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Question
Determine the series limit of Balmer, Paschen and Brackett series, given the limit for Lyman series is 911.6 Å.
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Solution
Data: `lambda_("L"∞)` = 911.6 Å
For hydrogen spectrum, `1/lambda = "R"_"H"(1/"n"^2 - 1/"m"^2)`
∴ `1/lambda_("L"∞) = "R"_"H"(1/1^2 - 1/∞) = "R"_"H"` ...(1)
as n = 1 and m = ∞
`1/lambda_("B"∞) = "R"_"H"(1/4 - 1/∞) = "R"_"H"/4` .....(2)
as n = 2 and m = ∞
`1/lambda_("Pa"∞) = "R"_"H"(1/9 - 1/∞) = "R"_"H"/9` .....(3)
as n = 3 and m = ∞
`1/lambda_("Br"∞) = "R"_"H"(1/16 - 1/∞) = "R"_"H"/16` ......(4)
as n = 4 and m = ∞
From Eqs. (1) and (2), we get,
`lambda_("B"∞)/lambda_("L"∞) = "R"_"H"/("R"_"H"//4) = 4`
∴ `lambda_("B"∞) = 4 lambda_("L"∞) = (4)(911.6)` = 3646 Å
This is the series limit of the Balmer series.
From Eqs. (1) and (3), we get,
`lambda_("Pa"∞)/lambda_("L"∞) = "R"_"H"/("R"_"H"//9)` = 9
∴ `lambda_("Pa"∞) = 9lambda_("L"∞) = (9)(911.6)` = 8204 Å
This is the series limit of the Paschen series.
From Eqs. (1) and (4), we get,
`lambda_("Br"∞)/lambda_("L"∞) = "R"_"H"/("R"_"H"//16)` = 16
∴ `lambda_("Br"∞) = 16 lambda_("L"∞) = (16)(911.6)` = 14585.6 Å
This is the series limit of the Brackett series.
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