Explanation Write the expression to obtain the energy of the photon at the upper state. E U = − 13.6 eV n 2 Here, E U is the energy of the photon at the upper state and n is the state. Write the expression to obtain the change in energy when a photon is excited to higher state from lower state. Δ E = E U − E L Here, Δ E is the change in energy when a photon is excited to upper state from lower state, E U is the energy of the photon at the upper state and E L is the energy of the photon at the lower state. Substitute − 13.6 eV n 2 for E U in the above equation. Δ E = − 13.6 eV n 2 − E L Further solve the above equation. − 13.6 eV n 2 = E L + Δ E n 2 = − 13.6 eV E L + Δ E n = − 13.6 eV E L + Δ E (I) Write the expression to obtain the wavelength of the emitted photon. 1 λ = R ( 1 ( n ' ) 2 − 1 ( n ) 2 ) (II) Here, λ is the wavelength of the emitted photon, R is the Rydberg constant, n is the lower state and n ' is the upper state. Conclusion: Substitute − 13.6 eV for E L and 12.75 eV for Δ E in equation (I) to calculate n . n = − 13.6 eV − 13.6 eV + 12.75 eV = − 13.6 eV − 0.85 eV = 16 = 4 Thus, the upper state is 4 . The possible transitions are n = 4 to n = 3 , n = 4 to n = 2 , n = 4 to n = 1 , n = 3 to n = 2 , n = 3 to n = 1 and n = 2 to n = 1 . Case 1: n = 4 to n = 3 . Substitute 1.097 × 10 7 m − 1 for R , 3 for n ' and 4 for n in equation (II) to calculate λ . 1 λ = 1.097 × 10 7 m − 1 ( 1 ( 3 ) 2 − 1 ( 4 ) 2 ) = 1.097 × 10 7 m − 1 ( 1 9 − 1 16 ) = 1.097 × 10 7 m − 1 ( 7 144 ) = 533263.8889 m − 1 Further solve the above equation. 1 λ = 533263.8889 m − 1 λ = 1 533263.8889 m − 1 = 1.875 × 10 − 6 m × 10 9 nm 1 m = 1875 nm Case 2: n = 4 to n = 2 . Substitute 1.097 × 10 7 m − 1 for R , 2 for n ' and 4 for n in equation (II) to calculate λ . 1 λ = 1.097 × 10 7 m − 1 ( 1 ( 2 ) 2 − 1 ( 4 ) 2 ) = 1.097 × 10 7 m − 1 ( 1 4 − 1 16 ) = 1.097 × 10 7 m − 1 ( 3 16 ) = 2056875 m − 1 Further solve the above equation. 1 λ = 2056875 m − 1 λ = 1 2056875 m − 1 = 4.86174 × 10 − 7 m × 10 9 nm 1 m = 486.2 nm Case 3: n = 4 to n = 1 . Substitute 1.097 × 10 7 m − 1 for R , 1 for n ' and 4 for n in equation (II) to calculate λ . 1 λ = 1.097 × 10 7 m − 1 ( 1 ( 1 ) 2 − 1 ( 4 ) 2 ) = 1

Physics for Scientists and Engineers with Modern Physics

4th Edition

ISBN: 9780131495081

Author: Douglas C. Giancoli

Publisher: Addison-Wesley

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Chapter 37, Problem 69P

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The photon wavelengths will be emitted by the gas for the given conditions.

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Chapter 37 Solutions