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Chapter 7 The Quantum-Mechanical Model of the Atom Chapter 7 Chapter 7 The Quantum-Mechanical Model of the Atom “When you have eliminated the impossible, whatever remains, no matter how improbable, must be the truth.” Sherlock Holmes Chapter 7 The Quantum-Mechanical Model of the Atom 7.1 7.2 7.3 7.4 7.5 7.6 Schrodinger’s Cat The Nature of Light Atoms, Spectroscopy and The Bohr Model The Wave Nature of Matter: The de Broglie Wavelength, the Uncertainty Principle and Indeterminacy Quantum Mechanics and the Atom The Shapes of Atomic Orbitals 3 Section 7.1 Schrodinger’s Cat Macroscopic vs Subatomic World • The laws of physics that govern the macroscopic world (apples and cars and trees and us) are called Newton’s Laws or Classical Physics – If something is travelling in a straight line – it will keep travelling that way unless a force acts on it – What goes up must come down – Things like this 4 Section 7.1 Schrodinger’s Cat Macroscopic vs Subatomic World • The Quantum Mechanical Model • The subatomic world is very strange, almost unimaginably strange. – Things don’t move in predictable ways – Things appear and disappear – Absolutely small particles (like electrons) can be in two different states at the same time • The scientists who developed the model were actually kind of shocked and dismayed by what it predicted. 5 Section 7.1 Schrodinger’s Cat Macroscopic vs Subatomic World • Schrödinger's Cat • This is an absurd example of trying to apply the rules of the subatomic world to the macroscopic world. • A cat in a container with a poison that is released by the emission of a radioactive particle • If the container is closed we don’t know if the poison has been released or not. • So the cat is both dead and alive at the same time • It is not til we open the container and make an observation that we force the cat into one state or the other by the act of observation. 6 Section 7.1 Schrodinger’s Cat Macroscopic vs Subatomic World • This chapter is about the Quantum Mechanical Model • We look at where the model came from • This is a model so it explains things • Like periodic behavior in the periodic table 7 Section 7.2 The Nature of Light Properties of Light • In the very beginning the experiments that would lead to the QM model of the atom began with an examination of the properties of light • So we are going to start there too. • First we will look at the wave nature of light. • The way light was first understood 8 Section 7.2 The Nature of Light The Wave Nature of Light • Light is electromagnetic radiation. • A type of energy characterized by oscillating electric and magnetic fields – Sounds awful doesn’t it. – It not – both are actually familiar. • Magnetic field is the space where a magnetic particle feels a force (area around a magnet) • Electric Field is the region of space where an electrically charged particle feels a force – Proton generates an electric field 9 Section 7.2 The Nature of Light The Wave Nature of Light • Light is electromagnetic radiation. – Oscillating electric and magnetic fields – Characterized by amplitude (intensity or brightness) and wavelength (distance between peaks) 10 Section 7.2 The Nature of Light The Wave Nature of Light • Wavelength and amplitude are related to the amount of energy in the wave • Imagine swimming at the beach – High waves (large amplitude) – Close together (short wavelength) • Very hard to swim against • Because they have lots of energy – Small waves (low amplitude) – Far apart (long wavelength) • Easy to swim against – low energy 11 Section 7.2 The Nature of Light The Wave Nature of Light • Light is also characterized by frequency (n) the number of waves that pass a certain point in a given period of time • The relationship between n and l is c n l • Where c = a constant (speed of light 3.00 x 108 m/s) • And l is the wavelength • Notice the inverse relationship between n and l 12 Section 7.2 The Nature of Light The Wave Nature of Light • For visible light, wavelength (l) or frequency (n) determines color. 13 Section 7.2 The Nature of Light Concept Check Determine the frequency of a type of electromagnetic radiation with a wavelength of 2.12 x 10 – 10 m. 14 Section 7.2 The Nature of Light The Electromagnetic Spectrum • Visible light is only a tiny portion of the entire electromagnetic spectrum. Fig 7-5 • Shortest wavelength have highest frequency (and energy) • Longest wavelength have the lowest energy 15 Section 7.2 The Nature of Light Interference and Diffraction • Electromagnetic radiation (light) moves in waves • Waves can interact with each other (interference) • They can cancel each (constructive interference) other or build each other up (destructive interference) 16 Section 7.2 The Nature of Light Interference and Diffraction • Waves that are ‘in phase” align so that the crests overlap • Waves that are ‘out of phase” overlap so that the crest of one overlaps with the trough of another 17 Section 7.2 The Nature of Light Interference and Diffraction Fig 7-6 • Another characteristic of light is diffraction. Light bends around the slit. Particles pass straight through. 18 Section 7.2 The Nature of Light Diffraction Patterns • Diffraction patterns arise from constructive and destructive interference of light passing through multiple slits. Fig 7-7 19 Section 7.2 The Nature of Light Matter vs Energy • At the end of the 19th century the structure of the atom figured out. • Matter and Energy considered distinct • Matter – Particles which have mass – Position in space can be specified • Energy – Form of light described as a wave, which is massless – Position in space cannot be localized 20 Section 7.2 The Nature of Light The Particle Nature of Light • So light is a wave and matter is a particle • The classical view of light is that it was purely a wave phenomenon – This view was particularly supported by the observation of the diffraction of light. • So of course it can’t be that simple • There was an observation that challenged this view • Called the Photoelectric Effect 21 Section 7.2 The Nature of Light The Photoelectric Effect • Many metals emit electrons when light shines on them Fig 7-8 22 Section 7.2 The Nature of Light The Photoelectric Effect • The explanation for the photoelectric effect (according to classical physics) was that the energy from the light was transferred to the metal which dislodged the electrons. • OK – so light is still a wave. 23 Section 7.2 The Nature of Light The Photoelectric Effect • According to this explanation the brighter the light (higher intensity) the more electrons should be dislodged • If you used a dim light there should be a lag time before you build up enough energy to dislodge an electron • But – that is not what the experimental results showed • Low intensity (dim) high frequency (energy) light produced electrons without the predicted lag time 24 Section 7.2 The Nature of Light The Photoelectric Effect • In addition the photoelectric effect shows a threshold frequency • Below the threshold no electrons are ejected no matter how long the light shines or how intense (bright) it is. Fig 7-9 25 Section 7.2 The Nature of Light The Photoelectric Effect • Low frequency (low energy) light does not eject electrons from metal – No matter how intense (bright) – No matter how long it shines • High frequency (high energy) light does eject electrons – Even if the light is dim • How to explain this? 26 Section 7.2 The Nature of Light The Photoelectric Effect • Albert Einstein explained it using the reasoning that light actually comes in packets (photons) • And that the amount of energy in a photon is given by the equation Ephoton hn where h is a constant (Planck' s constant) with a value of 6.626 x 10 sin ce n = c/l hc Ephoton l 34 Js 27 Section 7.2 The Nature of Light The Photoelectric Effect • This was (of course) a revolutionary idea. • Classical electromagnetic theory viewed light as a wave whose intensity was continuously variable • Einstein is suggested that light is quantized into particles and that a beam of light is not a wave but rather a shower of particles each with a discrete energy = hn 28 Section 7.2 The Nature of Light • Dual nature of light: Electromagnetic radiation (and all matter) exhibits wave properties and particulate properties. 29 Section 7.2 The Nature of Light The Photoelectric Effect • So lets assume Einstein is correct. • How does quantizing light explain the experimental observations of the photoelectric effect? • The existence of the threshold frequency and the lack of lag time with low intensity light suggest that the light energy does not add up to the point where the electron is ejected. • Rather you need a single event that provides the appropriate amount of energy. 30 Section 7.2 The Nature of Light How do Photons Explain The Photoelectric Effect • Low frequency (low energy) light does not eject electrons from metal – No matter how intense (bright) – No matter how long it shines – These photons don’t have enough energy and they don’t sum or add up • High frequency (high energy) light does eject electrons – Even if the light is dim – These photons have enough energy – right frequency – Remember amplitude (or intensity/brightness) is not energy – frequency/color is energy 31 Section 7.2 The Nature of Light The Photoelectric Effect • The emission of electrons from the metal surface depend on whether or not a single photon has sufficient energy (hn) to dislodge a single electron 32 Section 7.2 The Nature of Light The Photoelectric Effect • Think of photons like ping pong balls or baseballs and the ejection of electrons like breaking the pane of glass. – ping pong balls are below the threshold frequency – baseballs are above the threshold frequency • Any photons above the threshold frequency transfer the extra energy to the electron in the form of kinetic energy 33 Section 7.2 The Nature of Light Concept Check Determine the increment of energy (the energy of a photon of light) that is emitted by light with a frequency of 1.42 x 1018 /s. The value of Planck’s constant is 6.626 x 10 – 34 Js. 34 Section 7.2 The Nature of Light Conceptual Connection Light of three different wavelengths 325 nm, 455 nm and 632 nm shines on a metal surface. Match the wavelength with the observation. A. No photoelectrons were observed. B. Photoelectrons with a kinetic energy of 155 kJ/mole were observed C. Photoelectrons with a kinetic energy of 51 kJ/mole were observed 35 Section 7.3 Atomic Spectroscopy and the Bohr Model Atomic Spectroscopy • The discovery of the particle nature of light was a breakthrough that began to challenge the classical view that light was only a wave • Similarly, certain observations about atoms began to suggest a wave nature for particles • The most significant of the observations was atomic spectroscopy – the study of electromagnetic radiation 36 Section 7.3 Atomic Spectroscopy and the Bohr Model Atomic Spectroscopy • When atoms absorb energy (heat, light, electricity) they can re-emit that energy as light Fig 7-10 – Think of a neon sign • Different atoms emit light of a characteristic color Mercury, Helium and Hydrogen 37 Section 7.3 Atomic Spectroscopy and the Bohr Model Atomic Spectroscopy • If we pass the light emitted by an element through a prism we see that it is actually composed of several different wavelengths • The color of light is determined by its wavelength Fig 7-11 38 Section 7.3 Atomic Spectroscopy and the Bohr Model Atomic Spectroscopy • The series of bight lines is called an emission spectrum • Emission spectrum for an element is always the same • Can be used to identify an element • Used to identify the composition of distant stars Fig 7-11 39 Section 7.3 Atomic Spectroscopy and the Bohr Model Atomic Spectroscopy • Look at the difference between the white light spectrum (continuous) and the barium spectrum (discrete lines) • Classical physics could not explain this Fig 7-11 40 Section 7.3 Atomic Spectroscopy and the Bohr Model Atomic Spectroscopy • Classical physics actually predicts and continuous spectrum where an electron orbiting a nucleus would release light of every wavelength • An even bigger problem is that classical physics predicts an electron orbiting the nucleus would lose energy as it emits light and spiral into the nucleus. • According to classical physics the atom should not even be stable! Hmmmmm! 41 Section 7.3 Atomic Spectroscopy and the Bohr Model The Bohr Model • So basically what we need is a new kind of physics to explain the behavior of the atom. • This process had already begun with the breaking down of the barrier between light as a wave and matter as a particle. • Einstein showed that light behaves as a particle • The Bohr model is the beginning to the process of treating matter (electrons) as a wave. 42 Section 7.3 Atomic Spectroscopy and the Bohr Model The Bohr Model • There was a mathematician named Rydberg who looked at lots and lots of atomic spectra and he came up with an equation that predicted the wavelengths of the lines in the hydrogen emission spectra 1 1 1 = R 2 2 The Rydberg Equation m l n • His equation worked but it didn’t explain anything. He didn’t even know what m and n were. (We are going to see it again later) 43 Section 7.3 Atomic Spectroscopy and the Bohr Model The Bohr Model • So Niels Bohr developed a model for the atom that would explain atomic spectra. • Electrons travel around the nucleus in circular orbits • These orbits exist only at specific fixed distances – They are quantized – So they obey classical physics but also posses an unknown stability – Bohr called these orbits stationary states 44 Section 7.3 Atomic Spectroscopy and the Bohr Model The Bohr Model • Bohr also stated that while the electron is circling the nucleus in the stationary state, no radiation is emitted – Contradicts classical physics • In the Bohr Model the only time radiation is emitted or absorbed is when the electron transitions from one stationary state to another. 45 Section 7.3 Atomic Spectroscopy and the Bohr Model The Bohr Model • Wavelengths for electronic transitions for hydrogen Fig 7-12 • Electron moving from n= 5 orbit to n = 1 orbit released energy with a wavelenth of 434 nm 46 Section 7.3 Atomic Spectroscopy and the Bohr Model The Bohr Model • Transitions between stationary states in a hydrogen atom are not like what you imagine in the macroscopic world. • The electron does not exactly travel from n = 5 to n = 1 • It is never observed between states • It is only observed in one state or the other • This stuff is weird. 47 Section 7.3 Atomic Spectroscopy and the Bohr Model The Bohr Model • The Bohr Model explained the line spectrum of hydrogen • But also left a lot of unanswered questions • Was an intermediate between the classical view of the electron and the fully quantum-mechanical view. • Which is what we will do next. 48 Section 7.4 The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy The Wave Nature of the Electron • The heart of the quantum – mechanical model (which replaced the Bohr Model) is the wave nature of the electron. • The wave nature of the electron was first proposed by Louis de Broglie (we will look at what he did next) but was not confirmed until about 50 years later when scientists observed diffraction of electrons. 49 Section 7.4 The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy • de Broglie figured if light has characteristic of both waves and particles, what about matter? • Took the equation for the energy of a photon E hc l hv mv = or l 2 hv h m = 2 lv lv • Substituted mc2 for E and n for c because matter does not move at the speed of light (yet). • Allows us to calculate the wavelength of a particle. 50 Section 7.4 The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy The de Broglie Wavelength • So Louis de Broglie proposed that is was possible to determine the wavelength of an electron from its velocity h if m = then lv h l = mv • Where l = wavelength, h = Planck's constant 6.626 x 10 – 34 J s, m is the mass of the electron (in kg) and v is its velocity 51 Section 7.4 The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy Concept Check • Since quantum mechanical theory is universal that means even macroscopic objects have a wavelength. • Determine the wavelength of a car which has a mass of 1800 kg travelling at 19.4 m/s (appx 70 km/hour). 52 Section 7.4 The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy The Wave Nature of the Electron • The wave nature of the electron which was proposed by Louis de Broglie was not confirmed until about 50 years later when scientists observed diffraction of electrons. 53 Section 7.4 The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy The Wave Nature of the Electron • We have seen diffraction before. Light diffracts when it passes through a slit 54 Section 7.4 The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy The Wave Nature of the Electron • We would not expect to see diffraction of electrons if they were particles. We would Fig 7-16b expect something more like this. 55 Section 7.4 The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy The Wave Nature of the Electron • Instead a beam of electrons gives this pattern. Fig 7-16b • This same pattern appears when single electrons pass through the slit one at a time. 56 Section 7.4 The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy The Wave Nature of the Electron • How can a single electron passing through the slit produce a diffraction pattern • Two waves are required to produce interference and produce the diffraction pattern. • This is where things get really weird. • The single electron actually passes through both slits at the same time. • This is the proof of its wave nature. 57 Section 7.4 The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy The Uncertainty Principle • So Louis de Broglie proposed that electrons have a wave nature and we have seen the diffraction pattern. • But electrons also have a particle nature because they have mass. • How can the electron be both a particle and wave at the same time. • Lets return to the diffraction experiment. 58 Section 7.4 The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy The Uncertainty Principle • The diffraction pattern from the single electrons was assumed to arise from the electron passing through both slits at the same time and producing two new waves (thus interfering with itself) • Lets test this hypothesis • Well it turns out we can’t • The act of observing the electron forces it to go through one slit or the other • Diffraction pattern disappears. 59 Section 7.4 The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy The Uncertainty Principle • If this laser beam detector is turned on the diffraction pattern disappears. 60 Section 7.4 The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy The Uncertainty Principle • We cannot simultaneously observe both the particle and wave nature of the electron. • When we try to observe which slit the electron passes through (particle nature) we lose the diffraction patter (wave nature) • When we observe the diffraction pattern (wave nature) we cannot determine which slit the electron passes through (particle nature) • Wave and particle nature are said to be complementary properties 61 Section 7.4 The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy The Uncertainty Principle • Complementary properties exclude one another. • The more we know about one the less we know about the other. • Which property we observe depends on the experiment • In quantum mechanics the event affect the outcome. • Kind of like Schrödinger's Cat 62 Section 7.4 The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy The Uncertainty Principle • The electron has other complementary properties – The velocity of an electron is related to its wave nature (de Broglie equation) – The position of an electron is related to its particle nature • So we cannot simultaneously know both velocity and position • Velocity and position are complementary properties • Werner Heisenberg formalized this idea with an equation 63 Section 7.4 The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy The Uncertainty Principle • The Heisenberg Uncertainty Principle h Dx mDv 4 • Where Dx is uncertainty in position Dv is uncertainty in velocity • h is Planck’s constant 64 Section 7.4 The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy Indeterminacy and the Probability Distribution Map • In classical physics particles move in a trajectory determined by position, velocity and forces acting on the particle. 65 Section 7.4 The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy Indeterminacy and the Probability Distribution Map • Classical physics is deterministic – the present determines the future. • Two baseballs hit the same way will travel the same path and land in the same place. • Electrons don’t behave this way. • We can’t know both their position and velocity. • In quantum mechanics trajectories are replaced by probability distribution maps. 66 Section 7.4 The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy Indeterminacy and the Probability Distribution Map • If a baseball behaved like an electron it would land in a different place very time. • This behavior of the electron is called indeterminacy. – If we were able to observe hundreds of electron baseballs however we would be able to observe a statistical pattern of where the electron-baseball was likely to land 67 Section 7.4 The Wave Nature of Matter: The de Broglie Wavelength, The Uncertainty Principle and Indeterminacy Indeterminacy and the Probability Distribution Map • So how do we apply the probability distribution map to actual electrons. • In the next section we introduce the concept of quantum-mechanical orbitals • Probability distribution maps for electrons as they exist in atoms. 68 Section 7.5 Quantum Mechanics and the Atom Orbitals • The electrons position is described in terms of an orbital which is a probability distribution map where the electron is likely to be found. – A mathematically description of the probability of finding an electron at a given point in space around the nucleus • Not a very satisfying definition • Notice it says – likely. • Why does an orbital describe a probability? • Why isn’t an orbital a real physical thing? 69 Section 7.5 Quantum Mechanics and the Atom Orbitals • Remember the Heisenberg Uncertainty Principle. • Says we can’t know both position and velocity of an electron at the same time • The orbital doesn’t define position very accurately but it turns out we don’t need to know the position very accurately. • The next few slides explain why. 70 Section 7.5 Quantum Mechanics and the Atom Position vs Energy • Position and velocity are complementary properties – He more accurately we know one the less accurately we know the other • Velocity is directly related to energy (KE = ½ mv2), • So position and energy are also complementary properties. • So we can only define one accurately • Either Position or Energy. 71 Section 7.5 Quantum Mechanics and the Atom Energies • The properties of elements depend primarily on the energies of its electrons. – Whether and electron is transferred from one atom to another to form an ionic bond depends on the relative energies of the two atoms. • Since the properties of elements depend on energy this is the one we need to be well defined 72 Section 7.5 Quantum Mechanics and the Atom The Schrodinger Equation • So a probability distribution map gives us an idea of where the electron is (the orbital) and each of these electrons has an very specific energy. • What are the values of these energies? • The mathematical derivation of energies and orbitals comes from solving the Schrodinger Equation. 73 Section 7.5 Quantum Mechanics and the Atom The Schrodinger Equation • The Schrodinger Equation. HYEY H is a set of mathematical operations that represent the total energy (kinetic and potential) of the electron – E is the actual energy of the electron Y is called the wave function – describes the wavelike nature of the electron • A plot of Y2 represents an orbital – a position probability distribution map of the electron 74 Section 7.5 Quantum Mechanics and the Atom Plot of Y2 represents an orbital • Imagine every one of these dots representing the position of the electron at a given point in time. • Notice how there are more dots closer to the origin (nucleus) Fig 7-23a 75 Section 7.5 Quantum Mechanics and the Atom Quantum numbers • Plot of Y2 represents an orbital. • The electrons in each orbital have very specific energies. • Each of these orbitals can be described by a series of numbers (quantum numbers) which describe various properties of the orbital. • Kind of an orbital bookkeeping system. 76 Section 7.5 Quantum Mechanics and the Atom Quantum numbers • Principle quantum number (n) – overall size and energy of the orbital • Angular momentum quantum number (l) – shape of orbital • Magnetic quantum number (ml) – orientation of orbital • Spin quantum number (ms)– direction of spin of the electron 77 Section 7.5 Quantum Mechanics and the Atom Principle Quantum Number (n) • Principle quantum number (n) • Integer that determines the overall size and energy of the orbital • n = 1, 2, 3, …. • The energy of the orbital is given by En 2.178 x 10 –18 1 J 2 n 78 Section 7.5 Quantum Mechanics and the Atom Principle Quantum Number • Why is this a negative value? • An unbound electron would have an energy of zero En 2.178 x 10 –18 1 J 2 0 • The energy of an electron bound to a nucleus would be lower (more stable) compared to this reference state – Thus – a negative sign in the equation 79 Section 7.5 Quantum Mechanics and the Atom Angular Momentum Quantum Number • • • • • • Angular momentum quantum number Integer that determines the shape of the orbital Indicated by the letter l Possible values of l are from 0 to n-1 For example if n = 1 the only value of l = 0 If n = 2 the values of l are 0 and 1 80 Section 7.5 Quantum Mechanics and the Atom Angular Momentum Quantum Number • The different values of l have letter designations • These are the familiar orbital designations. 81 Section 7.5 Quantum Mechanics and the Atom Magnetic Quantum Number • • • • • • Magnetic quantum number Integer that determines the orientation of the orbital Indicated by the letter ml Possible values of ml are from –l to +l. For example if n = 1 the only value of l = 0 so ml = 0 If l = 2 the values of l are 0 and 1 – For l = 0 ml = 0 – For l = 1 ml = –1, 0 and 1 82 Section 7.5 Quantum Mechanics and the Atom Spin Quantum Number • • • • Spin quantum number Specifies the direction of spin of the electron Indicated by the letter ms Electrons either spin up + ½ or down – ½ 83 Section 7.5 Quantum Mechanics and the Atom Summary of Quantum Number • Principle (n) 0, 1, 2, etc. • Shape (l) 0 to n minus 1 • Orientation (ml) – l to + l • Electron Spin + ½ (up) – ½ (down) 84 Section 7.5 Quantum Mechanics and the Atom Quantum Numbers Specify Orbitals • n = principle – Indicates energy • l = sublevel or subshell – Indicates shape • ml - indicates orientation 85 Section 7.5 Quantum Mechanics and the Atom Quantum Numbers Specify Orbitals • n=1 • l=0 • ml = 0 86 Section 7.5 Quantum Mechanics and the Atom Quantum Numbers Specify Orbitals • n=2 • l=1 • ml = 0 87 Section 7.5 Quantum Mechanics and the Atom Quantum Numbers Specify Orbitals • n=3 • l=2 • ml = 1 88 Section 7.5 Quantum Mechanics and the Atom Concept Check For principal quantum level n = 3, determine the number of allowed subshells (different values of l), and give the designation (number and letter) of each. 89 Section 7.5 Quantum Mechanics and the Atom Concept Check For l = 2, determine the magnetic quantum numbers (ml) and the number of orbitals. 90 Section 7.5 Quantum Mechanics and the Atom Concept Check Each set of quantum numbers below is supposed to specify an orbital. However each set contains one quantum number that is not allowed. Replace the quantum number that is not allowed with one that is allowed A. n = 3, l = 3, ml = +2 B. n= 2, l = 1, ml = –2 C. n = 1, l = 1, ml = 0 91 Section 7.5 Quantum Mechanics and the Atom Atomic Spectroscopy Explained • Quantum theory explains the observations made from atomic spectra in Section 7.3 • Each line in the emission spectra is due the the emission of a specific wavelength of light when an electron transitions between quantum mechanical orbitals. 92 Section 7.5 Quantum Mechanics and the Atom Atomic Spectroscopy Explained • When atom absorbs energy electron is excited to higher energy orbital. But this atoms is unstable • Electron quickly falls back (relaxes) to a lower energy orbital – releases a photon of light. Energy of that photon is exactly equal to the difference in energy between the two levels 93 Section 7.5 Quantum Mechanics and the Atom Atomic Spectroscopy Explained • The energy of an orbital in a hydrogen atom with a principal energy level of n is En 2.178 x 10 –18 • The difference between the two energy levels DE = Efinal – Einitial DE 2.178 x 10 –18 DE 2.178 x 10 –18 1 J 2 2.178 x 10 –18 n f 1 J 2 n 1 J 2 n i 1 1 J 2 – 2 look - its the Rydberg Equation n f n i 94 Section 7.5 Quantum Mechanics and the Atom Atomic Spectroscopy Explained • This equation allows us to calculate changes in energy of an electron when the electron changes orbits. From n = 6 to n = 1 DE 2.178 x 10 –18 1 1 J 2 – 2 n f n i DE 2.178 x 10 –18 1 1 J 2 – 2 1 6 DE 2.178 x 10 –18 J1 – .028 DE 2.178 x 10 –18 J x 0.972 DE 2.117 x 10 18 J 95 Section 7.5 Quantum Mechanics and the Atom Atomic Spectroscopy Explained • The DE is negative because the atom emits the energy DEatom = – DEphoton • Once we have the change in energy we can determine the wavelength of light l hc DE 96 Section 7.5 Quantum Mechanics and the Atom Atomic Spectroscopy Explained l l hc DE l hc 2.118 x 10 18 J 6.626x1034 J s3.00x108 m s 2.117 x 10 18 J 9.390x10 8 m • Notice DE in this calculation is not negative. Not relative to the atom anymore. 97 Section 7.5 Quantum Mechanics and the Atom Exercise What color of light is emitted when an excited electron in the hydrogen atom falls from: a) n = 5 to n = 2 b) n = 4 to n = 2 c) n = 3 to n = 2 98 Section 7.5 Quantum Mechanics and the Atom The Shapes of Atomic Orbitals • The shapes of orbitals are important because covalent bonding involves the sharing of electrons that occupy these orbitals. • In one model of bonding a bond consists of the overlap of atomic orbitals on adjacent atoms. • The shapes of atomic orbitals are determined primarily by the angular momentum quantum number (l). 99 Section 7.5 Quantum Mechanics and the Atom The Shapes of Atomic Orbitals • • • • Orbitals with l = 0 are called s orbitals Orbitals with l = 1 are called p orbitals Orbitals with l = 2 are called d orbitals Orbitals with l = 3 are called f orbitals 100 Section 7.5 Quantum Mechanics and the Atom s Orbitals (l = 0) • The lowest energy orbital is the spherically symmetrical 1s orbital. s orbital in 1st principal energy level. • There are a couple of different ways to visualize an orbital. • But always remember – an orbital is a region of space where there is a probability of finding an electron. • It is not a real physical thing. 101 Section 7.5 Quantum Mechanics and the Atom Physical Meaning of a Wave Function • The wave function (ψ itself has no physical meaning • The square of the wave function (ψ2 indicates the probability of finding an electron near a particular point in space (this is what we call an orbital). – Most commonly represented as a probability density – intensity of color is used to indicate the probability value near a given point in space. 102 Section 7.5 Quantum Mechanics and the Atom Visualizing Orbitals • Picture an orbital as a three-dimensional probability density map. 103 Section 7.5 Quantum Mechanics and the Atom s Orbitals (l = 0) • Three dimensional plot of the wave function squared y2. This represents a probability density. Probability (per unit volume) of finding the electron at a point in space Fig 7-23a 104 Section 7.5 Quantum Mechanics and the Atom s Orbitals (l = 0) • Imagine every one of these dots representing the position of the electron at a given point in time. • Notice how there are more dots closer to the origin (nucleus) Fig 7-23a 105 Section 7.5 Quantum Mechanics and the Atom s Orbitals (l = 0) Fig 7-24 • This representation is the surface of a sphere that encompasses the volume where the electrons is found 90% of the time. 106 Section 7.5 Quantum Mechanics and the Atom Radial Probability Distribution for 1s Orbital Notice now the probability of finding an electron actually increases a little bit out from the nucleus because the volume of the cross section is larger. Fig B 107 Section 7.5 Quantum Mechanics and the Atom 2s Orbital • Notice in the s orbital in the 2nd principal energy level (n= 2) there is a place on both the radial probability distribution where probability is 0. • Electron cannot exist here. • Called a node • The probability of finding the electron here is zero. 108 Section 7.5 Quantum Mechanics and the Atom The 3s Orbital • The s orbital in the 3rd principal energy level (n = 3) has 2 nodes. 109 Section 7.5 Quantum Mechanics and the Atom Two Representations of the Hydrogen 1s, 2s, and 3s Orbitals a = electron probability distribution b = surface that contains 90% of the total electron probability (size of orbital) As n increases the number of nodes increases 110 Section 7.5 Quantum Mechanics and the Atom p orbitals (l = 1) • Each principle energy level with n = 2 or greater has three p orbitals. • l = 1 so ml = –1, 0 and 1 • The three p orbitals are not spherical but have a dumbbell shape with a node in the center. • The three orbitals are arranged on the x, y and z axis. 111 Section 7.5 Quantum Mechanics and the Atom p orbitals (l = 1) • The 2p orbitals and their radial distribution function. 3p, 4p and higher p orbitals are larger and contain more nodes. Fig 7-27 112 Section 7.5 Quantum Mechanics and the Atom d orbitals (l = 2) • Each principle energy level with n = 3 or greater has five d orbitals. • l = 2 so ml = –2, –1, 0, 1 and 2 • These orbitals have more complicated cloverleaf shapes 113 Section 7.5 Quantum Mechanics and the Atom d orbitals (l = 2) • The 3d orbitals. The higher d orbitals are larger and contain more nodes. Fig 7-28 114 Section 7.5 Quantum Mechanics and the Atom f orbitals (l = 3) • Each principle energy level with n = 3 or greater has seven f orbitals. • l = 3 so ml = –3,–2, –1, 0, 1, 2 and 3 • These orbitals are very complex with many more lobes and nodes than d orbitals. 115 Section 7.5 Quantum Mechanics and the Atom f orbitals (l = 3) • The 4f orbitals Fig 7-29 116 Section 7.5 Quantum Mechanics and the Atom The Phases of Orbitals • The orbitals we have been looking at are actually 3 dimensional waves. • If we look at a 1 dimensional wave. • We see the wave on the left has positive amplitude over the entire length where the wave on the right has positive amplitude over the first half and negative amplitude over the second half. 117 Section 7.5 Quantum Mechanics and the Atom The Phases of Orbitals • The sign of the amplitude of a wave is known as its phase • Blue is positive phase and red is negative. 118 Section 7.5 Quantum Mechanics and the Atom The Phases of Orbitals • We can also represent phase of a three dimensional quantummechanical orbital. • Blue is positive phase and red is negative. • Phase is important in bonding. 119 Section 7.5 Quantum Mechanics and the Atom The Shapes of Atoms Fig 7-30 • Atoms are represented as spheres because when we superimpose all the different orbitals we get a roughly spherical shape. 120