A 12.5 eV electron beam is used to bombard gaseous hydrogen at room temperature. What series of wavelengths will be emitted?

Explanation

Ground state energy of hydrogen gas at room temperature = - 13.6 eV.
When the energy beam used to bombard gaseous hydrogen then its energy becomes = - 13.6 + 12.5 eV = -1.1 eV.
Now energy in nth orbit
`E_(n)=-(13.6)/(n^(2))eV`
`:.-1.1eV=(13.6)/(n^(2))eV`
`:. n^(2)=(13.6)/(1.1)=12.36`
`:. n=3.51`
but possibel orbits n=3

`:.` Spectral line of emiision spectra
`=(n(n-1))/(2)`
`=(3(-1))/(2)`
=3
Hence in transition from n = 3 to n = 1 an n = 2 to n = 1 two lines of Lyman series and i transition n = 3 to n = 2, one line of Balme series are obtained.
`rArr` In transition from n = 3 to n = 1 th wavelength of `beta`-line emitted in Lyman series.
`lambda_(31)=(hc)/(E_(1)-E_(3))=(6.62xx10^(-34)xx3xx10^(8))/((-13.6-(-1.51))`
`:. lambda_(31)=(19.875xx10^(-26))/(12.09xx1.6xx10^(-19))`
`=1.0274xx10^(-7)m`
`~~103xx10^(-9)m=103 m`
In transition from n = 2 to n = 1, the wavelength of c-line emitted in Lyman series,
`lambda_(21)=(hc)/(E_(1)-E_(2))=(6.625xx10^(-34)xx3xx10^(8))/([-13.6-(3.4)]eV)`
`=(19.875xx10^(-26))/(10.2xx1.6xx10^(-19))`
`=1.2178xx10^(-7)m`
`~~122xx10^(-9)m=122 mm`
`rArr` In transition from n = 3 to n = 2. wavelength of a-line emitted in Balmer series,
`lambda_(32)=(hc)/(E_(2)-E_(3))=(6.625xx10^(-34)xx3xx10^(8))/([-3.4-(-1.51)]eV)`
`:. lambda_(32)=(19.875xx10^(-26))/(1.89xx1.6xx10^(-19))`
`6.5724xx10^(-7)m`
`=657xx10^(-9)m`
`657nm`