Question

The Bohr model of the atom was an early attempt, by Danish theoretical physicist Niels Bohr in 1913, to understand the properties of atoms by incorporating the wavelike nature of electrons into an otherwise classical description. The basic assumptions of the model are:

a. An electron in an atom moves in circular orbits about a positively charged nucleus, with the Coulomb attraction providing the centripetal force.

b. Since electrons are waves, an electron may only (stably) orbit the nucleus at radii for which an integer number of wavelengths fit around the circumference. Otherwise the electron wave destructively interferes with itself.

c. When an electron makes a transition from one stable orbit to another, its energy changes by a finite amount. This must be accompanied by absorption or emission of a photon of energy equal to the change in energy of the electron.

(a) For a hydrogen atom, the nucleus is a proton (charge e) with one orbiting electron (charge ?e). Using assumption i., show that the radius of an orbit is related to the mass m and speed v of an electron by: r = e^2/(4??0mv^2)

(b) Using assumption ii., show that the angular momentum of an electron in an atom is restricted in value to (positive) integer multiples of ?: L = mvr = n? n = 1, 2, 3, …

(c) Combine (a) and (b) to show that the radii of the stable orbits of the Bohr hydrogen atom are rn = a0n^2 , n = 1, 2, 3, …, where the Bohr radius, a0, is given by: a0 = (4??0?^2)/(me ^2)= 0.0529 nm .

(d) Show that the energies of the stable orbits are given by: En = ?(?^2)/(2ma0^2) 1/n^2 = ?13.6 eV/n^2 , n = 1, 2, 3, …

(e) Using assumption iii., determine the wavelength of the photon emitted by a Bohr hydrogen atom when it makes each of the following transitions: n = 2 ? n = 1, n = 3 ? n = 1, and n = 3 ? n = 2.

Answer 1

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