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The Bohr model: success and failure Applying the photon model to electronic structure The emergence of the quantum world

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Learning objectives Describe the basic principles of the Bohr model Distinguish between the “classical” view and the “quantum” view of matter Describe Heisenberg Uncertainty principle and deBroglie wave-particle duality Calculate wavelengths of particles

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Bohr’s theory of the atom: applying photons to electronic structure Electrons occupy specific levels (orbits) and no others Orbits have energy and size Electron excited to higher level by absorbing photon Electron relaxes to lower level by emitting photon Photon energy (hν) exactly equals gap between levels –Gap ↑, ν ↑ Larger orbits are at higher energy – larger radius

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Size of energy gap determines photon energy Small energy gap, low frequency, long wavelength (red shift) High energy gap, high frequency, short wavelength (blue shift)

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The full spectrum of lines for H Each set of lines in the H spectrum comes from transitions from all the higher levels to a particular level. The lines in the visible are transitions to the second level

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Successes and shortcomings of Bohr Could not explain why these levels were allowed Only successful agreement with experiment was with the H atom Introduced connection between spectra and electron structure Concept of allowed orbits is developed further with new knowledge Nonetheless, an important contribution, worthy of the Nobel prize

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Electrons are waves too! Life at the electron level is very different Key to unlocking the low door to the secret garden of the atom lay in accepting the wave properties of electrons De Broglie wave-particle duality All particles have a wavelength – wavelike nature. –Significant only for very small particles – like electrons –As mass increases, wavelength decreases Electrons have wavelengths about the size of an atom –Electrons are used for studying matter – electron microscopy

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De Broglie relation E = mc 2 m = E/c 2 But, E = hc/λ, so m = h/cλ –For electron: m = 9 x 10 -31 kg, v = 2 x 10 6 ms -1 –λ = 3 x 10 -10 m (0.3 nm) The electron’s wavelength is of the order of the atomic diameter

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Wavelengths of large objects Should we be concerned about the wave- particle nature of large objects? Consider a baseball pitched at 100 mph. –m = 120 g and v = 45 m/s –λ = 10 -34 m For normal size objects, the wavelength will be immeasurably and irrelevantly small

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Relating this to Bohr: Standing waves and strings Strings of fixed length can only support certain wavelengths. These are standing waves.

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The Bohr orbits revisited The allowed orbits have a circumference equal to a fixed number of wavelengths All others disappear via destructive interference

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Heisenberg Uncertainty Principle: the illusive electron We can exactly predict the motion of a ball –Newton’s laws are deterministic But not an electron

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Heisenberg Uncertainty Principle The position and momentum of a particle cannot be measured simultaneously to unlimited accuracy Δx Δp > 0

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Locating the electron: catching a goldfish in a bowl The act of “seeing” an electron using photons changes electron’s energy, thereby changing its position As the object increases in size, the impact of the photon decreases Limits precision of determining position and momentum

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Heisenberg Uncertainty Relation In mathematical terms, If the position is known precisely, Δx is small and the uncertainty in momentum is large If the velocity is known precisely, there is a high uncertainty in the position The electron will appear as a blur rather than a sharp point

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Implications for the electron m = 9 x 10 -31 kg, v = 2 x 10 6 m/s If uncertainty in v is 10%, –mΔv = (9 x 10 -31 ) x (2 x 10 5 ) kgm/s – = 18 x 10 26 kgm/s –Δx ≥ h/4π18 x 10 -26 m – ≥ 6 x 10 -34 /4π18 x 10 -26 m – ≥ 3 x 10 -10 m or 300 pm Diameter of the H atom is about 100 pm

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Quantum effects: when should we care? The Correspondence Principle states that quantum effects disappear when Planck’s constant is small compared to other physical quantities

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Quantum Theory of the Atom

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