Using the Rydberg formula, calculate the wavelengths of the first four spectral lines in the Lyman series of the hydrogen spectrum.

Explanation

The Rydberg formula is
`hc//lamda_("if")=(me^(4))/(8 epsi_(0)^(2)h^(2))((1)/(n_(1)^(2))-(1)/(n_(1)^(2)))`
The wavelengths of the first four lines in the Lyman series correspond to transitions from `n_(i) = 2, 3, 4, 5 to n_(f) = 1.` We know that
`(me^(4))/(8epsi_(0)^(2)h^(2))=13.6 eV =21.76 xx 10^(-19)J`
Therefore,
`lamda_("if")=(hc)/(21.76xx10^(-19)((1)/(1)-(1)/(n_(1)^(2)))m`
`=(6.625xx10^(-34)xx3xx10^(8)xxn_(i)^(2))/(21.76xx10^(-19)xx(n_(1)^(2)-1))m=(0.1934 n_(i)^(2))/((n_(1)^(2)-1))xx10^(-7) m`
`=913.4 n_(i)^(2)//(n_(1)^(2)-1)Å`
Substituting, `n_(i)=2,3,4,5" we get "lamda_(21)=1218 Å, lamda_(31)=1028 Å, lamda_(41)=974.3Å and lamda_(51)=951.4 Å.`