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In a hydrogen atom, electron is excited from ground state to 3rd orbit, find the emitted wavelength. (R is the Rydberg constant.)
1. \({4 \over {3R}}\)
2. \({3 \over {4R}}\)
3. \({8 \over {9R}}\)
4. \({9 \over {8R}}\)

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Correct Answer - Option 4 : \({9 \over {8R}}\)

CONCEPT:

  • When atoms are excited they emit light of certain wavelengths that correspond to different colors due to the electron making transitions between two energy levels in an atom.
    • The wavelength related to these emissions is given by the Rydberg formula.
    • The energy differences between levels in the Bohr model are given by the Rydberg formula.

The wavelengths of emitted/absorbed photons is given by:

\({\displaystyle {1 \over λ }=Z^{2}R_{∞ }\left({1 \over {n_{1}}^{2}}-{1 \over {n_{2}}^{2}}\right)}\)

  • Given that atom is hydrogen so Z = 1;
  • It is excited from ground state n1 = 1 to third orbit n2 = 3

Where Z is the atomic number, n1 is the lower energy level orbit, n2 is the upper energy level orbit, and R∞  is the Rydberg constant (1.09677 × 107 m−1 for hydrogen and 1.09737 × 107 m−1 for heavy metals).

EXPLANATION:

\({\displaystyle {1 \over λ }=Z^{2}R\left({1 \over {n_{1}}^{2}}-{1 \over {n_{2}}^{2}}\right)}\)

\({\displaystyle {1 \over λ }=R\left({1 \over {1}^{2}}-{1 \over {3}^{2}}\right)}\)

\({\displaystyle {1 \over λ }=R\left({1}-{1 \over {9}}\right)}\)

\({\displaystyle {1 \over λ }=R\left({8 \over {9}}\right)}\)

\({\displaystyle { λ }=\left({9 \over {8R}}\right)}\)

So the correct answer is option 4.

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