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Ionization energy of hydrogen 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 = ∞ Chapter 8 Electrons in Atoms Dr. Peter Warburton [email protected] http://www.chem.mun.ca/zcourses/1050.php ∆E=hν=R H 1 1 − = R H 1−0 =R H ni 2 nf 2 All media copyright of their respective owners Ionization energy of hydrogen-like ions Ionization energy of hydrogen-like ions − = If we consider ions that have only one electron like He+, Li2+, Be3+ 3 then we can use the Bohr models to predict the energy levels and the ionization energy of the last electron 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! − = where Z is the atomic number for the ion All media copyright of their respective owners 59 60 All media copyright of their respective owners 61 1 Problem Problem answer 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. All media copyright of their respective owners 62 Problem λ = 80.13 nm. This is actually ultraviolet light. All media copyright of their respective owners 63 Problem answer 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? All media copyright of their respective owners 64 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+. All media copyright of their respective owners 65 2 Inadequacies of the Bohr model Building a new theory 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 66 Wave-particle duality All media copyright of their respective owners 67 Wave-particle duality Starting from Einstein’s equation telling us that matter and energy are interconvertible This means hν = mc2 E = mc2 which we can rearrange to de Broglie related this to the energy of a photon hν/c = mc = p where p is the momentum (mass times velocity) of the photon E = hν All media copyright of their respective owners 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.” 68 All media copyright of their respective owners 69 3 Wave-particle duality de Broglie wavelength Since h/p = h/(mu) = λ λν = c or ν/c = 1/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! then p = h/λ = mu where u is the speed of any particle All media copyright of their respective owners 70 de Broglie wavelength 71 de Broglie wavelength 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. h/p = h/(mu) = λ 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 How? We’ll see soon! λ = h/(meu) = 43.3 pm All media copyright of their respective owners All media copyright of their respective owners 72 All media copyright of their respective owners 73 4 X-rays can be in the pm range! X-ray diffraction Boats and waves Electron beam diffraction 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 74 All media copyright of their respective owners 75 All media copyright of their respective owners 77 More SEM Scanning electron microscopy (SEM) 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 5 Building a new theory Heisenberg uncertainty principle 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) 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 78 All media copyright of their respective owners 79 Wave packet 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. 80 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) All media copyright of their respective owners 81 6 Types of waves Types of waves 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 n=1 λ = 2L/n n=2 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 82 Standing waves in a Bohr orbit 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! Standing waves have a wavelength that is an integer fraction of twice the total length 83 All media copyright of their respective owners 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! Allowed standing wave All media copyright of their respective owners n=3 84 All media copyright of their respective owners Disallowed standing wave 85 7 de Broglie vs Bohr 3D description of an electron 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. All media copyright of their respective owners 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. 86 Particle in a 1D box 87 Particle in a 1D box 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 All media copyright of their respective owners 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 2 nπx ψn = sin L L where n = 1,2,3,3 88 All media copyright of their respective owners 89 8 Particle in a 1D box Particle in a 1D box h2 n2 h2 Ek = = 2 8mL2 2mλ 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 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 h2 n2 h2 Ek = = 2 8mL2 2mλ All media copyright of their respective owners 90 Particle in a 1D box 91 What is the wavefunction? h2 n2 h2 Ek = = 2 8mL2 2mλ 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. 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 All media copyright of their respective owners 92 All media copyright of their respective owners 93 9 Wavefunction analogy 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. 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. All media copyright of their respective owners 94 Particle in a 1D box All media copyright of their respective owners 95 Particle in a 1D box The probability of finding a particle is the quantum equivalent to the intensity, and depends on the “square” of the wavefunction ψ2 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 All media copyright of their respective owners 97 10 Particle in 3D box Problem 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 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? ℎ " % & = + + 8! #" #% #& All media copyright of their respective owners 98 Problem answer 99 All media copyright of their respective owners 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. λ = 0.52 nm All media copyright of their respective owners 100 All media copyright of their respective owners 101 11 Schrödinger equation Wavefunction of the H atom Schrödinger equation – 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 ' 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 103 Wavefunction of the H atom This allows us to break the wavefunction down into two parts: radial wavefunction R(r) angular wavefunction Y(θ, φ) This results in 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. All media copyright of their respective owners All media copyright of their respective owners ψ(r, θ, φ) = R(r) Y(θ, φ) 104 All media copyright of their respective owners 105 12 Wavefunction of the H atom 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 connected3 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. All media copyright of their respective owners 106 Quantum numbers 107 Problems Principal (angular momentum) quantum number n where n = 1, 2, 3, 3 Orbital angular momentum quantum number l where l = 0, 1, 2, 3, 3, n-1 Magnetic quantum number ml where ml = -l, (-l+1), 3, -2, -1, 0, 1, 2, l-1, l All media copyright of their respective owners All media copyright of their respective owners 108 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 13 Problem answers Physical interpretation of the quantum numbers 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. All media copyright of their respective owners 110 Shells and subshells All media copyright of their respective owners 111 Subshells 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. All media copyright of their respective owners 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 112 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 14 Subshells Total orbitals in a subshell 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) All media copyright of their respective owners 114 Total orbitals in a subshell All media copyright of their respective owners 115 Orbital energies 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 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) 116 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 117 15 Orbital energies 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. All media copyright of their respective owners 118 16