First slide

Quantum mechanical model of the atom

Question

When an electron makes transition from (n + 1) state to n state the wavelength of emitted radiations is related to n (n>>>1) according to $λ \propto n^{x}$ . The value of x is

Moderate

Solution

$\begin{array}{l} \frac{1}{λ} = {RZ}^{2} (\frac{1}{n^{2}} - \frac{1}{{(n + 1)}^{2}}) \\ \frac{1}{λ} = {RZ}^{2} \frac{{(n + 1)}^{2} - n^{2}}{n^{2} {(n + 1)}^{2}} \\ \frac{1}{λ} = {RZ}^{2} \frac{(n + 1 + n) (n + 1 - n)}{n^{2} {(n + 1)}^{2}} \\ \frac{1}{λ} = {RZ}^{2} \frac{(2 n + 1)}{n^{2} {(n + 1)}^{2}} \\ λ \propto n^{3} \\ x = 3 \end{array}$

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