The Quantum- Mechanical Model of the Atom. Atoms and Energy All energy is either kinetic energy (the energy of motion), or potential energy (energy based

The Quantum-The Quantum-Mechanical Mechanical Model of the Model of the

AtomAtom

Atoms and EnergyAtoms and EnergyAll energy is either All energy is either kinetickinetic energyenergy

(the energy of motion), or (the energy of motion), or potential potential energyenergy (energy based on position). (energy based on position).

In chemical systems, In chemical systems, thermal thermal energyenergy is of interest as particles is of interest as particles move, collide, and exchange energy.move, collide, and exchange energy.

In individual atoms, In individual atoms, electrostatic electrostatic energyenergy, arising from the attraction or , arising from the attraction or repulsion of charges is of interest.repulsion of charges is of interest.

Coulomb’s LawCoulomb’s Law

Attraction or replusion between charged Attraction or replusion between charged particles can be calculated using Couomb’s particles can be calculated using Couomb’s Law:Law:

E = (QE = (Qaae)(Qe)(Qbbe) e) αα Q QaaQQbb

44πεπεoorrabab rrabab

QQaa and Q and Qbb = charges on the particles = charges on the particles

e= charge of an electron (-1.602 x 10e= charge of an electron (-1.602 x 10-19-19C)C)

44πεπεoo=permittivity of vacuum (1.1127 x 10=permittivity of vacuum (1.1127 x 10-10-10JJ-1-1CC22mm--

11))

rrabab = distance between the particles = distance between the particles

Electrostatic EnergyElectrostatic Energy

With oppositely charged With oppositely charged particles, the energy is negative, particles, the energy is negative, indicating attraction between the indicating attraction between the particles and a lowering of potential particles and a lowering of potential energy as the particles are closer to energy as the particles are closer to each other.each other.

E α (QE α (QaaQQbb)÷ d)÷ d

Electrostatic EnergyElectrostatic Energy

E α (QE α (QaaQQbb)/d)/d

It is important to note that the It is important to note that the attraction between opposite charges attraction between opposite charges increases dramatically as the increases dramatically as the magnitude of the charges on the magnitude of the charges on the particles increases.particles increases.

Units of EnergyUnits of Energy

There are many different units used There are many different units used for energy depending on the application. for energy depending on the application. The SI unit for energy is joules (J), The SI unit for energy is joules (J), which is the energy exerted when a which is the energy exerted when a force of 1 newton over a distance of 1 force of 1 newton over a distance of 1 meter. A newton (N) is a force of 1kg meter. A newton (N) is a force of 1kg (m/s(m/s22). Although this unit is related to ). Although this unit is related to kinetic energy (a force acting over a kinetic energy (a force acting over a distance), it can be used for other types distance), it can be used for other types of energy.of energy.

Matter and EnergyMatter and Energy

By 1900, physicists thought that the By 1900, physicists thought that the nature of energy and matter was well nature of energy and matter was well understood and distinct. understood and distinct.

Matter, a collection of particles, has Matter, a collection of particles, has mass and a defined position in space. mass and a defined position in space. Radiant (light) energy, as waves, is massless Radiant (light) energy, as waves, is massless and delocalized.and delocalized.

It was also believed that particles of It was also believed that particles of matter could absorb or emit any energy, matter could absorb or emit any energy, without restriction.without restriction.

Matter and EnergyMatter and Energy

After Rutherford, Geiger and After Rutherford, Geiger and Marsden proposed the nuclear model Marsden proposed the nuclear model of the atom, scientists focused on how of the atom, scientists focused on how the electrons are arranged around the the electrons are arranged around the nucleus. Since electrons could not be nucleus. Since electrons could not be observed directly, scientists studied observed directly, scientists studied the light that matter emits when it is the light that matter emits when it is stimulated by heat or an electric stimulated by heat or an electric discharge.discharge.

Atomic SpectroscopyAtomic Spectroscopy

The study of the light emitted or The study of the light emitted or absorbed by matter is a branch of absorbed by matter is a branch of chemistry called chemistry called spectroscopyspectroscopy. .

Atomic spectroscopy allows Atomic spectroscopy allows scientists to understand the nature scientists to understand the nature of the electrons in atoms. Molecular of the electrons in atoms. Molecular spectroscopy provides information spectroscopy provides information about the bonds in molecules.about the bonds in molecules.

Electromagnetic Electromagnetic RadiationRadiation

Early atomic scientists studied Early atomic scientists studied the interaction of matter with the interaction of matter with electromagnetic radiationelectromagnetic radiation, or light. , or light.

Electromagnetic radiation, or Electromagnetic radiation, or radiant energy, includes visible light, radiant energy, includes visible light, infrared, micro and radio waves, and infrared, micro and radio waves, and X-rays and ultraviolet light.X-rays and ultraviolet light.


Light consists of oscillating Light consists of oscillating electric and magnetic fields that electric and magnetic fields that travel through space at the rate of 3 travel through space at the rate of 3 x 10x 1088m/s. m/s.

The oscillating fields interact The oscillating fields interact with electrons in the atom.with electrons in the atom.


This drawing This drawing represents a represents a “snapshot” “snapshot” of an of an electro-electro-magnetic magnetic wave at a wave at a given given instant.instant.


Electromagnetic radiation Electromagnetic radiation travels in waves. travels in waves.

The waves of radiant energy The waves of radiant energy have three important characteristics:have three important characteristics:

1. Wavelength - 1. Wavelength - λλ - (lambda) - (lambda)

2. Frequency – 2. Frequency – νν – (nu) – (nu)

3. Speed – c – the speed of light3. Speed – c – the speed of light

WavelengthWavelength

Wavelength, Wavelength, λλ, , is the distance is the distance between two between two adjacent peaks or adjacent peaks or troughs in a wave.troughs in a wave.

The units may The units may range from range from picometers to picometers to kilometers kilometers depending upon the depending upon the energy of the wave.energy of the wave.

FrequencyFrequency

Frequency, Frequency, νν, is the number , is the number of waves (or of waves (or cycles) that pass cycles) that pass a given point in a given point in space per second.space per second.

The units are The units are cycles/s, scycles/s, s-1-1 or or hertz (Hz).hertz (Hz).

The Speed of LightThe Speed of Light

All electromagnetic radiation All electromagnetic radiation travels at the same speed. The travels at the same speed. The speed of light ( c ) is:speed of light ( c ) is:

c = 2.9979 x 10c = 2.9979 x 1088 m/s m/s

Wavelength and Wavelength and FrequencyFrequency

Wavelength Wavelength and frequency are and frequency are inversely related. inversely related. That is, waves That is, waves with a low with a low frequency have a frequency have a long wavelength. long wavelength. Waves with a high Waves with a high frequency have frequency have short wavelengths.short wavelengths.


The relationship between The relationship between wavelength and frequency is:wavelength and frequency is:

λνλν = c = c

Properties of Light - Properties of Light - AmplitudeAmplitude

DiffractionDiffraction

Waves of electromagnetic radiation Waves of electromagnetic radiation are bent or are bent or diffracteddiffracted with they a passed with they a passed through an obstacle or a slit with a size through an obstacle or a slit with a size comparable to their wavelength.comparable to their wavelength.

Interference PatternsInterference Patterns

The Failure of Classical The Failure of Classical PhysicsPhysics

Observations of the behavior of Observations of the behavior of sub-atomic particles in the early sub-atomic particles in the early 1900s could not be predicted or 1900s could not be predicted or explained using classical physics.explained using classical physics.

Very small particles such as Very small particles such as electrons appear to interact with electrons appear to interact with electromagnetic radiation (light) electromagnetic radiation (light) differently than objects we can see differently than objects we can see and handle.and handle.

Black Body RadiationBlack Body Radiation

Physicists focused on interactions Physicists focused on interactions between light (electromagnetic between light (electromagnetic radiation) and matter to try to better radiation) and matter to try to better understand the nature of the atom.understand the nature of the atom.

When objects are heated, they emit When objects are heated, they emit light in relation to their temperature. light in relation to their temperature. Iron rods glow red, and will glow Iron rods glow red, and will glow yellow at higher temperatures.yellow at higher temperatures.

Black Body RadiationBlack Body Radiation

Classical physics, when applied to Classical physics, when applied to black body radiation, predicted that the black body radiation, predicted that the intensity of the radiation emitted would intensity of the radiation emitted would dramatically increase at shorter and dramatically increase at shorter and shorter wavelengths. The result was that shorter wavelengths. The result was that any hot body should emit intense UV any hot body should emit intense UV radiation, and even x-rays. Even a human radiation, and even x-rays. Even a human body at 37body at 37ooC would glow in the dark. This C would glow in the dark. This discrepancy between theory and discrepancy between theory and observation is called “The Ultraviolet observation is called “The Ultraviolet Catastrophe.”Catastrophe.”

The Ultraviolet The Ultraviolet CatastropheCatastrophe

The failure The failure of classical of classical physics is seen physics is seen in the shorter in the shorter wavelength wavelength ultraviolet ultraviolet regionregion

Planck & Black Body Planck & Black Body RadiationRadiation

Max Planck (1858-1947) studied Max Planck (1858-1947) studied the radiation emitted by objects the radiation emitted by objects heated until they glowed. In order to heated until they glowed. In order to explain the observations, he explain the observations, he proposed (in 1900) that the energy proposed (in 1900) that the energy emitted was not continuous, but emitted was not continuous, but instead was released in multiples of instead was released in multiples of hhνν. . h h is known as Planck’s constant.is known as Planck’s constant.


∆∆E = nhE = nhννwhere n=integerwhere n=integer

νν = frequency = frequencyh = 6.626 x 10h = 6.626 x 10-34 -34 J-sJ-s

Planck’s work showed that when Planck’s work showed that when matter and energy interact, the energy matter and energy interact, the energy is is quantizedquantized, and can occur only in , and can occur only in discrete units or bundles with energy of discrete units or bundles with energy of hhνν. .


∆∆E = nhE = nhνν

Each packet or bundle of energy Each packet or bundle of energy is called a is called a quantumquantum. A fraction of a . A fraction of a quantum is never emitted. A quantum is never emitted. A quantum is the smallest amount of quantum is the smallest amount of energy that can be emitted or energy that can be emitted or absorbed in the form of absorbed in the form of electromagnetic radiation.electromagnetic radiation.

Planck’s LawPlanck’s Law

Planck’s Planck’s approach shows approach shows good agreement good agreement between the between the observed observed spectrum (in spectrum (in blue) and the blue) and the calculated calculated values (in red).values (in red).

Planck’s LawPlanck’s Law

Planck’s law was based on Planck’s law was based on empiricalempirical data data. . He found a He found a mathematical relationship that fits mathematical relationship that fits the observations. It is important to the observations. It is important to note that Planck did not explain the note that Planck did not explain the reason for the relationship.reason for the relationship.

The concept of energy being The concept of energy being quantized rather than continuous quantized rather than continuous was quite revolutionary.was quite revolutionary.


Planck received the Nobel Prize Planck received the Nobel Prize for his work in 1918 (at the age of for his work in 1918 (at the age of 42).42).

Einstein – Photoelectric Einstein – Photoelectric EffectEffect

Albert Einstein (1879-1955) won a Albert Einstein (1879-1955) won a Nobel Prize for his explanation of the Nobel Prize for his explanation of the photoelectric effectphotoelectric effect. .

When light of sufficient energy When light of sufficient energy strikes the surface of a metal, electrons strikes the surface of a metal, electrons are emitted from the metal surface. are emitted from the metal surface. Each metal has a characteristic Each metal has a characteristic minimum frequency, minimum frequency, ννo o , called the , called the threshold frequencythreshold frequency, needed for , needed for electrons to be emitted.electrons to be emitted.

The Photoelectric EffectThe Photoelectric Effect

ObservationsObservations

1. No electrons are emitted if the 1. No electrons are emitted if the frequency of light used is less than frequency of light used is less than ννoo, , regardless of the intensity of the light.regardless of the intensity of the light.

2. For light with a frequency≥ 2. For light with a frequency≥ ννo o , electrons , electrons are emitted. The number of electrons are emitted. The number of electrons increases with the intensity of the light.increases with the intensity of the light.

3. For light with a frequency > 3. For light with a frequency > ννoo , the , the electrons are emitted with greater electrons are emitted with greater kinetic energy.kinetic energy.

ExplanationExplanation

Einstein proposed that light is Einstein proposed that light is quantized, consisting of a stream of quantized, consisting of a stream of “particles” called “particles” called photonsphotons. .

If the photon has sufficient energy, it If the photon has sufficient energy, it can “knock off” an electron from the can “knock off” an electron from the metal surface. If the energy of the metal surface. If the energy of the photon is greater than that needed to photon is greater than that needed to eject an electron, the excess energy is eject an electron, the excess energy is transferred to the electron as kinetic transferred to the electron as kinetic energy.energy.


EEphotonphoton= h= hνν = hc/ = hc/λλ

If incident radiation with a frequency If incident radiation with a frequency ννii is is used:used:

KEKEelectronelectron = h = hννii -h -hννoo = ½ mv = ½ mv22

The kinetic energy of the electron The kinetic energy of the electron equals the energy of the incident radiation equals the energy of the incident radiation less the minimum energy needed to eject an less the minimum energy needed to eject an electron.electron.


The frequency The frequency hhννoo is the is the minimum energy needed to eject an minimum energy needed to eject an electron from a specific metal. This electron from a specific metal. This energy is called the energy is called the binding energy binding energy of of the emitted electron.the emitted electron.

Binding energy is often Binding energy is often expressed in electron volts (eV), with expressed in electron volts (eV), with 1 eV = 1.602 x 101 eV = 1.602 x 10–19–19 J. J.

Particle-Wave DualityParticle-Wave Duality

Einstein’s work suggested that Einstein’s work suggested that the incident photon behaved like a the incident photon behaved like a particle. If it “hits” the metal particle. If it “hits” the metal surface with sufficient energy (hsurface with sufficient energy (hννii), ), the excess energy of the photon is the excess energy of the photon is transferred to the ejected electron.transferred to the ejected electron.

In the atomic scale, waves of In the atomic scale, waves of radiant energy have particle-like radiant energy have particle-like properties.properties.


Einstein also combined his Einstein also combined his equations:equations:

E=mcE=mc22

with with

EEphotonphoton= hc/= hc/λλto obtain the “mass” of a photon:to obtain the “mass” of a photon:

m= m=

m=m=

E c2

= hc/hc/λλc2

h λc

Albert EinsteinAlbert Einstein


The apparent mass of radiant The apparent mass of radiant energy can be calculated. Although a energy can be calculated. Although a wave lacks any mass at rest, at times, wave lacks any mass at rest, at times, it behaves as if it has mass.it behaves as if it has mass.

Einstein’s equation was Einstein’s equation was confirmed by experiments done by confirmed by experiments done by Arthur Compton in 1922. Collisions Arthur Compton in 1922. Collisions between X-rays and electrons between X-rays and electrons confirmed the “mass” of the radiation.confirmed the “mass” of the radiation.


Arthur Compton attempted to Arthur Compton attempted to study the collision of a light quantum study the collision of a light quantum with an electron moving freely with an electron moving freely through space. However, creating a through space. However, creating a collision between a beam of light collision between a beam of light and a beam of electrons isn’t and a beam of electrons isn’t feasible, since it would take an feasible, since it would take an extremely long time for such a extremely long time for such a collision to occur.collision to occur.

Arthur ComptonArthur Compton

Compton solved this problem by using Compton solved this problem by using extremely high energy x-rays to bombard extremely high energy x-rays to bombard small atoms. Since the energy of the small atoms. Since the energy of the radiation was so high, the electrons in the radiation was so high, the electrons in the atoms were viewed as “free” by comparison. atoms were viewed as “free” by comparison.

Compton viewed the collision as if Compton viewed the collision as if between two elastic spheres, and perfectly between two elastic spheres, and perfectly predicted the scattering of the x-rays and predicted the scattering of the x-rays and the decrease in frequency as a result of the the decrease in frequency as a result of the collision.collision.

Arthur ComptonArthur Compton

Compton received the Nobel Compton received the Nobel prize in 1927.prize in 1927.

Emission Spectrum of Emission Spectrum of HydrogenHydrogen

When atoms When atoms are given extra are given extra energy, or energy, or excitedexcited, they , they give off the give off the excess energy excess energy as light as they as light as they return to their return to their original energy, original energy, or or ground stateground state. .

Hg He

H2


Scientists expected atoms to be able Scientists expected atoms to be able to absorb and emit a continuous range of to absorb and emit a continuous range of energies, so that a continuous spectrum energies, so that a continuous spectrum of wavelengths would be emitted.of wavelengths would be emitted.


A continuous spectrum in the visible A continuous spectrum in the visible range, would look like a rainbow, range, would look like a rainbow, with all colors visible.with all colors visible.

Instead, hydrogen, and other excited Instead, hydrogen, and other excited atoms emit only specific atoms emit only specific wavelengths of light as they return wavelengths of light as they return to the ground state. A to the ground state. A line spectrumline spectrum results.results.



Instead, only a few wavelengths of light Instead, only a few wavelengths of light are emitted, creating a are emitted, creating a line spectrumline spectrum. . The spectrum of hydrogen contains four The spectrum of hydrogen contains four very sharp lines in the visible range.very sharp lines in the visible range.


The discrete lines in the spectrum The discrete lines in the spectrum indicate that the energy of the atom indicate that the energy of the atom is is quantizedquantized. Only specific energies . Only specific energies exist in the excited atom, so only exist in the excited atom, so only specific wavelengths of radiation are specific wavelengths of radiation are emitted.emitted.

The Bohr Atomic ModelThe Bohr Atomic Model

In 1913, Neils Bohr (1885-1962) In 1913, Neils Bohr (1885-1962) proposed that the electron of hydrogen proposed that the electron of hydrogen circles the nucleus in circles the nucleus in allowed orbitsallowed orbits. .

That is, the electron is in its That is, the electron is in its ground state in an orbit closest to the ground state in an orbit closest to the nucleus. As the atom becomes excited, nucleus. As the atom becomes excited, the electron is promoted to an orbit the electron is promoted to an orbit further away from the nucleus.further away from the nucleus.


Classical physics Classical physics dictates that an dictates that an electron in a circular electron in a circular orbit must constantly orbit must constantly lose energy and emit lose energy and emit radiation. radiation.

Bohr proposed a Bohr proposed a quantum modelquantum model, as , as the spectrum showed the spectrum showed that only certain that only certain energies are energies are absorbed or emitted.absorbed or emitted.


Bohr’s model of the hydrogen Bohr’s model of the hydrogen atom was consistent with the atom was consistent with the emission spectrum, and explained emission spectrum, and explained the distinct lines observed.the distinct lines observed.

The Bohr Atomic ModelThe Bohr Atomic ModelBohr’s orbits existed at specific fixed Bohr’s orbits existed at specific fixed

distances from the nucleus. Thus the distances from the nucleus. Thus the energy of each orbit was fixed or energy of each orbit was fixed or quantizedquantized. Bohr called these stable . Bohr called these stable orbits orbits stationary states.stationary states.

Electrons can transition from Electrons can transition from one orbit to another, but they are one orbit to another, but they are never observed between states.never observed between states.

The Balmer SeriesThe Balmer Series

The emissions of hydrogen in The emissions of hydrogen in the visible region (the Balmer the visible region (the Balmer Series) produces four lines with the Series) produces four lines with the following frequencies:following frequencies:

νν1 1 =4.569 x 10=4.569 x 101414secsec-1-1

νν2 2 =6.168 x 10=6.168 x 101414secsec-1-1

νν3 3 =6.908 x 10=6.908 x 101414secsec-1-1

νν4 4 =7.301 x 10=7.301 x 101414secsec-1-1

The Balmer SeriesThe Balmer Series

The emission frequencies of The emission frequencies of hydrogen in the visible region (the hydrogen in the visible region (the Balmer Series) can be calculated Balmer Series) can be calculated using the formula:using the formula:

ννm,n m,n =3.289 x 10=3.289 x 1015 15 [1/4 -1/[1/4 -1/m m 22]sec]sec--

11

where where m m has the value of 3, 4, 5 has the value of 3, 4, 5 or 6or 6

The Rydberg EquationThe Rydberg Equation

Johannes Rydberg suggested a Johannes Rydberg suggested a different form of the equation that lead different form of the equation that lead to future discovery.to future discovery.

1/λ α [1/21/λ α [1/222 – 1/n – 1/n22] where n = 3,4,5,…] where n = 3,4,5,…

This equation was adapted for lines This equation was adapted for lines found in the infrared and ultraviolet found in the infrared and ultraviolet spectrum of hydrogen.spectrum of hydrogen.

The Rydberg EquationThe Rydberg Equation

The general form of the equation is:The general form of the equation is:

ν = ν = RR [1/n[1/n1122 – 1/n – 1/n22

22] where n] where n11 = 1,2,3,.. = 1,2,3,..

and nand n22 =n =n11+1, n+1, n11+2, ….+2, ….

R R is determined experimentally and isis determined experimentally and is

3.29 x 103.29 x 1015 15 HzHz



Bohr also developed an equation, Bohr also developed an equation, using the spectrum of hydrogen, that using the spectrum of hydrogen, that calculates the energy levels an calculates the energy levels an electron may have in the hydrogen electron may have in the hydrogen atom:atom:

E=-2.178 x 10E=-2.178 x 10-18-18J(ZJ(Z22/n/n22))

Where Z = atomic numberWhere Z = atomic number

n = an integern = an integer


Bohr also calculated the radius of Bohr also calculated the radius of the lowest energy orbit in the hydrogen the lowest energy orbit in the hydrogen atom. He proposed that the lowest atom. He proposed that the lowest energy orbit had a radius of 52.9 pm. (1 energy orbit had a radius of 52.9 pm. (1 pm = 10pm = 10-12-12 m) m)

Although the concept of circular Although the concept of circular orbits is incorrect, the value of the Bohr orbits is incorrect, the value of the Bohr radius is consistent with calculations radius is consistent with calculations based on quantum mechanics.based on quantum mechanics.


The Bohr model didn’t work for The Bohr model didn’t work for atoms other than hydrogen. It also failed atoms other than hydrogen. It also failed to explain the fine splitting of the lines of to explain the fine splitting of the lines of the emission spectrum. Though limited, the emission spectrum. Though limited, Bohr’s approach did attempt to explain Bohr’s approach did attempt to explain the quantized energy levels of electrons.the quantized energy levels of electrons.

Later developments showed that any Later developments showed that any attempt to define the path of the electron attempt to define the path of the electron is incorrect.is incorrect.

Neils BohrNeils Bohr

““If quantum If quantum mechanics hasn't mechanics hasn't profoundly profoundly shocked you, you shocked you, you haven't haven't understood it.”understood it.”

Bohr won Bohr won the Nobel prize the Nobel prize in 1922.in 1922.

Louis de BroglieLouis de Broglie

Einstein showed that waves can Einstein showed that waves can behave like particles. In 1923, Louis behave like particles. In 1923, Louis de Broglie (1892-1987) proposed de Broglie (1892-1987) proposed that moving electrons have wave-like that moving electrons have wave-like properties.properties.


In 1924, de Broglie (1892-1987) In 1924, de Broglie (1892-1987) came up with an explanation of why only came up with an explanation of why only certain orbits (and energy levels) for the certain orbits (and energy levels) for the electrons in an atom exist. Not only electrons in an atom exist. Not only does electromagnetic radiation have does electromagnetic radiation have particle-like properties, he proposed that particle-like properties, he proposed that moving electrons have wave-like moving electrons have wave-like properties.properties.


The electron in an The electron in an atom was viewed atom was viewed as a as a standing standing wavewave. For an . For an energy level to energy level to exist, the wave exist, the wave must reinforce must reinforce itself via itself via constructive constructive interferenceinterference..


Using Einstein’s equation:Using Einstein’s equation:

m=h/m=h/λλvvwhere v is the velocity of the particle, where v is the velocity of the particle,

de Broglie rearranged the equation de Broglie rearranged the equation to calculate the wavelength to calculate the wavelength associated with any moving object.associated with any moving object.


λλ=h/mv=h/mv

de Broglie’s equation was tested de Broglie’s equation was tested using a stream of electrons directed at using a stream of electrons directed at a crystal. A diffraction pattern, due to a crystal. A diffraction pattern, due to the interaction of waves, resulted. The the interaction of waves, resulted. The experiment showed that electrons have experiment showed that electrons have wave-like properties.wave-like properties.

Particle BeamsParticle Beams

Wave-Like Nature of the Wave-Like Nature of the ElectronElectron


De Broglie was awarded the Nobel De Broglie was awarded the Nobel prize in 1929.prize in 1929.


It is important to note that the It is important to note that the wave-like properties of moving particles wave-like properties of moving particles are insignificant in our everyday world. are insignificant in our everyday world. A moving object such as a car or a A moving object such as a car or a tennis ball has an insignificant radiation tennis ball has an insignificant radiation component associated with it.component associated with it.

It is on the atomic scale that the It is on the atomic scale that the dual nature of particles and light dual nature of particles and light become significant.become significant.

The Heisenberg The Heisenberg Uncertainty PrincipleUncertainty PrincipleWerner Heisenberg showed that, Werner Heisenberg showed that,

due to the wave nature of the electron, due to the wave nature of the electron,

It is impossible toIt is impossible to knowknow both the both the precise position and the momentum of precise position and the momentum of the electron at the same time.the electron at the same time.

This is known as the Heisenberg This is known as the Heisenberg Uncertainty Principle.Uncertainty Principle.

The Heisenberg The Heisenberg Uncertainty PrincipleUncertainty PrincipleIt is impossible toIt is impossible to knowknow both the both the

precise position and the momentum of precise position and the momentum of the electron at the same time.the electron at the same time.

In mathematical terms, the principle is:In mathematical terms, the principle is:

((ΔΔx) (x) (ΔΔmv) ≥ (h/4mv) ≥ (h/4ππ))

The Heisenberg The Heisenberg Uncertainty PrincipleUncertainty PrincipleIt is impossible toIt is impossible to knowknow both the both the

precise position and the momentum of precise position and the momentum of the electron at the same time.the electron at the same time.

The Heisenberg The Heisenberg Uncertainty PrincipleUncertainty Principle

((ΔΔx) (x) (ΔΔmv) ≥ (h/4mv) ≥ (h/4ππ))

There is a limit to how well we can There is a limit to how well we can determine position (x), if mass and determine position (x), if mass and velocity are known precisely.velocity are known precisely.

For large particles, the uncertainty For large particles, the uncertainty is insignificant. However, on the atomic is insignificant. However, on the atomic scale, scale, we cannot know the exact motion we cannot know the exact motion of an electronof an electron..

The Heisenberg The Heisenberg Uncertainty PrincipleUncertainty Principle

((ΔΔx) (x) (ΔΔmv) ≥ (h/4mv) ≥ (h/4ππ))

For an electron in a hydrogen For an electron in a hydrogen atom, the uncertainty in the position atom, the uncertainty in the position of the electron is similar in size to of the electron is similar in size to the entire hydrogen atom. Thus the the entire hydrogen atom. Thus the location of the electron cannot be location of the electron cannot be determined.determined.

Werner HeisenbergWerner Heisenberg

““The problems of The problems of language here are language here are really serious. We really serious. We wish to speak in wish to speak in some way about some way about the structure of the the structure of the atoms. But we atoms. But we cannot speak about cannot speak about atoms in ordinary atoms in ordinary language.”language.”

Werner HeisenbergWerner Heisenberg

Werner Heisenberg won the Werner Heisenberg won the Nobel prize in 1932. During world Nobel prize in 1932. During world war II, he lead the German research war II, he lead the German research team that was developing nuclear team that was developing nuclear fission.fission.

The Quantum The Quantum Mechanical ModelMechanical ModelThe quantum mechanical atomic The quantum mechanical atomic

model was developed based on the model was developed based on the theories of Werner Heisenberg theories of Werner Heisenberg (1901-1976), Louis de Broglie (1892-(1901-1976), Louis de Broglie (1892-1987) and Erwin Schrödinger (1887-1987) and Erwin Schrödinger (1887-1961).1961).

They focused on the wave-like They focused on the wave-like nature of the moving electron.nature of the moving electron.

The Quantum The Quantum Mechanical ModelMechanical ModelErwin Schrödinger developed Erwin Schrödinger developed

complex equations called complex equations called wave wave functions functions ( ( ΨΨ). The wave functions ). The wave functions can be used to calculate the energy can be used to calculate the energy of electrons, not only in hydrogen, of electrons, not only in hydrogen, but in other atoms. but in other atoms.

The Quantum The Quantum Mechanical ModelMechanical ModelThe wave functions also The wave functions also

describe various volumes or describe various volumes or spaces where electrons of a spaces where electrons of a specific energy are likely to be specific energy are likely to be found. These spaces are called found. These spaces are called orbitalsorbitals..

The Quantum The Quantum Mechanical ModelMechanical ModelOrbitals are not orbitsOrbitals are not orbits. .

The wave functions provide no The wave functions provide no information about the path of the information about the path of the electron. Instead, the equations (electron. Instead, the equations (ΨΨ22) ) provide the space in which there is a provide the space in which there is a high probability (90%) of finding an high probability (90%) of finding an electron with a specific energy.electron with a specific energy.

Erwin SchrödingerErwin Schrödinger

Schrodinger won the Nobel Schrodinger won the Nobel prize in 1933.prize in 1933.

OrbitalsOrbitals

The Schrödinger equation is used The Schrödinger equation is used to describe the space in which it is to describe the space in which it is likely to find an electron with a specific likely to find an electron with a specific energy. energy.

The equation provides us with a The equation provides us with a probability distributionprobability distribution, or an , or an electron electron density mapdensity map. It is important to . It is important to remember that the resulting shape does remember that the resulting shape does not give us any information about the not give us any information about the path of the electrons.path of the electrons.

OrbitalsOrbitals

Each orbital described by the Each orbital described by the Schrodinger equations is associated Schrodinger equations is associated with three interrelated with three interrelated quantum quantum numbersnumbers which relate to the energy which relate to the energy of electrons in the orbital and the of electrons in the orbital and the probability of finding the electron probability of finding the electron within a particular volume.within a particular volume.

Quantum NumbersQuantum Numbers

The The principal quantum number, n, principal quantum number, n, determines the overall size and energy of determines the overall size and energy of an orbital. It is an integer with values of an orbital. It is an integer with values of 1, 2, 3, etc.1, 2, 3, etc.

The The angular momentum quantum angular momentum quantum number, l, number, l, determines the shape of the determines the shape of the orbital. It is related to the more familiar orbital. It is related to the more familiar designations of designations of s, p, d s, p, d and and ff. The value . The value of of ll is 0 for an is 0 for an s s orbital, 1 for a orbital, 1 for a pp orbital, orbital, 2 for a 2 for a d d orbital, and 3 for an orbital, and 3 for an f orbital.f orbital.


For a given value of For a given value of nn, , ll is an is an integer with values from 0 up to integer with values from 0 up to n-1n-1. .

For For nn=1, =1, ll can only = 0 [a 1s can only = 0 [a 1s orbital].orbital].

For For nn=2, =2, ll can be 0 or 1 [a 2s or can be 0 or 1 [a 2s or 2p subshell].2p subshell].

For For nn=3, =3, ll can be 0, 1 or 2 [a 3s, can be 0, 1 or 2 [a 3s, 3p or 3d subshell].3p or 3d subshell].


The magnetic quantum number, The magnetic quantum number, mmll , describes the spatial orientation of , describes the spatial orientation of the orbital. For a given value of the orbital. For a given value of ll, , mmll may have the value of: may have the value of:

––l,…0,…+ll,…0,…+l

Thus, a Thus, a pp subshell consists of three subshell consists of three pp orbitals (p orbitals (pxx, p, pyy, p, pzz) with ) with mmll values of -values of -1, 0 and +1.1, 0 and +1.


The orbital is described by The orbital is described by quantum numbers quantum numbers n, l, n, l, and and mmll . . To To describe the electrons within an orbital, describe the electrons within an orbital, a fourth quantum number, the electron a fourth quantum number, the electron spin quantum number, spin quantum number, mmss is needed.is needed.

The quantum number relates to the The quantum number relates to the direction of spin of an electron around direction of spin of an electron around its own axis, and it has the values of its own axis, and it has the values of either +½ or -½ .either +½ or -½ .

Electron SpinElectron Spin

Each orbital, regardless of type, can Each orbital, regardless of type, can contain zero, one or two electrons. If contain zero, one or two electrons. If two electrons occupy the same orbital, two electrons occupy the same orbital, they must spin in opposite directions.they must spin in opposite directions.

The spin is quantized, and can be The spin is quantized, and can be expressed using quantum numbers, or expressed using quantum numbers, or simply specifying the spin as up or down simply specifying the spin as up or down or clockwise and counter-clockwise.or clockwise and counter-clockwise.

The Pauli Exclusion The Pauli Exclusion PrinciplePrinciple

Quantum mechanics dictates Quantum mechanics dictates that no two electrons in an atom can that no two electrons in an atom can have the same four quantum have the same four quantum numbers. Another way of stating the numbers. Another way of stating the Pauli Exclusion PrinciplePauli Exclusion Principle is that if is that if electrons occupy the same orbital, electrons occupy the same orbital, they must have opposite spins.they must have opposite spins.

Multi-electron AtomsMulti-electron Atoms

Orbitals of Orbitals of any type can be any type can be empty, or have 1 empty, or have 1 or two electrons. or two electrons.

Experimental Experimental data indicate that data indicate that if two electrons if two electrons are in the same are in the same orbital, they will orbital, they will spin in opposite spin in opposite directions.directions.

Energy LevelsEnergy LevelsIn any atom or ion In any atom or ion with only 1 with only 1 electronelectron, the , the principal quantum principal quantum number, n, number, n, determines the determines the energy of the energy of the electron. For n=2, electron. For n=2, the 2s and 2p the 2s and 2p orbitals all have the orbitals all have the same energy.same energy.

Energy LevelsEnergy Levels

Likewise, Likewise, the 3s, 3p and the 3s, 3p and 3d orbitals are 3d orbitals are all degenerate, all degenerate, with the same with the same energy.energy.


In a multi-electron atom, there In a multi-electron atom, there is interaction between electrons. As is interaction between electrons. As a result of this interaction, the a result of this interaction, the various subshells of a principal various subshells of a principal quantum level will vary in energy.quantum level will vary in energy.




The energy The energy diagram for the diagram for the first three first three quantum levels quantum levels shows the shows the splitting of splitting of energies. energies.


For a given For a given value of n, the value of n, the energies of the energies of the subshells is as subshells is as follows:follows:

ns<np<nd<nfns<np<nd<nf


The subshells The subshells have different have different energies due to energies due to the the penetrating penetrating abilityability for each for each type of orbital.type of orbital.

Electrons in a Electrons in a 2s orbital can get 2s orbital can get nearer to the nearer to the nucleus than those nucleus than those in a 2p orbital. in a 2p orbital.

Energy Energy LevelsLevels

The electrons The electrons in the 3s orbital in the 3s orbital (top diagram) have (top diagram) have higher probability higher probability to be found near to be found near the nucleus, and the nucleus, and thus greater thus greater penetrating ability penetrating ability than those in 3p or than those in 3p or 3d orbitals.3d orbitals.

OrbitalsOrbitals

The orbital of lowest energy is The orbital of lowest energy is the 1s orbital. The probability the 1s orbital. The probability density, or probability of finding an density, or probability of finding an electron per unit volume, shows electron per unit volume, shows electron density in all directions, electron density in all directions, creating a spherical shape. creating a spherical shape.

The probability density The probability density decreases with greater distance decreases with greater distance from the nucleus.from the nucleus.

OrbitalsOrbitals

OrbitalsOrbitals

Radial Distribution Radial Distribution FunctionFunction

The radial distribution function The radial distribution function is a graphical representation of the is a graphical representation of the probability of finding an electron in probability of finding an electron in a thin spherical shell a specific a thin spherical shell a specific distance from the nucleus.distance from the nucleus.

It shows that there is zero It shows that there is zero probability that the electron will be probability that the electron will be at the nucleus, and also indicates the at the nucleus, and also indicates the most probable distance the electron most probable distance the electron will have from the nucleus.will have from the nucleus.

Radial Distribution Radial Distribution FunctionFunction

The maximum at The maximum at 52.9 pm is 52.9 pm is consistent with consistent with Bohr’s radius for Bohr’s radius for the hydrogen atom. the hydrogen atom. It more correctly It more correctly indicates the indicates the most most probableprobable distance distance between the between the electron and electron and nucleus.nucleus.

OrbitalsOrbitals

The first energy level of hydrogen The first energy level of hydrogen (n=1) consists of a 1s orbital.(n=1) consists of a 1s orbital.

The second energy level of The second energy level of hydrogen (n=2) consists of a 2s orbital hydrogen (n=2) consists of a 2s orbital and 2p orbitals.and 2p orbitals.

The third energy level of hydrogen The third energy level of hydrogen (n=3) consists of a 3s orbital, 3p (n=3) consists of a 3s orbital, 3p orbitals, and 3d orbitals.orbitals, and 3d orbitals.

OrbitalsOrbitals

As the value As the value of n increases, of n increases, the orbitals, on the orbitals, on average, average, become larger, become larger, with more with more electron density electron density farther from the farther from the nucleus.nucleus.

OrbitalsOrbitals

The “white The “white rings” in the rings” in the drawings are drawings are nodesnodes. This is . This is the region where the region where the wave function the wave function goes from a goes from a positive value to positive value to a negative value.a negative value.

The 2s and 3s Orbitals

OrbitalsOrbitalsp orbitals are “dumbbell” p orbitals are “dumbbell”

shaped, with two lobes. In one lobe, shaped, with two lobes. In one lobe, the wave function is positive, in the the wave function is positive, in the other lobe, it is negative.other lobe, it is negative.

OrbitalsOrbitalsp orbitals come in sets of three, called a p orbitals come in sets of three, called a

subshellsubshell. The three orbitals are designated . The three orbitals are designated as pas pxx, p, pyy and p and pzz, because the electron density , because the electron density lies primarily along either the x, y or z axis.lies primarily along either the x, y or z axis.

OrbitalsOrbitalsAll three orbitals have the exact All three orbitals have the exact

same energy. Orbitals with the same energy. Orbitals with the same energy are called same energy are called degenerate.degenerate.

Orbital PhaseOrbital Phase

The drawings of orbitals is an The drawings of orbitals is an attempt to visualize three-attempt to visualize three-dimensional waves. Waves can dimensional waves. Waves can undulate from positive to negative undulate from positive to negative amplitudes. The sign of the amplitudes. The sign of the amplitude is known as its amplitude is known as its phasephase..

The phase of a sine wave The phase of a sine wave fluctuates between positive and fluctuates between positive and negative.negative.



The phase of the wave functions The phase of the wave functions or orbitals is quite important when or orbitals is quite important when atoms bond together. The orbitals atoms bond together. The orbitals must be of the same phase to must be of the same phase to overlap and form covalent bonds.overlap and form covalent bonds.

OrbitalsOrbitalsThe n=3 The n=3 level level contains contains s, p and d s, p and d orbitals. orbitals. The d The d orbitals orbitals are are shown.shown.

OrbitalsOrbitals

The The n=4 level n=4 level contains s, contains s, p, d and f p, d and f orbitals. orbitals. The f The f orbitals are orbitals are shown.shown.


Electron configurations are a Electron configurations are a way of noting which subshells of an way of noting which subshells of an atom contain electrons. atom contain electrons. Although Although much of the periodic table was much of the periodic table was developed before the concept of developed before the concept of electron configurations, it turns out electron configurations, it turns out that the position of an element on that the position of an element on the periodic table is directly related the periodic table is directly related to its electron configuration.to its electron configuration.


Electron ConfigurationsElectron Configurations

Write the complete electron Write the complete electron configurations for nitrogen and zinc.configurations for nitrogen and zinc.

How many unpaired electrons does How many unpaired electrons does each atom have?each atom have?

What is the short hand notation for What is the short hand notation for each element?each element?

Hund’s RuleHund’s Rule

When electrons occupy When electrons occupy degenerate orbitals, they occupy degenerate orbitals, they occupy separate orbitals with parallel spins.separate orbitals with parallel spins.

This is the lowest energy, or This is the lowest energy, or ground state, configuration.ground state, configuration.


The electron configurations for Cr The electron configurations for Cr and Cu differ from that expected and Cu differ from that expected based on their positions in the based on their positions in the periodic table.periodic table.

Documents

The Quantum- Mechanical Model of the Atom. Atoms and Energy All energy is either kinetic energy (the energy of motion), or potential energy (energy based