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Welcome to the “quantum” world Chapter 8 The behavior of “large”-scale matter in our everyday life is predictable: the motions of the planets around the Sun, the falling of an apple to the ground, or the collisions of balls on a pool table all can be described effectively using classical physics (or mechanics). Electrons in Atoms Dr. Peter Warburton [email protected] http://www.chem.mun.ca/zcourses/1050.php All media copyright of their respective owners Welcome to the “quantum” world Welcome to the “quantum” world Partly this is because these small objects are constantly awash in a sea of electromagnetic radiation which can affect how the objects move and interact with each other. Also, the simple act of measuring things like the position and speed of these objects becomes more complicated, since we must scale the atomic scale events to provide responses we can detect in the “large” world. However, as objects get smaller and smaller, it turns out that the behavior of some of the smallest pieces of the universe, like atoms and the electrons, protons and neutrons they are made of becomes less predictable – classical physics no longer works well in describing what’s going on. All media copyright of their respective owners 2 3 All media copyright of their respective owners 4 1 Welcome to the “quantum” world Electromagnetic radiation Ultimately, it is unreasonable to expect the small pieces of the universe to behave like things do in our “large” experience of the universe. We need a special way to deal with the “small” parts of the universe – quantum mechanics! All media copyright of their respective owners This is a form of energy transmission via the wave propagation of electric AND magnetic fields. Unlike waves in our every day experience (sound, water, seismic) these waves do not require a medium for propagation – electromagnetic radiation can travel through a vacuum that contains no matter! 5 Waves All media copyright of their respective owners 6 Waves Wavelength λ – The distance between adjacent peaks (crests) in the wave SI unit: the meter (m) though we sometimes see the nonSI angstrom (Å) where 1 Å = 10-10 m For electromagnetic radiation in chemistry we often deal with wavelengths ranging in length from kilometers (103 m - radio waves) to picometers (10-12 m – X-rays) Waves transmit energy via a cyclic motion, either of the medium itself or of the electric and magnetic fields. Frequency ν – The number of cycles or events that happen in a given time period. Has units of inverse time like s-1. SI unit: 1 Hz (hertz) = 1 s-1 All media copyright of their respective owners 7 All media copyright of their respective owners 8 2 Waves Electromagnetic radiation Amplitude A – The maximum height a point is displaced from it’s “rest” position as the wave propagates. In a medium like the water of the ocean, a larger amplitude means a greater amount of water is displaced – implying greater energy. In fact E Ñ A2. However, we’ll see for electromagnetic radiation the energy is determined by the wavelength and NOT the amplitude – one way where the quantum world is different! All media copyright of their respective owners 9 Electromagnetic radiation All media copyright of their respective owners 10 Electromagnetic spectrum Electromagnetic radiation is light! Speed of light (c) in a vacuum: c = 2.99792458 x 108 m s-1 Higher frequency means shorter wavelength Relationship of the electromagnetic waves to the speed of light: speed of light = frequency x wavelength c=νxλ All media copyright of their respective owners 11 All media copyright of their respective owners 12 3 Problem Problem answer ν = 4.34 x 1014 s-1 = 4.34 x 1014 Hz The light from red LEDs (light emitting diodes) is commonly seen in many electronic devices. A typical LED produces 690 nm light. What is the frequency of this light? All media copyright of their respective owners 13 Wave interference 14 Constructive interference If equal wavelength waves are in phase, crests line up perfectly with crests, and the added wave has twice the amplitude as the individual waves. This is called constructive interference. Regardless of the type of wave, when two waves interact with each other, their behavior is additive. However, the result of the additive behavior can be quite variable, depending on the phase of the interacting waves. All media copyright of their respective owners All media copyright of their respective owners 15 All media copyright of their respective owners 16 4 Destructive interference Wave interference If equal wavelength waves are out of phase, crests line up perfectly with troughs, and the added wave has zero amplitude compared to the individual waves. This is called destructive interference. All media copyright of their respective owners 17 The visible spectrum 18 The visible spectrum We’ve seen Speed of light (c) in a vacuum: c = 2.99792458 x 108 m s-1 However, when light travels through a physical medium (air, glass, water, etc.) the speed of light is slightly less than in a vacuum, and each wavelength of light will be refracted (bent from a straight path) by a different amount that depends on wavelength. All media copyright of their respective owners All media copyright of their respective owners 19 What we think of as white light actually contains light of all wavelengths. Refraction of the white light through a prism or a raindrop will cause the white light to separate into the individual wavelengths, resulting in the continuous visible spectrum of colors! All media copyright of their respective owners 20 5 Discontinuous spectra Discontinuous spectra What happens when we refract light of a specific color? That specific color, like white light, may be made up of more than one wavelength, but there are many wavelengths that are not components of the spectrum we get. Since it is the atoms of a substance that are somehow responsible for the wavelengths of light we see, we often call discontinuous spectra by the name atomic (or line) spectra All media copyright of their respective owners 21 Atomic spectrum of helium Atomic spectrum of hydrogen Here the purple/pink light of a helium discharge tube is refracted through a spectroscope and results in the atomic spectrum of helium we see below. We see the purple color is actually a combination of specific wavelengths of red, yellow, green, blue and violet! All media copyright of their respective owners 22 All media copyright of their respective owners When Johann Balmer looked at the atomic spectrum for hydrogen, he was able to detect a pattern! Of the four lines he was able to see in the visible range he was able to deduce that the frequencies of the four lines could be predicted by the equation = 3.288110 − where n > 2 n=6 23 n=5 n=4 All media copyright of their respective owners n=3 24 6 Blackbody radiation Blackbody radiation Hot objects give off light – we call this blackbody radiation. We see a hot object has a specific colour based on the distribution of wavelengths of light it emits. This distribution changes with temperature! All media copyright of their respective owners 25 Blackbody radiation 26 Energy is quantized Classical theory predicts the intensity of radiation increases indefinitely with wavelength – the ultraviolet catastrophe. A system may only possess very specific amounts of energy, not a continuum! To go between two of the allowed states of the system required the absorbtion or emission of a quantum of energy. Max Planck – “Energy, like matter, is discontinuous.” All media copyright of their respective owners All media copyright of their respective owners 27 All media copyright of their respective owners 28 7 Energy of matter is quantized Other evidence of quantization ε = nhν Photoelectric effect – Light shining on the surface of some metals causes the ejection of electrons. However, electrons are only ejected when the frequency ν of the light is above some threshold ν0, regardless of the intensity of the light. This is not classical behaviour! The number of electrons ejected does depend on the light intensity. where n is a positive integer h = 6.62607 x 10-34 J s ν is frequency of absorbed or emitted radiation All media copyright of their respective owners 29 Photoelectric effect (Einstein) All media copyright of their respective owners 30 Energy of a photon is quantized E = hν Light can be treated as particles called photons. The energy of a photon depends on the light’s frequency when treated as a wave (see slide 30), while a greater intensity of light implies a greater number of photons. All media copyright of their respective owners The slope of this graph is Planck’s constant! 31 where h = 6.62607 x 10-34 J s Planck’s constant ν is frequency All media copyright of their respective owners 32 8 Explaining the photoelectric effect Explaining the photoelectric effect When a photon hits an bound electron in an atom, the electron can absorb the photon energy. If the photon energy is greater than the work function (the amount of energy required to “just” unbind the electron from the atom), then the remaining energy goes into determining the kinetic energy of the now unbound electron. To measure the kinetic energy of the electron, we set up a stopping voltage Vs between two metal plates. The voltage is adjusted until the electron stops moving, which happens when Ek = ½ mev2 = eVs 33 All media copyright of their respective owners Explaining the photoelectric effect Ek = ½ me v2 = 34 Work function and threshold frequency We saw there was a threshold ν0 in the photoelectric effect which defined the minimum frequency (and therefore minimum energy of a photon since E = hν) required to eject the electron. Any energy above the threshold is what is measured by the stopping voltage. eVs Here me is the electron rest mass of 9.109 x 10-34 kg and e is the electron charge 1.602 x 10-19 C All media copyright of their respective owners All media copyright of their respective owners 35 All media copyright of their respective owners 36 9 Work function and threshold frequency Work function and threshold frequency The work function can be expressed By energy conservation, the energy of the photon must be used to overcome the work function, and provide the kinetic energy of the electron, so Ew = hν0 where ν0 is the minimum frequency of light required to free an electron, which depends on the metal. All media copyright of their respective owners Ew + Ek = hν 37 38 Problem Work function and threshold frequency The minimum energy (the work function) required to cause the photoelectric effect in potassium metal is 3.69 x 10-19 J. Will photoelectrons be produced when blue light of wavelength 400 nm is shone on the metal? If they are ejected, what is the velocity of the electrons? Therefore Ek = hν - Ew All media copyright of their respective owners All media copyright of their respective owners 39 All media copyright of their respective owners 40 10 Problem answer Photochemistry Electrons will be ejected by 400 nm light, with a velocity of 1.68 x 107 m s-1. This is about 5.6% of the speed of light. Photons of light can provide enough energy to break chemical bonds by changing the distribution of electrons in the molecule. We can treat hν as a reactant! O2 + hν ↓ 2 O All media copyright of their respective owners 41 Problem All media copyright of their respective owners 42 Problem answer Chlorophyll absorbs light energies of 3.056 x 10-19 J photon-1 and 4.414 x 10-19 J photon-1. To what color, frequency and wavelength do these absorptions correspond, and can you use these results to explain why chlorophyll appears green? All media copyright of their respective owners 43 E = 3.056 x 10-19 J photon-1 ν = 4.612 x 1014 s-1 and λ = 650.4 nm Absorption of red light and E = 4.414 x 10-19 J photon-1 ν = 6.662 x 1014 s-1 and λ = 450.3 nm Absorption of blue light When you absorb red and blue, the color of visible light that is reflected (and seen!) is around a wavelength of 550 nm, which is green! All media copyright of their respective owners 44 11 The Bohr atom The Bohr atom The problem with this image is that an orbiting electron is always accelerating – and accelerating charges give off light (and therefore lose energy!)– the electron should lose energy and “death spiral” into the nucleus! Often our vision of the electrons in an atom is one of the electrons “orbiting” the nucleus like the planets orbit around the Sun. This would be classical mechanics behaviour. All media copyright of their respective owners 45 The Bohr atom 46 The Bohr atom Second, an electron is only allowed to have of fixed set of orbits called stationary states, which depend on the angular momentum of the electron, which depends π, where n is the on nh/2π principal quantum number, and can only be a non-zero integer, so n = 1, 2, 3, - Niels Bohr proposed a different model with three main properties: First, the orbit of an electron is circular like in classical physics. All media copyright of their respective owners All media copyright of their respective owners 47 All media copyright of their respective owners 48 12 The Bohr hydrogen atom The Bohr atom Radius of an allowed orbit is rn = n2a0 a0 is the Bohr radius where a0 = 53 pm = 0.53 Å Third, an electron can only move from one stationary state to another. This transition requires the absorption or emission of a photon with an energy matching the difference of the energy of the electron in the two stationary states. If we treat the infinitely separated nucleus and electron as zero in energy, the energy of each orbit is = − where RH (the Rydberg constant) is 2.179 x 10-18 J 49 All media copyright of their respective owners Energy level diagram All media copyright of their respective owners Electronic transitions in a hydrogen atom The photon absorbed or emitted must have an energy that exactly matches the energy difference between the two electronic states See slide 24! Second excited state n=3 First excited state n=2 ∆E=hν=R H Ground state n = 1 All media copyright of their respective owners 50 51 1 1 − ni2 nf 2 All media copyright of their respective owners 52 13 Problem Problem answer The energy of an electron in a hydrogen atom is -4.45 x 10-20 J. What energy level does the electron occupy? All media copyright of their respective owners n=7 53 Problem 54 All media copyright of their respective owners Problem answer λ = 486.5 nm Determine the wavelength of light absorbed in an electron transition from n = 2 to n = 4 in a hydrogen atom. n=6 n=5 n=4 n=3 The Balmer series represents all the transitions to or from the n = 2 state! All media copyright of their respective owners 55 All media copyright of their respective owners 56 14 Absorption spectra Helium absorption spectrum The line spectra we saw earlier are emission spectra, where light is given off. Now we’ve seen that light can also be absorbed, which means we can have absorption spectra where specific wavelengths are absorbed (but usually only from the ground state). All media copyright of their respective owners 57 Ionization energy of hydrogen 58 If we consider ions that have only one electron like He+, Li2+, Be3+ P then we can use the Bohr models to predict the energy levels and the ionization energy of the last electron − = 1 1 − 2 = R H 1−0 =R H 2 ni nf All media copyright of their respective owners All media copyright of their respective owners Ionization energy of hydrogen-like ions To completely remove the electron from a hydrogen atom is an example of ionization. The Bohr model allows us to predict the ionization energy of an electron in a hydrogen atom from the ground state ni = 1 to an unbound electron where nf = ∞ ∆E=hν=R H Helium was actually discovered from the absorption spectrum of the Sun in 1868. Norman Lockyer predicted the existence of helium as an element 27 years before it was isolated on Earth! where Z is the atomic number for the ion 59 All media copyright of their respective owners 60 15 Problem Ionization energy of hydrogen-like ions − = Determine the wavelength (and color if it falls in the visible spectrum) of light emitted in an electron transition from n = 5 to n = 3 in a Be3+ ion. As the atomic number goes up, the ionization energy increases because we are trying to remove a negatively charged electron from a nucleus with a total Z+ charge. The electron is more tightly electrostatically bound to a nucleus with larger positive charge! All media copyright of their respective owners 61 Problem answer 62 Problem λ = 80.13 nm. This is actually ultraviolet light. All media copyright of their respective owners All media copyright of their respective owners The frequency of the n = 3 to n = 2 transition for an unknown hydrogen-like ion occurs at a frequency 16 times that for hydrogen. What is the identity of the ion? 63 All media copyright of their respective owners 64 16 Problem answer Inadequacies of the Bohr model Since there is a Z2 dependence for hydrogen-like ions, Z must be 4 for a 16 times increase in frequency (and energy of transition). This means the ion is Be3+. We only talk about the Bohr model for hydrogen and hydrogen-like ions because the model cannot handle more than one electron! Two or more electrons in an atom, molecule or ion will interact with each other, changing the energy levels in comparison to a single electron interacting with a nucleus. A new theory was required! All media copyright of their respective owners 65 Building a new theory 66 Wave-particle duality Wave-particle duality - we saw in the photoelectric effect that Einstein treated light (which we think of waves) as particles called photons. Louis de Broglie proposed that matter and energy actually have “dual” nature where “Small particles of matter may at times display wave-like properties.” All media copyright of their respective owners All media copyright of their respective owners Starting from Einstein’s equation telling us that matter and energy are interconvertible E = mc2 de Broglie related this to the energy of a photon E = hν 67 All media copyright of their respective owners 68 17 Wave-particle duality Wave-particle duality This means Since λν = c or ν/c = 1/λ hν = mc2 which we can rearrange to then hν/c = mc = p p = h/λ = mu where p is the momentum (mass times velocity) of the photon where u is the speed of any particle All media copyright of their respective owners 69 de Broglie wavelength All media copyright of their respective owners de Broglie wavelength h/p = h/(mu) = λ h/p = h/(mu) = λ Any wave can be described as a particle in motion, and any particle in motion can be described with a wavelength. However, we see from the equation, that since h is very small, then the wavelength for a particle must also be very small – UNLESS it’s mass is very small as well! Consider the problem we did on slides 40 and 41 where an electron was ejected from a potassium atom with a speed of 1.68 x 107 m s-1. The moving electron can be treated as a wave with All media copyright of their respective owners 70 71 λ = h/(meu) = 43.3 pm All media copyright of their respective owners 72 18 de Broglie wavelength X-rays can be in the pm range! We also saw on slide 50 that the Bohr radius of an electron in the ground state of a hydrogen atom is 53 pm! If moving electrons have wavelengths in the picometer range, we should be able to use them to probe the structure of matter. X-ray diffraction Electron beam diffraction How? We’ll see soon! All media copyright of their respective owners 73 Boats and waves 74 Scanning electron microscopy (SEM) To observe an object requires the waves interacting with the object to be similar in size, or smaller than the object. If the waves are too big, the small object gets “overwhelmed” by the wave. All media copyright of their respective owners All media copyright of their respective owners 75 Using the wave-like properties of a beam of electrons, we can get very detailed images of small objects that we cannot get using lenses. Here we have a white blood cell (pink) infected by HIV viruses (blue) at 13400x magnification! All media copyright of their respective owners 76 19 More SEM Building a new theory Heisenberg uncertainty principle - the idea that very small particles have wave-like qualities means we can never be absolutely certain about it’s behavior, since a particle is seen as discrete and localized and a wave is variable and “spread out” (delocalized) All media copyright of their respective owners 77 Heisenberg uncertainty principle 78 Analogy with camera shutter speed Is the car moving? Short time frame of the picture means we can’t be absolutely sure. We know x but aren’t sure about p. Here, the car is definitely moving. But the long time frame of the picture means we don’t know exactly where it was. We know p but not x. If we try and describe a particle by two variables (usually it’s position x and momentum p), it turns out that if we are dealing with a small enough particle, then we cannot accurately measure both things at the same time. The uncertainty of the measurements ∆x and ∆p are related! ∆x∆p ≤ h/4π All media copyright of their respective owners All media copyright of their respective owners 79 80 20 Wave packet Types of waves To “localize” a small particle, we must describe it as a group of waves - a wave packet. The more waves we have, the better we know x, but the harder it is to know p, and vice versa (less waves to get p, but don’t know x) Traveling waves – like ocean waves or sound waves, the crests and troughs move through space Standing waves – like a plucked guitar string, the crests and troughs DO NOT move through space 81 All media copyright of their respective owners Types of waves n=1 λ = 2L/n n=2 All media copyright of their respective owners 82 Standing waves in a Bohr orbit Standing waves have a wavelength that is an integer fraction of twice the total length Standing waves also have nodes where the displacement is zero. The number of such nodes is n + 1. All media copyright of their respective owners Imagine a guitar string of length 2L wrapped around to form a circle of circumference 2L. Standing waves in the string must still obey λ = 2L/n! n=3 83 Allowed standing wave All media copyright of their respective owners 84 21 Standing waves in a Bohr orbit If the standing wave doesn’t obey the relationship, it will destructively interfere with itself, cancelling out to zero. The standing wave is not allowed! de Broglie vs Bohr The electrons in an atom set up standing waves to achieve the allowed stationary states for each n. Still not good enough! A guitar string is 1D, and we need a 3D wave to describe the electron in all of the space around the nucleus. Disallowed standing wave All media copyright of their respective owners 85 3D description of an electron 86 Particle in a 1D box Before we can understand how to describe an electron in 3D, we need a better understanding of how to describe a standing wave in 1D. To do this we solve a model called particle in a box. All media copyright of their respective owners All media copyright of their respective owners 87 The standing wave for an electron in 1D can be treated exactly like the guitar string. There must be nodes at each end of the string, while the wave must be described by some sort of cyclic function, like a sine function! We give the cyclic function description a special name – the wavefunction! All media copyright of their respective owners 88 22 Particle in a 1D box Particle in a 1D box For a 1D box (along the x direction) of size L the wavefunctions ψn that describe stationary states of a particle in a 1D box can be described as The energy of each stationary state of a wave can be found since (using de Broglie) Ek = ½ mv2 = p2/2m = h2/2mλ2 For a particle in a 1D box h2 n2 h2 Ek = = 2 8mL2 2mλ 2 nπx ψn = sin L L where n = 1,2,3,P All media copyright of their respective owners 89 Particle in a 1D box All media copyright of their respective owners Particle in a 1D box h2 n2 h2 Ek = = 2 8mL2 2mλ n2 h2 Ek = = 2 8mL2 2mλ The energy of the particle can never be zero (the zero-point energy means never at rest!) Smaller L (better knowledge of x) implies larger Ek and therefore a less certain p Most importantly, the energy of the particle in the box is quantized, based on the allowed values for n! All media copyright of their respective owners 90 h2 91 All media copyright of their respective owners 92 23 What is the wavefunction? Wavefunction analogy The wavefunction is a mathematical description of how a particle will behave under certain conditions (like a box of a certain size). Ultimately, it must contain ALL the information of particle behavior under ALL conditions. However, the wavefunction is not something physical we can measure. All media copyright of their respective owners Consider all of the information available to us through the internet. We certainly can’t know all of it in our own heads, but under the appropriate circumstances (a Google search), we can find the information we want. We’ll see this analogy extend further in a few slides. 93 Particle in a 1D box All media copyright of their respective owners 94 Particle in a 1D box Since the particle behaves like a wave, we can not exactly say where the particle is, but we can evaluate the probability of finding the particle at a given point in the box. All media copyright of their respective owners 95 We saw that the energy of a classical wave depends on its intensity, which corresponds to the square of the amplitude. All media copyright of their respective owners 96 24 Particle in a 1D box Particle in 3D box The probability of finding a particle is the quantum equivalent to the intensity, and depends on the “square” of the wavefunction ψ2 All media copyright of their respective owners In 3D space, we can treat the wavefunction as a combination of three 1D wavefunctions, each with its own principal quantum number n. When we do this, we can get the energies of a particle in a 3D box &' ( ℎ + . / = + + 8* ,+ ,. ,/ 97 Problem All media copyright of their respective owners 98 Problem answer What is the wavelength of the photon emitted when an electron in a 1D box with a length of 5.0 x 101 pm falls from the n = 5 level to the n = 3 level? All media copyright of their respective owners 99 λ = 0.52 nm All media copyright of their respective owners 100 25 Schrödinger equation Schrödinger equation We’ve seen that the wavefunction must contain information about the particle in all circumstances, and therefore we should be able to use it to find many different properties of the particle. To get the specific property we’re interested in, we apply an operator to the wavefunction which should return the wavefunction multiplied by an observable such as energy. All media copyright of their respective owners 101 Wavefunction of the H atom 0 ψ=Eψ H The operator H-hat is much like a Google search of the internet. It “pulls out” the property information we are interested in from the large collection of information that is the wavefunction. All media copyright of their respective owners 102 Wavefunction of the H atom When we saw the energy solutions of a particle in a 3D box, we did them in terms of Cartesian coordinates (x, y, z). It turns out the math is more conveniently done in what is called spherical polar coordinates (r, θ, φ). φ All media copyright of their respective owners Schrödinger equation – In spherical polar coordinates, we define the electrons position in an atom by its distance r from the nucleus, as well as two angles θ and φ to some arbitrary line. 103 All media copyright of their respective owners 104 26 Wavefunction of the H atom Wavefunction of the H atom Solutions of the Schrödinger equation for the hydrogen atom define orbitals that describe the motion of the electron in the H atom (not Bohr orbits). These orbitals will have features that depend on both the radial wavefunction and the angular wavefunction. This allows us to break the wavefunction down into two parts: radial wavefunction R(r) angular wavefunction Y(θ, φ) This results in ψ(r, θ, φ) = R(r) Y(θ, φ) All media copyright of their respective owners 105 Quantum numbers 106 Quantum numbers We’ve already seen the principal quantum number n (based on the angular momentum of the electron). The name is interesting, though. It implies there must be other quantum numbers for the electron to help describe other aspects of the electron in the atom. It turns out solutions for the Schrödinger equation for the hydrogen atom require two more quantum numbers! However, they are connectedP All media copyright of their respective owners All media copyright of their respective owners 107 Principal (angular momentum) quantum number n where n = 1, 2, 3, P Orbital angular momentum quantum number l where l = 0, 1, 2, 3, P, n-1 Magnetic quantum number ml where ml = -l, (-l+1), P, -2, -1, 0, 1, 2, l-1, l All media copyright of their respective owners 108 27 Problems Problem answers a) Yes. For n = 3 the values of l can be either 0, 1 or 2. For ANY of these values of l, ml can be 0, since ml must fall in the range of –l to +l in all cases. b) If n = 3 then the values of l can be 0, 1 or 2. But since ml must fall between –l and +l, a ml value of 1 eliminates l = 0 as a possibility, but leaves l = 1 or l = 2 as allowed values. a) Can an orbital have the quantum numbers n = 3, l = 0 and ml = 0? b) For an orbital with n = 3 and ml = 1, what is (are) the possible value(s) of l? All media copyright of their respective owners 109 110 Shells and subshells Physical interpretation of the quantum numbers The quantum numbers tell us something about the orbitals the electrons move within n determines the energy and average (most probable) distance from the nucleus l determines the angular shape of the orbital ml determines the orientation (direction) of the orbital All media copyright of their respective owners All media copyright of their respective owners Since we see the quantum numbers are tied together, it makes sense to logically group them together in some way. Orbitals that have the same value of n are said to belong to the same shell. Orbitals that have the same n and l are said to belong to the same subshell. 111 All media copyright of their respective owners 112 28 Subshells Subshells We saw that l can take integer values ranging from 0 to n-1, and therefore for a given shell (given value of n) there can be a variable number of subshells. When n = 1 l can only be 0 (1 subshell) When n = 2 l can be 0 or 1 (2 subshells) When n = 3 l can be 0, 1 or 2 (3 subshell) When n = 4 l can be 0, 1, 2 or 3 (4 subshell) All media copyright of their respective owners 113 Total orbitals in a subshell All media copyright of their respective owners 114 Total orbitals in a subshell Since ml values can range from –l to +l, for each subshell, there will exist 2l+1 orbitals of a given name in a subshell. These should be similar in shape (same l) but different in their direction (different ml) All media copyright of their respective owners We very quickly see the number of allowed subshells in a shell equals the principal quantum number for the shell. Based on this, we give orbitals special names to describe the shape of the subshell When l = 0 (s orbital) For higher values When l = 1 (p orbital) of l the names When l = 2 (d orbital) continue as g, h, i, j, k, When l = 3 (f orbital) 115 There can only be 1 s orbital when l = 0 since ml can only be 0 There can be 3 p orbitals when l = 1 since ml = -1, 0 or 1 There can be 5 d orbitals when l = 2 since ml = -2, -1, 0, 1 or 2 There can be 7 f orbitals when l = 3 since ml = -3, -2, -1, 0, 1, 2 or 3. All media copyright of their respective owners 116 29 Orbital energies Orbital energies The energies of an electron in the orbitals in each subshell for a hydrogen atom are given by the energy of the shell (E depends on n only)! En = -RH (1/n2) All media copyright of their respective owners If there is no “favorite” direction determined by an external electric or magnetic field, this means each orbital of a subshell has the same energy. We call orbitals at the same energy degenerate. 117 What do orbitals tell us? 118 Radial vs angular wavefunctions We’ve already seen it’s more convenient to break the wavefunction into radial and angular parts. The radial part tells us something about the most probable distance that we will find an electron from the nucleus when it is in a given orbital. The orbitals that result from the solutions of the Schrödinger equation for the hydrogen atom represent the stationary states for the electron described by the appropriate set of quantum numbers for the orbital we are interested in. The “square” of the orbital will tell us the probability of finding an electron within some part of the orbital. All media copyright of their respective owners All media copyright of their respective owners 119 All media copyright of their respective owners 120 30 Radial vs angular wavefunctions s orbitals for hydrogen-like atoms If we look at the table, we see that the angular wavefunction for any s orbital, regardless of the value of n, is a constant! This tells us that no direction is “special” for an s orbital. No special direction implies a spherical shape for an s orbital in 3D. The angular wavefunction tells us something about the directions in which an electron is most likely to be found, when it is in a given orbital. All media copyright of their respective owners 121 1s orbital is when n = 1 and l = 0 122 s orbitals when n > 1 Here we see three different ways of representing the probability of finding an electron in the 1s orbital. No direction is “special”, so we see “spherical distributions”, but the radial wavefunction tells us we are more likely to see the electron close to the nucleus, and less likely to see it far away. All media copyright of their respective owners All media copyright of their respective owners 123 Here we see the 1s, 2s and 3s orbitals represented by contours within which we are likely to find the electron 95% of the time. Since the angular wavefunction is always constant for s orbitals, s orbitals are always spherical. The main differences is s orbitals when n = 1 or n = 2 or n = 3 comes from the radial part of the wavefunction. All media copyright of their respective owners 124 31 s orbitals when n > 1 s orbitals when n > 1 When we look at the radial wavefunctions for the 1s, 2s and 3s orbitals, we see two interesting features involving σ, which depends on the value of n and the distance r: For 2s there is (2-σ) For 3 s there is (6-6σ+σ2) If we think carefully, we realize there can be certain distances r where the interesting feature can BECOME ZERO in the radial wavefunction, which implies a zero probability of finding the electron at that distance! 125 All media copyright of their respective owners Radial nodes All media copyright of their respective owners 126 Angular wavefunction of p orbitals Radial nodes! This means, for s orbitals, there will exist n – 1 radial nodes at specific distances where the electron CAN NEVER BE FOUND! It turns out that regardless of orbital shape, there will always be n - l – 1 radial nodes for an orbital. All media copyright of their respective owners For p orbitals, we see that the angular wavefunction is not a constant. Depending on the direction considered (x, y or z) we will see a dependence on one or possibly two angles! 127 All media copyright of their respective owners 128 32 Angular (or planar) nodes Angular wavefunction of p orbitals More correctly, we see that we have dependencies on the sine or cosine of an angle(s). Because the sine or cosine of a given angle can be ZERO in certain cases, this implies there will be directions in a given p orbital where the electron CANNOT be found (probability of 0) All media copyright of their respective owners 129 A 2p orbital Here we see the angular part of p orbital lying along the z axis. The “square” of the orbital, gives a probability of finding an electron within this p orbital. In either case we see we will NEVER find the electron lying in the xy plane. This is an angular (or planar) node for the orbital. All media copyright of their respective owners 130 Angular (or planar) nodes We’ve seen that there are three allowed values for ml in what we call a p orbital for which l = 1. That is ml = -1, 0 or 1. This implies there are three different p orbitals P one in each direction, each with its own planar node to consider. However, notice that regardless of the p orbital we consider, there is zero chance of finding the electron at the nucleus! All media copyright of their respective owners 131 All media copyright of their respective owners 132 33 Angular (or planar) nodes Total number of nodes for an orbital It turns out that the number of angular nodes a given orbital has is given by l. For s orbitals l = 0 so no planar nodes For p orbitals l = 1 so one planar node Fro d orbitals l = 2 so two planar nodes and so onP All media copyright of their respective owners We’ve seen the number of radial nodes for an orbital is n - l – 1 and the total number of angular nodes is l. Therefore the total number of nodes of either type for a given orbital is n-l–1+l=n–1 133 d orbitals All media copyright of their respective owners 134 d orbitals We cannot see d orbitals until n is equal or greater than 3, and l = 2. If we consider the 3d orbitals, there should be 5 possible orbitals, each containing zero radial nodes and two planar nodes. All media copyright of their respective owners 135 Once again we see the planar nodes mean that we can never find a d orbital electron at the nucleus, because there are two planar nodes that pass through it! All media copyright of their respective owners 136 34 The 4th quantum number – ms d orbitals The two planar nodes are easy to identify in the first 4 orbitals seen below. What about the last one? It turns out that in that orbital, the two planar nodes are “wrapped around” to give a “double cone” Notice that table 8.1 gives the angular and radial wavefunctions for hydrogen-like atoms (with only 1 electron). It turns out that the hydrogen line spectrum can’t be completely explained unless we add another concept to the mix: electron spin All media copyright of their respective owners 137 Stern-Gerlach experiment 138 Electron spin If we pass a beam of silver atoms through a nonuniform magnetic field, the beam splits into two beams. This implies that the electrons in the atom interact with the magnetic field (via their own magnetic field caused by the electron spin), which can either work with the external field, or against it (two options). All media copyright of their respective owners All media copyright of their respective owners 139 1. The electron spin generates a magnetic field. 2. A pair of electrons with opposing spin has no net magnetic field. All media copyright of their respective owners 140 35 Electron spin and ms Electron spin and ms 3. The silver beam splits because there are an odd number of electrons – we can pair 23 sets of electrons, but we have one unpaired electron left over. 4. The unpaired electron can either be spin up (ms = +½ or ↑) or spin down (ms = -½ or ↓) spin up (ms = +½ or ↑) or spin down (ms = -½ or ↓) Notice that the electron spin is a property of the electron and does not depend on which orbital the electron is found in. It DOES NOT DEPEND on any of the other three quantum numbers. All media copyright of their respective owners 141 Shorthand notation 1s1 is when n = 1, l = 0 and one unpaired electron or 2p2 n = 2, l = 1 and two paired electrons or 3d1 n = 3, l = 2 and one unpaired electron :;<=>?@=A=BC>?DE?><ECFA such as 1s1 or 2p2 or 3d1 All media copyright of their respective owners 142 Shorthand notation To describe any electron in an atom, we can provide the first three quantum numbers (we can’t say what spin we have without an experiment, we can only tell if the spins are paired or not). However, its easier to use a shorthand notation. ℎ122 3456782ℎ891 All media copyright of their respective owners 143 All media copyright of their respective owners 144 36 Problem Problem answers Identify the error in each set of quantum numbers given in the form (n, l, ml, ms) below: a) (2, 1, 1, 0) b) (1, 1, 0, ½) c) (3, -1, 1, -½) d) (0, 0, 0, -½) e) (2, 1, 2, ½) All media copyright of their respective owners a) b) c) d) e) 145 Multielectron atoms All media copyright of their respective owners 146 Multielectron atoms The orbitals of multielectron atoms are somewhat different than those for the hydrogen atom. Where the radial and angular wavefunctions are mathematically similar, in multielectron atoms, the larger nuclear charge means that an orbital of a given type in a given shell will be lower in energy than the same orbital in a hydrogen atom. All media copyright of their respective owners (2, 1, 1, 0) – ms must be +½ or -½ (1, 1, 0, ½) – l must be n-1 or less (3, -1, 1, -½) – l cannot be less than 0 (0, 0, 0, -½) – n cannot be less than 0 (2, 1, 2, ½) – ml must be between –l and +l 147 Also, the orbitals of different types within the same shell are no longer degenerate – they differ in energy from each other! The reason for this is simple. Hydrogen only has one electron, and therefore each orbital of a given shell only “sees” the nucleus. In multielectron atoms, the presence of other electrons creates effects that will make the orbital energies nondegenerate. All media copyright of their respective owners 148 37 Effects of many electrons Electron penetration Electron shielding of the nucleus – the electrons found in lower shells are closer to the nucleus. This means the electrons in higher shells don’t “see” the full nuclear charge, but rather they “see” an effective nuclear charge, since the inner electrons shield part of the nuclear charge. A lower effective nuclear charge for a given orbital will increase the orbital energy (less electrostatic attraction) and increase the average distance an electron in that orbital will be found from the nucleus. All media copyright of their respective owners 149 Radial probability distributions All media copyright of their respective owners 150 Radial probability distributions To get the radial probability distribution, we multiply the square of the orbital wavefunction by 4πr2 All media copyright of their respective owners s orbital electrons are better at shielding the nucleus than electrons in any other orbital type because s electrons are most often found at the nucleus, while the planar nodes of other orbitals guarantee the electrons will never be found at the nucleus. To understand this difference of electron penetration, it is best to look at radial probability distributions. 151 What we then see are the distances from the nucleus where the electron is most probably going to be found. Larger “humps” closer to the nucleus indicate a greater electron penetration. All media copyright of their respective owners 152 38 Non-degenerate orbitals Multielectron atom energy diagrams Now we can explain why the orbitals of a given shell in a multielectron atom are nondegenerate. Consider the n = 2 shell. Any electrons in the 2s orbital will penetrate close to the nucleus and shield the 2p electrons, increasing its orbital energy compared to the 2s orbital. However, the p orbital electrons do not penetrate, and do not shield the 2s electrons. All media copyright of their respective owners 153 Electron configurations All media copyright of their respective owners 154 Electron configurations Imagine you can build an atom. You can start with a nucleus with a given atomic number Z and add in electrons one at a time until you have added the number of electrons you want. This is the aufbau (building up) process. Z electrons will give you a neutral atom. More than Z electrons will give you a negative ion, and less than Z electrons will give you a positive ion. All media copyright of their respective owners Generally, more electrons means a more complicated pattern of shielding, meaning the orbital energies within each shell tend to “spread out” more with increasing numbers of electrons. 155 As you place each electron into the atom or ion, it seems reasonable that it will go into the lowest energy orbital available, placing it as close to the nucleus as the other electrons present will allow. However, every time you add in a new electron, you are making the shielding on the higher energy orbitals more complicated. The orbitals of a given shell become non-degenerate. All media copyright of their respective owners 156 39 Electron configurations The “rules” of electron configurations In reality we cannot build atoms this way. However, we can use this idea as a good way to understand which orbitals we will find the electrons of atoms or ions in. All we need to do now is figure out the rules to describe the complicated pattern of shielding and the effect it has on orbital energies, so we know which available orbital is the lowest energy orbital. We call the pattern of electron filling the electron configuration. All media copyright of their respective owners 1. Electrons fill the orbital of lowest energy that is not already filled. This guarantees the lowest possible energy for the atom. 2. The Pauli exclusion principle – no two electrons in an atom can have exactly the same set of quantum numbers. This implies an orbital can hold AT MOST two electrons. 3. Hund’s rule – When degenerate orbitals of the same type are available in a shell, a single electron must be placed in each before we start to pair the electrons. 157 Order of orbital filling 158 Three different notations The complicated effect of shielding of orbitals that affects the energy of the higher orbitals AND THEREFORE THE ORDER THE ORBITALS ARE FILLED is seen here. In general, the orbitals are usually filled in the order spdf notation (condensed) 1s22s22p2 Only shows the total electrons of a subshell, but not how they are spread over degenerate orbitals. spdf notation (expanded) 1s22s22px12py1 Not only shows the total electrons of a subshell, but also shows how they are spread over degenerate orbitals (Hund’s rule). orbital diagram 1s, 2s, 2p, 3s, 2p, 4s, 3d, 4p, 5s, P All media copyright of their respective owners All media copyright of their respective owners Much like expanded notation, where we see the filling of degenerate orbitals, we now also show electron spin. 159 All media copyright of their respective owners 160 40 Orbital diagram Three different notations orbital diagram spdf notation (condensed) spdf notation (expanded) 1s22s22p2 1s22s22px12py1 orbital diagram We see here that the spins in the degenerate orbital filling are parallel. It turns out that parallel spins give the lowest energy atom. Any non-parallel spins in degenerate orbitals will give a higher energy! All media copyright of their respective owners 161 Electron configuration shortcuts All media copyright of their respective owners 162 Electron configuration shortcuts Here we see why we said [Ar] is 1s22s22p63s23p6 that the orbital filling usually follows the order found on slide 159. There are some exceptions, such as Cr and Cu, which show the importance of parallel spins being lower in energy. This tends to happen in d orbitals totalling 5 or 10 electrons. All media copyright of their respective owners Each representation above represents the lowest possible energy of a carbon atom (with 6 electrons) by orbital filling following the “rules”. The lowest energy representations are the ground-state electron configuration. Orbital diagrams that “break the rules” of filling represent excited-state electron configurations of an atom that is higher in energy than it is in its ground state. 163 Since the order of orbital [Ar] is 1s22s22p63s23p6 filling is the same for all atoms, we can express the orbital diagrams for atoms with a great number of electrons by only looking at the orbital filling that occurs for electrons after a certain point – the noble gas atom orbital diagram All media copyright of their respective owners 164 41 Problems Problem answers a) Configurations (a) and (c) are equivalent b) This is an excited state. The ground state configuration would look like The spins are parallel in the highest energy degenerate orbitals All media copyright of their respective owners 165 The periodic table 166 Electron configurations by group Niels Bohr emphasized that elements in the same group (column) of the periodic table have similar electron configurations. This implies a similarity of the distribution of the outermost valence electrons, which leads to a similarity in physical properties and chemistry of those elements! We’ll see more laterP All media copyright of their respective owners All media copyright of their respective owners 167 Because the order of filling rules are known, we can emphasize on the periodic table which block of orbitals was the last to be filled by the electrons for the element. All media copyright of their respective owners 168 42 Problems Problem answers a) Identify the element having the electron configuration 1s22s22p63s23p63d24s2 b) Represent the electron configuration of iron with an orbital diagram c) For an atom of Sn, indicate the number of: electronic shells that are filled or partially filled, the number of 3p electrons, the number of 5d electrons, and the number of unpaired electrons. All media copyright of their respective owners 169 a) Ti (see slide 163 for confirmation) b) Done in class c) Sn has 5 filled or partially filled shells, has six 3p electrons, has zero 5d electrons and has two unpaired electrons. All media copyright of their respective owners 170 43