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The Bohr Model (of the Hydrogen atom) Planetary Model • Force Applied to the electron: • One deduces: v = e / (√4πε0mr) • Total energy with K = (1/2)mv2: Failure of the classical (planetary) atomic model • Atom (neutral) = nucleus (+q) + q electrons • Assuming the Hydrogen atom: – The electron is attracted by the nucleus – Even in circular motion around the nucleus, the electron loses energy: • Radial acceleration: ar = v2/R • Classical e.m. theory: an accelerating charge continuously radiates energy, r decreases… The electron would eventually crash into the nucleus Failure of the planetary model • Doomed because the electron radiates energy, while orbiting around the nucleus • But: – There is some “truth” to it, since Rutherford was successful in describing the scattering experiment • In 1913, Niels Bohr 1885-1962 postulates that the electrons may be in stable (non-radiating) circular orbits, called stationary orbits Niels Bohr’s postulates • Existence of “stationary” states for the orbiting electrons • Transitions between stationary states: E = E1 – E2 = hν • Classical laws of physics do not apply to transitions between stationary states • The electron can only exist in orbits for which its angular momentum is given by: L= | r x p | = mvr = nh/2π with n=1,2,3… Bohr Radius • From Bohr’s postulate: v = nħ/mr • Since: v = e / (√4πε0mr) [Planetary Model] • One can deduce the diameter of the hydrogen atom for stationary states : with a0, the Bohr radius Calculation With: h = 6.6 x 10-34 J.s ε0 = 8.85 x 10-12 C2/N.m2 m = 9.1 x 10-31 kg e = 1.6 x 10-19 C • Diameter of the Hydrogen Atom: n=1: ground state (lowest energy) orbit of the hydrogen atom Energy Levels of the Hydrogen atom • Energy of the stationary states: with E0 = -13.6 eV Emission of light: hν = Enu – Enl Using: 1/λ = ν/c, 1 λ R∞, the Rydberg constant for Hydrogen Spectral Lines of Hydrogen n>l, l=1 & n=2,3,4,… Lyman Series l=2 & n=3,4,5,… Balmer Series l=3 & n=4,5,6,… Pashen Series By 1913, some of these lines were not observed. Still they were predicted by Bohr’s model Emission & Absorption • When shining a white light (all visible frequencies), one observes a spectrum containing dark lines Absorption • These lines are the same than those observed in the Emission spectrum. • Hydrogen in its ground state: – Transition induced by incoming photon hν Absorption – Then, the atom goes back to its ground state by emitting photon(s) Emission Hydrogen Absorption/Emission The Sun Quantum to Classical Mechanics ? • The Correspondance Principle (Bohr): In the limits where classical and quantum theories should agree, the quantum theory must reduce to the classical result. Limitations • The Bohr model was a great step of the new quantum theory, but it had its limitations: Works only to single-electron atoms. Could not account for the intensities or the fine structure of the spectral lines. Could not explain the binding of atoms into molecules. Exercise The electron in a hydrogen atom in the n=2 state absorbs a photon of wavelength λ=327 nm. Does the hydrogen atom become ionized ? Energy of the n=2 state: En = -13.6/n2 eV E2 = -13.6/(2)2 = -3.4 eV The H atom will ionize, if the photon brings an energy higher than 3.4 eV. Ephoton = hν = hc/λ = (6.6 x 10-34) x (3.0 x 108) / (327 x 10-9) = 6.1 x 10-19 J with 1 eV = 1.6 x 10-19 J, Ephoton = 6.1 x 10-19 / 1.6 x 10-19 = 3.8 eV The Hydrogen atom will become ionized. Exercise Calculate the 3 first transitions of: - The Lyman series (l=1): n=2 n=3 n=4 λ= λ= λ= n=3 n=4 n=5 λ= λ= λ= - The Pashen series (l=3): n=4 n=5 n=6 λ= λ= λ= - The Balmer series (l=2):