Name the different series obtained in hydrogen spectrum and give formulas for finding the wave number.

Explanation

In 1885, the first such series was observed by a Swedish school teacher Johann Jakob Balmer in the visible region of the hydrogen spectrum which is shown in the figure.

The line with the longest wavelength 656.3 nm in the red is called `H_(alpha)`.
The 486.1 nm wavelength of line that appears in blue-green region is called `H_(beta)`.
The 434.1 nm wavelength of line that appears in violet region is called `H_(gamma)`.
As the wavelength decreases, the lines appear closer together and are weaker in intensity.
Balmer found a simple empirical formula for the observed wavelengths.
`(1)/(lambda)=R[(1)/(2^(2))-(1)/(n^(2))]`....(1)
where `lambda`= wavelength
R= Rydberg constant `=1.097xx10^(7) m^(-1)`
n= Integal values 3,4,5...
Taking n = 3 in this equation one obtains the wavelength of the `H_(alpha)` line (maximum wavelength)
`:.(1)/(lambda)=1.097xx10^(7)[(1)/(2^(2))-(1)/(3^(2))]`
`=1.097xx10^(7)((5)/(36))`
`:. (1)/(lambda)0.1524x10^(7)m^(-1)`
`:. lambda=6.52xx10^(-7)m`
or `lambda=656.2nm`
`rArr` Taking n = 4, one obtain the wavelength of `H_(beta)`
`(1)/(lambda)=R[(1)/(2^(2))-(1)/(4^(2))]`
`(1)/(lambda=1.097xx10^(7))[(3)/(16)]`
`:. (1)/(lambda)=0.2057xx10^(7)m^(-1)`
`:. lambda=4.861xx10^(-7)m`
`:. lambda486.1nm`
`rArr` Taking `n=oo, "inifinite"^(th)` line obtained whose wavelength is the shortest (small).
`(1)/(lambda)=R[(1)/(1^(2)-(1)/(oo^(2)))]=(R)/(4)`
`:. lambda=(4)/(R)=(4)/(10.97xx10^(7))=3.646xx10^(-7)m`
`:. lambda=364.6 nm`
Beyond this limit, no further distinct lines appear, instead only a faint continuous spectrum is seen.
Moreover, the other discovered spectra lines are named by their inventor and their formulas are as follows:
Lyman series : Found in the ultraviolet region.
`(1)/(lambda)=R[(1)/(1^(2))-(1)/(n^(2))]` where n=2,3,4,...
If n=2 then the `H_(alpha)` line of the Lyman series is obtained.
If n=3 then the `H_(beta)` line of the Lyman series is obtained.
If n=4 then the `H_(gamma)` line of the Lyman series is obtained.
Paschen series : Found in the infrared region.
`(1)/(lambda)=R[(1)/(3^(2))-(1)/(n^(2))]` where n=4, 5,6,...
If n=4 then the `H_(alpha)` line of the Paschen series is obtained.
If n=5 then the `H_(beta)` line of the Paschen series is obtained.
If n=6 then the `H_(gamma)` line of the Paschen series is obtained.
Brackett series : Found in infrared region.
`(1)/(lambda)=R[(1)/(4^(2))-(1)/(n^(2))]` where n=5,6,7,.....
If n=5, then the `H_(alpha)` line of the Brckett series is obtained.
If n=5, then the `H_(beta)` line of the Brckett series is obtained.
If n=6, then the `H_(beta)` line of the Brckett series is obtained.
If n=7 then the `H_(gamma)` line of the Brckett series is obtained.
Pfund series : Found in infrared region.
`(1)/(lambda)R[(1)/(5^(2))-(1)/(n^(2))]` where n=6,7,8,...
If n=6, then the `H_(alpha)` line of P fund series is obtained.
If n=7, then the `H_(beta)` line of P fund series is obtained.
If n=8 then the `H_(gamma)` line of P fund series is obtained.
The general formula for all the above series is a follows.
`(1)/(lambda)=R[(1)/(m^(2))-(1)/(n^(2))]` where m=n-1
If `m=1 rArr` Lyman sereis, m=2, Balmer series.
m = 3, Paschen series, m = 4, Brackett series and m = 5, Pfund series formulas are obtain.