Suppose in an imaginary world the angular momentum is quantized t… - Vidyadip

Question

Suppose in an imaginary world the angular momentum is quantized to be even integral multiples of $\frac{\text{h}}{2\pi}$ What is the longest possible wavelength emitted by hydrogen atoms in visible range in such a world according to Bohr's model?

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Answer

Even quantum numbers are allowed,
$\text{n}_1=2,\text{n}_2=4\xrightarrow{\ \ \ }$ For minimum energy or for longest possible wavelength.
$\text{E}=13.6\bigg(\frac{1}{\text{n}^2_1}-\frac{1}{\text{n}^2_2}\bigg)$
$\text{E}=13.6\Big(\frac{1}{2^2}-\frac{1}{4^2}\Big)=2.55$
$2.55=\frac{\text{hc}}{\lambda}$
$\lambda=\frac{\text{hc}}{2.55}=\frac{1242}{2.55}$
$\lambda=487.05\text{nm}=487\text{nm}$

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