A gas of hydrogen - like ion is perpendicular in such a way that ions are only in the ground state and the first excite state. A monochromatic light of wavelength `1216 Å` is absorbed by the ions. The ions are lifted to higher excited state and emit emit radiation of six wavelength , some higher and some lower than the incident wavelength. Find the principal quantum number of the excited state identify the nuclear charge on the ions . Calculate the values of the maximum and minimum wavelengths.

Explanation

The energy state of hydrogen-like atom is given by
`E_(n) = - (B)/(n^(2))`
where `B = Z^(2) Rhc`.
The emission of six wavelength implies that the higher excited state corresponds to quantum number `n = 4` . As some wavelength are lower and some wavelength are higher , it implies that the initial excited state is `n = 2` . Thus the principal quantum number of excited states are `n = , n = 3, n = 4`.

`:.(hc)/(lambda) = B ((1)/(2^(2)) - (1)/(4^(2))) implies (hc)/(lambda) = (3 B)/(16)`
`:. B = (1.6 xx 6.62 xx 10^(-34) xx 3 xx 10^(8))/(3 xx (1216 xx 10^(-10))`
`= 8.71 xx 10^(-18)J`
If `Z` is atimic number of necleus, then `B = Z^(2) Rhc`
`:. Z^(2) = (B)/(Rhc) = (8.17 xx 10^(-18)J)/(13.6 xx 1.6 xx 10^(-19) = 4`
`:. Z = 2`
The maximum wavelength of emitted redition corresponds to transition `n = 4 to n = 3` and is given by
`(hc)/(lambda _(max)) = B ((1)/(3^(2)) - (1)/(4^(2)))`
`implies lambda _(max) = (144 hc)/(7 B) = (144 xx 6.62 xx 10^(-34) xx 3 xx 10^(8))/(7 xx 8.71 xx 10^(-18))`
`= 4.69 xx 10^(-7) m`
The minimum wavelength of emitted rediction corresponds to transition `n = 4 to n = 1` and is given by
`(hc)/(lambda _(max)) = B ((1)/(1^(2)) - (1)/(4^(2)))`
`implies lambda _(max) = (16 hc)/(15 B) = (16 xx 6.62 xx 10^(-34) xx 3 xx 10^(8))/(15 xx 8.71 xx 10^(-18))`
`= 0.2 xx 10^(-7) m`