De Broglie’s explanation for certain fixed orbits for the hydrogen atom.

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Chapter 9, Problem 30SA

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De Broglie’s explanation for certain fixed orbits for the hydrogen atom.

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De Broglie explained the Bohr model of the hydrogen atom by attributing a wavelength > to a particle, with the value of λ coming from the momentum p = mv as λ = h/p. i) Calculate the de Broglie wavelength of a bird (a racing pigeon) that weighs 0.350 kg and flies at 100 km per hour. ii) Equating the centripetal force on an electron orbiting around a proton with the Coulomb force gives the expression v² = e²/(4πomer). Calculate the speed of an electron or- biting at the Bohr radius, B = 0.053 nm. iii) Calculate the momentum and the de Broglie wavelength of an electron moving at the speed you obtain in (ii). iv) Compare the wavelength you obtain in (iii) with the circumference of the orbit. Com- ment on this comparison. Explain briefly what it implies about the possible orbits of the Bohr model and how the higher orbits might be predicted.

a) An electron in a hydrogen atom has energy E= -3.40 eV, where the zero of energy is at the ionization threshold. In the Bohr model, what is the angular momentum of the electron? Express your result as a multiple of ħ. Ans. b) What is the deBroglie wavelength of the electron when it is in this state? Ans. c) When the electron is in this state, what is the ratio of the circumference of the orbit of the electron to the deBroglie wavelength of the electron? Ans. d) The electron makes a transition from the state with energy E= -3.40 eV to the ground state, that has energy -13.6 eV. What is the wavelength of the photon emitted during this transition? Ans.

When an electron in hydrogen atom jumps from the third excited state to the ground state, how would the de Broglie wavelength associated with the electron change? Justify your answer.