Question

Calculate the wavelength of the first and limiting spectral line in the Lymann series of hydrogen spectrum.

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Solution

For the first spectral line in the Lyman series

straight n subscript 1 space equals space 1 comma space space space space straight n subscript 2 space equals space 2 space space straight v with bar on top space equals space 1 over straight lambda space equals space straight R open square brackets fraction numerator 1 over denominator straight n subscript 1 superscript 2 end fraction minus fraction numerator 1 over denominator straight n subscript 2 superscript 2 end fraction close square brackets or space space space 1 over straight lambda space equals space 109677 open square brackets fraction numerator 1 over denominator left parenthesis 1 right parenthesis squared end fraction minus fraction numerator 1 over denominator left parenthesis 2 right parenthesis squared end fraction close square brackets or space space space space space 1 over straight lambda space equals space 109677 open square brackets 1 minus 1 fourth close square brackets or space space space space space 1 over straight lambda space equals space 109678 space cross times space 3 over 4 space equals space 82259 space cm to the power of negative 1 end exponent therefore space space space space straight lambda space equals space 1.2157 space cross times space 10 to the power of negative 5 end exponent space cm

For the limiting spectral line,

straight n subscript 1 space equals space 1 comma space space space straight n subscript 2 space equals space infinity

or space space space 1 over straight lambda space equals 109677 space open square brackets fraction numerator 1 over denominator left parenthesis 1 right parenthesis squared end fraction minus fraction numerator 1 over denominator left parenthesis infinity right parenthesis squared end fraction close square brackets space space space space space space space equals 109677 space cm to the power of negative 1 end exponent therefore space space space space space straight lambda space equals space 9.1176 space cross times space 10 to the power of negative 6 end exponent space cm

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