First slide

Hydrogen spectrum

Question

Transitions between three energy levels in a particular atom give rise to three spectral lines of wavelengths, in increasing magnitudes, $λ_{1}, λ_{2}, and λ_{3}$ . Which one of the following equations correctly relates $λ_{1}, λ_{2}, and λ_{3} ?$

Moderate

A

$λ_{1} = λ_{2} - λ_{3}$
B

$λ_{1} = λ_{3} - λ_{2}$
C

$\frac{1}{λ_{1}} = \frac{1}{λ_{2}} + \frac{1}{λ_{3}}$
D

$\frac{1}{λ_{1}} = \frac{1}{λ_{3}} - \frac{1}{λ_{2}}$

Solution

Let the three energy levels be E₁, E₂, and E₃. The wavelengths $λ_{1}, λ_{2}, and λ_{3}$ of the spectral lines corresponding to the, three energy transitions are depicted as shown in figure.
$E = h f = \frac{h}{λ} or E \propto \frac{1}{λ}$
$(given λ_{1} < λ_{2} < λ_{3})$
Thus, for the three wavelengths, we have
$E_{3} - E_{2} = \frac{h}{λ_{3}} . . . . . (i)$
$\{\begin{cases} E_{2} - E_{1} = \frac{h}{λ_{2}} . . . . . . . . . . (i i) \\ E_{3} - E_{1} = \frac{h}{λ_{1}} . . . . . . . . . . (i i i) \end{cases}$
$Now, E_{3} - E_{1} = (E_{3} - E_{2}) + (E_{2} - E_{1})$
$\Rightarrow \frac{h}{λ_{1}} = \frac{h}{λ_{3}} + \frac{h}{λ_{2}} \Rightarrow \frac{1}{λ_{1}} = \frac{1}{λ_{2}} + \frac{1}{λ_{3}}$

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