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Chemistry Question on Developments Leading to the Bohr’s Model of Atom

What is the wavelength of light emitted when the electron in a hydrogen atom undergoes transition from an energy level with n = 4 to an energy level with n = 2?

Answer

The ni = 4 to nf = 2 transition will give rise to a spectral line of the Balmer series. The energy involved in the transition is given by the relation,
E=2.18×10−18[1ni2−1nf2]E = 2.18 × 10^{-18} [\frac {1}{n_i^2}- \frac {1}{n_f^2}]
Substituting the values in the given expression of E:
E=2.18×10−18[142−122]E = 2.18 × 10^{-18} [\frac {1}{4^2}- \frac {1}{2^2}]

E=2.18×10−18[1−416]E = 2.18 × 10^{-18} [\frac {1-4}{16}]

E=2.18×10−18(−316)E = 2.18 × 10^{-18} (\frac {-3}{16})

E=−(4.0875×10−19 J)E = - (4.0875×10^{-19}\ J)
The negative sign indicates the energy of emission.
Wavelength of light emitted λ=hcEλ = \frac {hc}{E}
Substituting the values in the given expression of λ:
λ=(6.626×10−34)(3×108)4.0875×10−19λ = \frac {(6.626×10^{-34})(3×10^8)}{4.0875×10^{-19}}
λ=4.8631×10−7mλ = 4.8631×10^{-7} m
λ=486.3×10−9mλ= 486.3×10^{-9} m
λ=486 nmλ= 486\ nm