Question

Suppose in an imaginary world the angular momentum is quantized to be even integral multiples of h/2π. What is the longest possible wavelength emitted by hydrogen atoms in visible range in such a world according to Bohr's model?

Answer 1

In the imaginary world, the angular momentum is quantized to be an even integral multiple of h/2$\mathrm{\pi }$.
Therefore, the quantum numbers that are allowed are n₁ = 2 and n₂ = 4.

We have the longest possible wavelength for minimum energy.
Energy of the light emitted (E) is given by
$$\begin{gathered} E=13.6\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right) \\ E=13.6\left[\frac{1}{{\left(2\right)}^2}-\frac{1}{{\left(4\right)}^2}\right] \\ E=13.6\left(\frac{1}{4}-\frac{1}{16}\right) \\ E=\frac{13.6\times 12}{64}=2.55\mathrm{eV} \end{gathered}$$
Equating the calculated energy with that of photon, we get
$$\begin{gathered} 2.55\mathrm{eV}=\frac{hc}{\mathrm{\lambda }} \\ \mathrm{\lambda }=\frac{hc}{2.55}=\frac{1242}{2.55}\mathrm{nm} \\ =487.05\mathrm{nm}=487\mathrm{nm} \end{gathered}$$