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27-12 The Bohr Model • Bohr studied Rutherford’s planetary model and found it had validity • But to make it work the newly developing quantum theory would have to be incorporated • Plank and Einstein had shown that in heated solids, the energy of oscillating electric charges must change from one discrete energy state to another with the emission of a quantum of light • Whole steps, not continuous • Bohr argued that the electrons in a atom also couldn’t lose energy continuously, but must do so in quantum “jumps” • Bohr assumed that the electrons move about in a certain circular orbit • Each orbit have a specific amount of energy • The electrons could move about in that orbit without radiating energy • He called the possible orbits stationary states • Light is emitted only when an electron jumps from a higher energy state to a lower energy state • When the “jump” happens a single photon of light is emitted whose energy is the difference between the two states • Bohr found that his theory was in line with the Balmer formula is he assumed that the electron’s angular momentum is quantized • His resulting equation didn’t have firm theoretical foundation • He searched for some quantum condition and ties to E=hf • But is didn’t give the desired results • His reason for using his equation was that it worked • His equation is explained by using Coulomb’s law, Newton’s law and much substitution • results 2 Z e mk 1 En 2 2 h n 2 2 4 2 Where E is the energy level n is the orbit Z is the number of positive charge e is the charge of an electron m is the mass of an electron k is Coulomb’s law constant h is Planck’s constant • Or converted to eV • The lowest energy level for hydrogen is -13.6eV 2 Z En (13.6eV ) 2 n • So both the orbit radii and the energy levels are quantized • The quantum number, n, labels the orbit radii and the energy levels • The lowest energy level is called the ground state • The higher energies levels are called excited states • The farther the energy levels are from the nucleus the higher the energy • The minimum energy required to remove an electron from the ground state is called the binding energy • Hydrogen is 13.6 eV • Corresponds to removing an electron from the lowest state to where it is free and E=0 • Once in an excited state an atom’s electron can jump down to a lower state and give off a photon in the process • This is the origin of the emission spectra of excited gases • The vertical arrows represent the transitions or jumps that correspond to the various observed spectral lines The lowest energy level is called the ground state; the others are excited states. • The success of Bohr’s model it can explain why atoms emit line spectra and correctly predicts the wavelengths or emitted light for hydrogen • The Bohr model can also explain absorption spectra • Photons of just the right wavelength can knock an electron from one energy level to a higher one • To conserve energy, only photons that have the right energy will be absorbed • Bohr model also guarantees the stability of the atoms • Bohr model is great for finding the binding energy of hydrogen, but not for other atoms • Bohr model was a very important start • The concepts of stationary states, the ground state, and the transitions between states are still used today Correspondence Principle • Bohr made radical assumptions to make his model work • Electrons in fixed orbits don’t radiate light • Was unable to say how electrons were raised to a higher energy level • No real reason why a tiny electron would behave like a regular sized object • The correspondence principle can predict classical results by overlapping the quantum theory with the macroscopic world • His theory does work for hydrogen with n=1, but not very well for n=100,000,000 • The well defined orbits defined by the Bohr model don’t really exist • This idea rejected a few years after Bohr proposed it • Today electrons for “probability clouds” 27-13 de Broglie’s Hypothesis • Bohr’s theory was based on assumptions that were made to have the theory agree with experiments • Bohr could give no reason why orbits were quantized • 10 years later Louis de Broglie proposed that all particles have wave nature • One of de Broglie’s original arguments supporting the wave nature of electrons was to provide an explanation of Bohr’s theory • De Broglie hypothesized that electrons have a wavelength • The orbits correspond to circular standing waves in which the circumference of the orbit equals whole number of wavelengths • Bohr’s theory worked well for hydrogen, but wasn’t successful for multi-electron atoms • A new theory was developed in the 1920s called quantum mechanics • The electrons being in a well defined orbit was replaced with electron “clouds”