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Q.

Let $ν_{1}$ be the frequency of the series limit of the Lyman series, $ν_{2}$ be the frequency of the first line of the Lyman series, and $ν_{3}$ be the frequency of the series limit of the Balmer series, then-

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a

$v_{1} + v_{2} = v_{3}$

b

$v_{2} - v_{1} = v_{3}$

c

$v_{3} = \frac{1}{2} (v_{1} - v_{3})$

d

$v_{1} - v_{2} = v_{3}$

answer is C.

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Detailed Solution

$\begin{array}{l} ν = R c Z^{2} (\frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}}) \\ ν_{1} = R c Z^{2} (\frac{1}{1^{2}} - \frac{1}{\infty^{2}}) = R c Z^{2} \\ ν_{2} = R c Z^{2} (\frac{1}{2^{2}} - \frac{1}{\infty^{2}}) = \frac{R c Z^{2}}{4} \\ ν_{3} = R c Z^{2} (\frac{1}{1^{2}} - \frac{1}{2^{2}}) = \frac{3 R c Z^{2}}{4} \\ ∴ ν_{1} - ν_{2} = ν_{3} \end{array}$

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Let ν1 be the frequency of the series limit of the Lyman series, ν2 be the frequency of the first line of the Lyman series, and ν3 be the frequency of the series limit of the Balmer series, then-