Sevy Lecture 10-24 PDF

What we observe is not nature itself, but nature exposed to our method of questioning. - Werner Heisenberg 1 CHEM 111...

What we observe is not nature itself, but nature exposed to our method of questioning. - Werner Heisenberg 1 CHEM 111 PRINCIPLES OF CHEMISTRY Prof. Eric T. Sevy Fall 2024 “Are you saying here’s something I don’t understand so the Gospel must not be true? Or are you willing to say, here’s something I don't understand, but I wonder what the Lord or his prophets will teach me about this? Are your questions asked with the assumption that there are answers? Are you willing to trust the Lord and give him the benefit of the doubt?” Sherry Dew Worth the Wrestle 2 IMPLICATIONS 1. 𝜓 is just a regular function (nothing mysterious). 2. 𝜓 2 can be zero (and at weird places). 3. Energy is quantized (related to an integer) 4. Boundary conditions and the need for a physically meaningful wavefunction lead to quantization 5. Can’t have zero energy. Zero point energy is the lowest. This consistent with uncertainty principle. 6. Probability density n=1 most probably in center, n=2 no probability in center 7. Nodes. More node – higher energy state. Increase by one with increasing state. 8. Energy decreases (and energy differences decrease) with increased box length. 3 STM 4 QUANTUM CORRAL 5 SURFACE DEFECTS 6 RECTANGULAR CORRAL 7 iCLICKER QUIZ Assuming all else is held constant, increasing the length of the box in the particle in a box model, will A. lower the energy of the states B. have no effect on the energy C. increase the energy of the states D. increase the number of nodes E. decrease the number of nodes 𝑛2 ℎ2 𝐸= 8𝑚𝐿2 8 iCLICKER QUIZ Which of the following are Not true about the particle in a box? A. We can know the position of the particle in a box and its momentum at the same time. B. The length of the box is quantized. C. The energy states that result from the particle in a box are not quantized. D. The lowest energy possible for a particle in box model is zero. E. None of these statements are true. 9 CORRESPONDENCE PRINCIPLE What happens to Energy as the box gets larger? What happens to Energy as the mass increases? As we approach macroscopic mass and length scales, quantum mechanics must reproduce the classical result. 10 iCLICKER QUIZ The square of the wavefunction indicates A. The color of the corresponding particle. B. Where the particle is located in space. C. The probability density for finding the particle. D. The charge on the particle. E. Nothing whatsoever. 11 iCLICKER QUIZ As we make the box smaller (decreasing L), A. Allowed energies get closer together. B. Allowed energies get farther apart. C. The particle becomes more “particle-like”. D. Both B and C. E. None of these. 𝑛2 ℎ2 𝐸= 8𝑚𝐿2 12 NANOPARTICLES, QUANTUM DOTS, & STAINED GLASS WINDOWS Ag & Au grains in cathedral stained glass. CdS particles that differ only in size. 13 2-D BOX x and y are independent & separable Wavefunctions are products of the 1-D solutions: 2 n x px 2 n ypy y (x, y) = sin ´ sin Lx Lx Ly Ly Energies are sums of the 1-D solutions: 𝑛𝑥2 ℎ2 𝑛𝑦2 ℎ2 𝐸 = 𝐸𝑥 + 𝐸𝑦 = 2 + 8𝑚𝐿𝑥 8𝑚𝐿2𝑦 Graphing? 14 GRAPHING THE 2-D BOX https://www.physicsforums.com/insights/visualizing-2-d-particle-box/ 15 A 3-D BOX? Starting to look like s and p orbitals! 16 LESSONS FROM THE PARTICLE IN A BOX 1. ψ is just a regular function (nothing mysterious). 2. ψ2 can be zero (and at weird places). 3. Energy is quantized (related to an integer) 4. Quantization is natural outcome 5. Can’t have zero energy. Zero point energy is the lowest. This consistent with uncertainty principle. 6. Nodes. More node – higher energy state. Increase by one with increasing state. 7. Energy decreases (and energy differences decrease) with increased box length. 8. Even flawed models can give insights. 17 MEANING OF “DEGENERATE” Physics definition: degenerate = have the same energy. Comes from symmetry. Must degenerate wavefunctions have the same shape? Orientation? 18 iCLICKER QUIZ What Makes the H Atom Different From a Particle In a 3-D Box? A. The kinetic energy term in the Schrödinger equation is different. B. The potential energy term is different. C. The Schrödinger equation for H atom can’t be solved. 19 THE SCHRÖDINGER EQUATION FOR H  The Hamiltonian is 3- dimensional.  Spherical polar coordinates take advantage of symmetry of atom and yield easier equations to solve.  What is the potential energy? 20 iCLICKER QUIZ What accounts for potential energy between electrons and the nucleus in an atom? A. Gravitational potential B. Electrostatic (Coulomb) potential C. Weak nuclear potential D. Strong nuclear potential E. None of these 21 THE SCHRÖDINGER EQUATION FOR H  The Hamiltonian is 3- dimensional.  Spherical polar coordinates take advantage of symmetry of atom and yield easier equations to solve.  What is the potential energy? Coulombic 22 iCLICKER QUIZ The Schrödinger equation for H gives allowed electron energies that A. are the same as those predicted by the Bohr model. B. are much more accurate than those predicted by the Bohr model. C. depend only on the electron’s angular momentum. D. begin at zero and go up to infinity. E. are proportional to 1/n. 23 ALLOWED ENERGIES FROM SCHRÖDINGER MODEL OF H J Exactly the same as predicted by Bohr (no surprise here). Note that n doesn’t have anything to do with angular momentum. Energies are quantized, as expected. Luckily, there are patterns in the solutions. 24 H WAVEFUNCTIONS Central electrostatic potential in H atom gives different solutions than PIB. 3-dimensional problem, so we need 3 quantum numbers. n, l, m Each combination of Q.N. leads to a unique wavefunction; called a “state”or “orbital”. 25 RADIAL AND ANGULAR PARTS All the H wavefunction solutions can be factored in the form Radial part Angular part Angular dependence determines shape of orbital. 26 WAVEFUNCTIONS FOR H ATOM 27 ORBITALS QM says electrons are not in circular orbits. In fact, QM says we can’t say with certainty where electrons are. Best we can do is give probabilities. Orbitals are stable standing wave patterns (from solution of Schrödinger wave equation). Patterns in these solutions.  Ψ (psi) Square of the orbital gives the probability density distribution for electron in atom. 28 iCLICKER QUIZ Orbitals are best described as A. Descriptions of the trajectories electrons take within atoms. B. Surfaces within which each type of electron must exist. C. One electron wavefunctions, the square of which gives the probability density for finding an electron at any given point around the atom. D. The imaginations of a frenzied mind E. More than one of these 29 ORBITALS n l ml n, l, ml define the energy level (size), shape, & orientation of an orbital. ↑ n = ↑ energy level = ↑ average distance from the nucleus All orbitals exist simultaneously in an H atom – they just aren’t all filled simultaneously (like rooms in an apt building) 30 QUANTUM NUMBERS: AN ELECTRON’S ADDRESS A silly (but surprisingly apt) comparison: n = energy level Particular floor you’re on l = shape of the room corner, hallway, jut-out… ml = orientation of room which corner, which jut-out ms = gender of occupant(s) male or female 31 QUANTUM NUMBERS Orbitals have 3 quantum #’s: Name Symbol Possible Values It tells you… Principle n Integers ≥ 1 Energy level (SHELL) Quantum # Average distance from the nucleus Orbital l Integers ranging from Shape of the orbital in that shell angular 0 to n-1 (SUBSHELL) momentum 0=s Quantum # 1=p 2=d 3=f Magnetic ml Integers ranging from Spatial orientation of the orbital Quantum # –l to +l (specific ORBITAL) ( – l,...-1, 0, 1,...l ) Spin ms +1/2 or -1/2 If the electron in the orbital is magnetic (for an electron) spin up or spin down Quantum # n, l, ml define the energy, shape, & orientation of an orbital. n and l = Shell and Subshell Adding a 4th quantum #, ms, defines an electron in that orbital 32 PRINCIPLE QUANTUM NUMBER, n Describes the energy level (shell) Values of n = 1, 2, 3, 4,... (integer ≥ 1) Corresponds to the orbit number “n” from Bohr, but the meaning is different.