Electronic Structure of the Atom Chapter 4, Part 4 PDF

THE ELECTRONIC STRUCTURE OF THE ATOM CHAPTER 4, PART 4 Wavefunctions Quantum Numbers Electron Orbitals DETERMINANT VS. INDETERMINANT: CLASSIC VS. QUANTUM  One of the...

THE ELECTRONIC STRUCTURE OF THE ATOM CHAPTER 4, PART 4 Wavefunctions Quantum Numbers Electron Orbitals DETERMINANT VS. INDETERMINANT: CLASSIC VS. QUANTUM  One of the fundamental principles of classical physics is that by knowing an object’s initial acceleration, velocity, and position, and any additional forces felt by the object over time, you can determine the exact path that object will take.  Under identical conditions, copies of that object will always follow the exact same path.  PROBLEM: we have showed that knowing an electron’s position and velocity isn’t possible, thus we cannot know its trajectory with absolute certainty. MAKIN’ WAVE(FUNCTION)S  Like velocity and position, energy and position are also affected by the uncertainty principle.  To get more precise information about the energy of the electron, we must give up some knowledge of its position.  A wavefunction, denoted by Ψ, is a mathematical description of the wavelike behavior of an electron.  Ψ2 represents a probability density map (where is the electron most likely to be?)  Ψ2 ≥ 0  More commonly referred to as an orbital THE PARTICLE IN A BOX  A simple example of a wavefunction is the following, which describes a particle in a box. 2 𝑛𝜋𝑥 𝜓𝑛 𝑥 = sin , n = 1, 2, 3, … 𝐿 𝐿 Where L is the length of the box and x is position  A free-moving particle that is bound on both sides by infinitely high potential walls (standing wave).  No potential within the well itself  Nodes at the box walls (0 and L) and at all fractions of L/n ENTER SCHRÖDINGER  Schrödinger’s equation allows us to calculate the energy of a system if we know the wavefunction that describes it. ෡ = 𝐸Ψ 𝐻Ψ  Ĥ is the Hamiltonian operator, a set of mathematical operations that represent the combined kinetic and potential energies of a system.  Performing those operations on Ψ will return Ψ multiplied by a real number, E.  For our PIAB system, the Hamiltonian looks like this ℏ2 𝜕 2 − 2𝑚 𝜕𝑥 2 THE SOLUTION  The solution to Schrödinger’s equation for the PIAB is 25ℎ2 ℎ2 𝑛2 8𝑚𝐿2 𝐸𝑛 = 2ℎ2 8𝑚𝐿2 𝑚𝐿2  What can we learn from this? 9ℎ2  Particles have a minimum energy (zero-point energy) 8𝑚𝐿2  Energy is inversely proportional to L ℎ2 2𝑚𝐿2 ℎ2  n is the first of three quantum numbers used to 8𝑚𝐿2 quantize the energy of an electron FROM ONE DIMENSION TO THREE  The PIAB is a great starter, but it only describes an electron moving in one dimension… atoms are three dimensional  The move from one dimension to three greatly complicates the wavefunction  Unlike PIAB, we also now need to account for coulombic potential  Expressing the wavefunction in terms of spherical coordinates makes things a bit easier  For us, we only need to know that the solution for the Hydrogen atom looks like Ψ 𝑟, 𝜃, 𝜙 = 𝑅(𝑟)Υ(𝜃, 𝜙)  𝑅(𝑟) is known as the radial wavefunction  Υ(𝜃, 𝜙) is known as the angular wavefunction  The size, shape, and orientation of the electron orbitals can be correlated to three integer numbers contained within the wavefunctions: n, ℓ, and 𝑚ℓ THE PRINCIPAL QUANTUM NUMBER  The first quantum number is designated as n  Comparable to the n we used in talking about the Bohr model’s energy levels  Determines the size of the orbital and energy of the electron  All energies are negative. E = 0 is defined as “vacuum” where the electron will simply leave the atom.  The larger the value of n…  The larger the orbital  The smaller the energy gap between orbitals 𝑚𝑒 𝑞 4 −18 𝐽 1 𝐸𝑛 = − = −2.18 × 10 32𝜋 2 𝜖02 ℏ2 𝑛2 𝑛2 THE ANGULAR MOMENTUM QUANTUM NUMBER  The 2nd quantum number is designated as ℓ  Describes the shape of the orbital  Must be a whole number (≥ 0)  For the nth energy level, the maximum value for ℓ is n-1  If n = 3, ℓ may have the values of 0, 1, and 2  The various orbital shapes (and ℓ values) correspond to a letter  ℓ = 0, 1, 2, 3 → s, p, d, f (respectively) THE MAGNETIC QUANTUM NUMBER  The 3rd quantum number is designated as 𝑚ℓ  Describes the spatial orientation of the orbital  Must be an integer (includes zero)  For any orbital, the value of 𝑚ℓ can go from −ℓ to +ℓ  For a p-orbital (ℓ = 1), the values of 𝑚ℓ are -1, 0, and 1  Orbitals that have the same value of n and ℓ are said to be in the same subshell  Each subshell contains 2ℓ + 1 degenerate orbitals  Degenerate orbitals are energetically equivalent  Especially important when talking about molecular bonding because those bonds are formed by overlapping atomic orbitals PUT IT ALL TOGETHER… GETTING IN SHAPE  The orbital shapes shown are the 1s, 2p, and 3d orbitals.  Orbital surfaces represent the volume in which there is a 90% probability of finding the electron. A CLOSER LOOK AT THE S-ORBITAL  The probability density, Ψ 2 , corresponds directly with the orbital surface.  By multiplying Ψ 2 by the volume of a shell at distance r from the nucleus, we get a radial distribution function.  For hydrogen, the max at 52.9 pm corresponds with the distance predicted by Bohr! INCREASING THE VALUE OF N  As mentioned previously, increasing the value of n increases the “size” of the orbital  Also introduces a feature called a radial node  Zero probability of finding the electron at that distance A CLOSER LOOK AT THE P-ORBITALS  The p cloud is composed of three orbitals, 𝑝𝑥 , 𝑝𝑦 , and 𝑝𝑧  Each one goes along its corresponding Cartesian axis  All two-lobed (sometimes referred to as dumbbells)  Node at the nucleus, referred to as an angular node  The off-axis plane (ex. the 𝑝𝑥 orbital’s node is the yz plane) A CLOSER LOOK AT THE D-ORBITALS  The d cloud is composed of five orbitals: 𝑑𝑥𝑦 , 𝑑𝑥𝑧 , 𝑑𝑦𝑧 , 𝑑𝑥 2 −𝑦 2 , and 𝑑𝑧 2  𝑑𝑥𝑦 , 𝑑𝑥𝑧 , and 𝑑𝑦𝑧 have lobes going out at a 45° offset on the specified plane  𝑑𝑥 2−𝑦 2 has lobes going along the x and y axes  𝑑𝑧 2 is a two-lobed shape along the z-axis with a toroid in the xy plane  Two angular nodes A CLOSER LOOK AT THE F-ORBITALS  The f cloud is composed of seven orbitals that are mostly eight-lobed with some two-lobed+toroid shapes  You do not need to know the subscripts!  Three angular nodes NODES – A SUMMARY  All orbitals other than the s orbital have angular nodes  # angular nodes = ℓ  s = 0, p = 1, d = 2, f = 3  Radial nodes arise when the wavefunction changes phase (+ to – or vice versa)  # of radial nodes = n - ℓ - 1  Total nodes = n - 1 FOR NEXT TIME… Read: 4.05 – 4.07.1B An “exam reflection” document is available on Brightspace. I recommend filling out the first page as much as possible, regardless of how you feel you did on the exam.