z^(2)R ((1)/(n_(1)^(2)-(1)/(n_(2)^(2))))=(2)^(2)R ((1)/(1^(2))-(1)/(2^(2))) =3R`
` lambda =(1)/(3R)=(1)/(3xx1.097xx10^(7))=0.303xx10^(-7) m=30.3 nm`
(ii) `r_(n)=0.53 (n^(2))/(Z)`Å
`r_(1)=0.53((1)^(2))/(2)=0.13` Å
(c)`DeltaE=2.55 eV` "for H- atom corresponds to" `n=2 to no=4 n_(4)=2, n_(B)=4`
(d) `E_(n)=(13.6)/(n^(2)) eV implies 0.85=(13.6)/(n^(2))`
`n^(2)=16 implies n=4`
Excitation energy 10.2 eV corresponds from n=1 to n=2
(i) 4 to 2
(ii) `(1)/(lambda)=R((1)/(2^(2))-(1)/(4^(2))=(3)/(16))`
`lambda=(16)/(3R)=(16)/(3xx1.097xx10^(7))`
=486 nm