27-12 The Bohr Model Bohr studied Rutherford’s planetary model and found it had validityBohr studied Rutherford’s planetary model and found it had validity But to make it work the newly developing quantum theory would have to be incorporatedBut 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 lightPlank 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 continuousWhole 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 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 orbitBohr assumed that the electrons move about in a certain circular orbit Each orbit have a specific amount of energyEach orbit have a specific amount of energy The electrons could move about in that orbit without radiating energyThe electrons could move about in that orbit without radiating energy He called the possible orbits stationary statesHe called the possible orbits stationary states

Light is emitted only when an electron jumps from a higher energy state to a lower energy stateLight 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 statesWhen 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 quantizedBohr 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 foundationHis resulting equation didn’t have firm theoretical foundation He searched for some quantum condition and ties to E=hfHe searched for some quantum condition and ties to E=hf But is didn’t give the desired resultsBut is didn’t give the desired results His reason for using his equation was that it workedHis reason for using his equation was that it worked His equation is explained by using Coulomb’s law, Newton’s law and much substitutionHis equation is explained by using Coulomb’s law, Newton’s law and much substitution resultsresults

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 eVOr converted to eV The lowest energy level for hydrogen is -13.6eVThe lowest energy level for hydrogen is -13.6eV

So both the orbit radii and the energy levels are quantizedSo both the orbit radii and the energy levels are quantized The quantum number, n, labels the orbit radii and the energy levelsThe quantum number, n, labels the orbit radii and the energy levels The lowest energy level is called the ground stateThe lowest energy level is called the ground state The higher energies levels are called excited statesThe higher energies levels are called excited states The farther the energy levels are from the nucleus the higher the energyThe 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 energyThe minimum energy required to remove an electron from the ground state is called the binding energy Hydrogen is 13.6 eVHydrogen is 13.6 eV Corresponds to removing an electron from the lowest state to where it is free and E=0Corresponds 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 processOnce 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 gasesThis 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 linesThe 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 hydrogenThe 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 spectraThe Bohr model can also explain absorption spectra Photons of just the right wavelength can knock an electron from one energy level to a higher onePhotons 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 absorbedTo conserve energy, only photons that have the right energy will be absorbed

Bohr model also guarantees the stability of the atomsBohr model also guarantees the stability of the atoms Bohr model is great for finding the binding energy of hydrogen, but not for other atomsBohr model is great for finding the binding energy of hydrogen, but not for other atoms Bohr model was a very important startBohr model was a very important start The concepts of stationary states, the ground state, and the transitions between states are still used todayThe 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 workBohr made radical assumptions to make his model work Electrons in fixed orbits don’t radiate lightElectrons in fixed orbits don’t radiate light Was unable to say how electrons were raised to a higher energy levelWas 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 objectNo 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 worldThe 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=100,000,000His 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 existThe well defined orbits defined by the Bohr model don’t really exist This idea rejected a few years after Bohr proposed itThis idea rejected a few years after Bohr proposed it Today electrons for “probability clouds”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 experimentsBohr’s theory was based on assumptions that were made to have the theory agree with experiments Bohr could give no reason why orbits were quantizedBohr could give no reason why orbits were quantized 10 years later Louis de Broglie proposed that all particles have wave nature10 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 theoryOne 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 wavelengthDe 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 wavelengthsThe 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 atomsBohr’s theory worked well for hydrogen, but wasn’t successful for multi-electron atoms A new theory was developed in the 1920s called quantum mechanicsA new theory was developed in the 1920s called quantum mechanics The electrons being in a well defined orbit was replaced with electron “clouds”The electrons being in a well defined orbit was replaced with electron “clouds”