27-12 The Bohr Model27-12 The Bohr Model• Bohr studied Rutherford’s planetary model Bohr studied Rutherford’s planetary model

and found it had validityand found it had validity• But to make it work the newly developing But to make it work the newly developing

quantum theory would have to be quantum theory would have to be incorporatedincorporated

• Plank and Einstein had shown that in Plank and Einstein had shown that in heated solids, the energy of oscillating heated solids, the energy of oscillating electric charges must change from one electric charges must change from one discrete energy state to another with the discrete energy state to another with the emission of a quantum of lightemission of a quantum of light

• Whole steps, not continuousWhole steps, not continuous

• Bohr argued that the electrons in a Bohr argued that the electrons in a atom also couldn’t lose energy atom also couldn’t lose energy continuously, but must do so in continuously, but must do so in quantum “jumps”quantum “jumps”

• Bohr assumed that the electrons Bohr assumed that the electrons move about in a certain circular move about in a certain circular orbitorbit

• Each orbit have a specific amount Each orbit have a specific amount of energyof energy

• The electrons could move about in The electrons could move about in that orbit without radiating energythat orbit without radiating energy

• He called the possible orbits He called the possible orbits stationary statesstationary states

• Light is emitted only when an Light is emitted only when an electron jumps from a higher electron jumps from a higher energy state to a lower energy energy state to a lower energy statestate

• When the “jump” happens a single When the “jump” happens a single photon of light is emitted whose photon of light is emitted whose energy is the difference between energy is the difference between the two statesthe two states

• Bohr found that his theory was in Bohr found that his theory was in line with the Balmer formula is he line with the Balmer formula is he assumed that the electron’s assumed that the electron’s angular momentum is quantizedangular momentum is quantized

• His resulting equation didn’t have His resulting equation didn’t have firm theoretical foundationfirm theoretical foundation

• He searched for some quantum He searched for some quantum condition and ties to E=hfcondition 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 His reason for using his equation

was that it workedwas that it worked• His equation is explained by using His equation is explained by using

Coulomb’s law, Newton’s law and Coulomb’s law, Newton’s law and much substitutionmuch substitution

• resultsresults

22

2422 12

nh

mkeZEn

Where E is the energy leveln is the orbitZ is the number of positive chargee is the charge of an electronm is the mass of an electronk is Coulomb’s law constanth is Planck’s constant

• Or converted to eVOr converted to eV• The lowest energy level for The lowest energy level for

hydrogen is -13.6eVhydrogen is -13.6eV

2

2

)6.13(n

ZeVEn

• So both the orbit radii and the So both the orbit radii and the energy levels are quantizedenergy levels are quantized

• The quantum number, n, labels the The quantum number, n, labels the orbit radii and the energy levelsorbit radii and the energy levels

• The lowest energy level is called The lowest energy level is called the ground statethe ground state

• The higher energies levels are The higher energies levels are called excited statescalled excited states

• The farther the energy levels are The farther the energy levels are from the nucleus the higher the from the nucleus the higher the energyenergy

• The minimum energy required to The minimum energy required to remove an electron from the remove an electron from the ground state is called the binding ground state is called the binding energyenergy

• Hydrogen is 13.6 eVHydrogen is 13.6 eV• Corresponds to removing an Corresponds to removing an

electron from the lowest state to electron from the lowest state to where it is free and E=0where it is free and E=0

• Once in an excited state an atom’s Once in an excited state an atom’s electron can jump down to a lower electron can jump down to a lower state and give off a photon in the state and give off a photon in the processprocess

• This is the origin of the emission This is the origin of the emission spectra of excited gasesspectra of excited gases

• The vertical arrows represent the The vertical arrows represent the transitions or jumps that correspond transitions or jumps that correspond to the various observed spectral linesto 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 The success of Bohr’s model it can explain why atoms emit line spectra explain why atoms emit line spectra and correctly predicts the and correctly predicts the wavelengths or emitted light for wavelengths or emitted light for hydrogenhydrogen

• The Bohr model can also explain The Bohr model can also explain absorption spectraabsorption spectra

• Photons of just the right wavelength Photons of just the right wavelength can knock an electron from one can knock an electron from one energy level to a higher oneenergy level to a higher one

• To conserve energy, only photons that To conserve energy, only photons that have the right energy will be absorbedhave the right energy will be absorbed

• Bohr model also guarantees the Bohr model also guarantees the stability of the atomsstability of the atoms

• Bohr model is great for finding the Bohr model is great for finding the binding energy of hydrogen, but binding energy of hydrogen, but not for other atomsnot for other atoms

• Bohr model was a very important Bohr model was a very important startstart

• The concepts of stationary states, The concepts of stationary states, the ground state, and the the ground state, and the transitions between states are still transitions between states are still used todayused today

Correspondence PrincipleCorrespondence Principle

• Bohr made radical assumptions to Bohr made radical assumptions to make his model workmake his model work

• Electrons in fixed orbits don’t Electrons in fixed orbits don’t radiate lightradiate light

• Was unable to say how electrons Was unable to say how electrons were raised to a higher energy levelwere raised to a higher energy level

• No real reason why a tiny electron No real reason why a tiny electron would behave like a regular sized would behave like a regular sized objectobject

• The correspondence principle can The correspondence principle can predict classical results by overlapping predict classical results by overlapping the quantum theory with the the quantum theory with the macroscopic worldmacroscopic world

• His theory does work for hydrogen with His theory does work for hydrogen with n=1, but not very well for n=1, but not very well for n=100,000,000n=100,000,000

• The well defined orbits defined by the The well defined orbits defined by the Bohr model don’t really existBohr model don’t really exist

• This idea rejected a few years after Bohr This idea rejected a few years after Bohr proposed itproposed it

• Today electrons for “probability clouds” Today electrons for “probability clouds”

27-13 de Broglie’s 27-13 de Broglie’s HypothesisHypothesis

• Bohr’s theory was based on assumptions Bohr’s theory was based on assumptions that were made to have the theory agree that were made to have the theory agree with experimentswith experiments

• Bohr could give no reason why orbits were Bohr could give no reason why orbits were quantizedquantized

• 10 years later Louis de Broglie proposed 10 years later Louis de Broglie proposed that all particles have wave naturethat all particles have wave nature

• One of de Broglie’s original arguments One of de Broglie’s original arguments supporting the wave nature of electrons supporting the wave nature of electrons was to provide an explanation of Bohr’s was to provide an explanation of Bohr’s theorytheory

• De Broglie hypothesized that De Broglie hypothesized that electrons have a wavelengthelectrons have a wavelength

• The orbits correspond to circular The orbits correspond to circular standing waves in which the standing waves in which the circumference of the orbit equals circumference of the orbit equals whole number of wavelengthswhole number of wavelengths

• Bohr’s theory worked well for Bohr’s theory worked well for hydrogen, but wasn’t successful hydrogen, but wasn’t successful for multi-electron atomsfor multi-electron atoms

• A new theory was developed in the A new theory was developed in the 1920s called quantum mechanics1920s called quantum mechanics

• The electrons being in a well The electrons being in a well defined orbit was replaced with defined orbit was replaced with electron “clouds”electron “clouds”