6.1 The Wave Nature of Light

6.1 The Wave Nature of Light • To understand the electronic structure of atoms, one must understand the nature of electromagnetic radiation • Electromagnetic radiation • A form of energy exhibiting wave-like behavior as it travels through space • Visible light, IR, and X-rays share certain fundamental characteristics • All types of electromagnetic radiation move through a vacuum at a speed of 3.00 × 108 m/s, the speed of light

6.1 The Wave Nature of Light • Characteristics of Electromagnetic waves • Wavelength (), amplitude, and frequency () • The distance between corresponding points on adjacent waves is the wavelength • The number of waves passing a given point per unit of time is the frequency • Relationship between the wavelength and the frequency c =  For waves traveling at the same velocity, the longer the wavelength, the smaller the frequency

6.1 The Wave Nature of Light • The Electromagnetic Spectrum • Different properties of electromagnetic radiation comes from their different wavelength

6.1 The Wave Nature of Light • Common Wavelength Units

6.1 The Wave Nature of Light SAMPLE EXERCISE 6.1 • Which wave has the higher frequency? • Visible light and IR • Blue light and red light

6.1 The Wave Nature of Light

6.2 Quantized Energy and Photons • Although the wave model of light explains many aspects of its behavior, this model cannot explain several phenomena: • Blackbody radiation • Photoelectric effect • Emission spectra

6.2 Quantized Energy and Photons • Hot Objects and the Quantization of Energy • As a body is heated, it begins to emit radiation and becomes first red, then orange, then yellow, then white as its temperature increases • The classical theory of radiation • predicts: • Intensity kT/4 • Max Planck explained it by assuming that energy comes in packets called quanta • E= h • h, Planck’s constant • 6.626  10−34 J-s

6.2 Quantized Energy and Photons • The Photoelectric Effect and Photons Light with a specific frequency or greater causes a metal to emit electrons, but light of lower frequency has no effect Einstein used Planck’s assumption to explain the photoelectric effect. He concluded that energy is proportional to frequency: E = h h, Planck’s constant 6.626  10−34 J-s.

6.2 Quantized Energy and Photons • The Photoelectric Effect and Photons

6.3 Line Spectra and the Bohr Model • Line Spectra • Radiation composed of a single wavelength is said to be monochromatic • Most common radiation sources contain many different wavelength

6.3 Line Spectra and the Bohr Model • Line Spectra • For atoms and molecules one does not observe a continuous spectrum, as one gets from a white light source. • Only a line spectrum of discrete wavelengths is observed. High voltage under reduced pressure of different gases produces different colors of light Rydberg equation RH, Rydberg constant Ne H2

6.3 Line Spectra and the Bohr Model • Bohr’s Model • The classical “microscopic solar system” model of the atom cannot explain the line spectrum • Bohr’s postulates • Electrons in an atom can only occupy certain orbits (corresponding to certain energies). • An electrons in a permitted orbit have a specific energy and is in an “allowed” state. An electron in an allowed energy state will not radiate energy and therefore will not spiral into the nucleus • Energy is emitted or absorbed by the electron only as the electron changes from one allowed energy state to another. This energy is emitted or absorbed as a photon, E= h

6.3 Line Spectra and the Bohr Model • The Energy States of the Hydrogen Atom • Bohr calculated the energies corresponding to each allowed orbit for the electron in the hydrogen atom • RHis the Rydberg constant • n is the principal quantum number • Ground state & excited state

6.3 Line Spectra and the Bohr Model • The Energy States of the Hydrogen Atom The equation is corresponding to the experimental equation

6.3 Line Spectra and the Bohr Model • The Energy States of the Hydrogen Atom

6.3 Line Spectra and the Bohr Model • Limitations of the Bohr Model • Significance of the Bohr model • Electrons exist only in certain discrete energy levels, which are described by quantum numbers • Energy is involved in moving an electron from one level to another • The model cannot explain the spectra of other atoms and why electrons do not fall into the positively charged nucleus

6.4 The Wave Behavior of Matter • Louis de Broglie suggested: “If radiant energy could, under appropriate conditions behave as though it were a stream of particles, could matter, under appropriate conditions, possibly show the properties of a wave?” • An electron moving about the nucleus of an atom behaves like a wave and therefore has a wavelength • de Broglie’s hypothesis is applicable to all matters

6.4 The Wave Behavior of Matter

6.4 The Wave Behavior of Matter • The Uncertainty Principle Heisenberg showed that the more precisely the momentum of a particle is known, the less precisely is its position known: In many cases, our uncertainty of the whereabouts of an electron is greater than the size of the atom itself! If we have 1% of uncertainty in the speed of electron in hydrogen atom (diameter, 1 × 10-10 m)

6.4 The Wave Behavior of Matter Whenever any measurement is made, some uncertainty exists This limit is not a restriction on how well instruments can be made; rather, it is inherent in nature No practical consequences on ordinary-sized objects Enormous implications when dealing with subatomic particles, such as electrons Short wavelength of light for accuracy in position High energy light will change the electron’s motion There is an uncertainty in simultaneously knowing either the position or the momentum of the electron that cannot be reduced beyond a certain minimum level

6.5 Quantum Mechanics and Atomic Orbitals Erwin Schrödinger developed a mathematical treatment into which both the wave and particle nature of matter could be incorporated. He treated the electron in a hydrogen atom like the wave on a guitar string Overtones vs higher- energy standing waves Nodes vs zero amplitude Solving Schrödinger’s equation for the hydrogen atom leads to a series of mathematical functions – wave functions

6.5 Quantum Mechanics and Atomic Orbitals These wave functions are represented by the symbol  (psi) The wave function has no direct physical meaning The square of the wave equation, 2, represents the probability that the electron will be found at that location – probability density This is just a kind of statistical knowledge – we cannot specify the exact location of an electron Figure 6.16 Electron-density distribution

6.5 Quantum Mechanics and Atomic Orbitals • Orbitals and Quantum Numbers Solving the wave equation gives a set of wave functions, or orbitals, and their corresponding energies. Each orbital describes a spatial distribution of electron density. An orbital is described by a set of three quantum numbers. • Principal quantum number The principal quantum number, n, describes the energy level on which the orbital resides. The values of n are integers ≥ 1.

6.5 Quantum Mechanics and Atomic Orbitals • Angular momentum quantum number This quantum number defines the shape of the orbital. Allowed values of l are integers ranging from 0 to n − 1. We use letter designations to communicate the different values of l and, therefore, the shapes and types of orbitals.

6.5 Quantum Mechanics and Atomic Orbitals • Magnetic quantum number The magnetic quantum number describes the three-dimensional orientation of the orbital. Allowed values of ml are integers ranging from −lto l: −l≤ ml≤ l Therefore, on any given energy level, there can be up to 1 s orbital, 3 p orbitals, 5 d orbitals, 7 f orbitals, etc.

6.5 Quantum Mechanics and Atomic Orbitals • Orbitals and Quantum Numbers Orbitals with the same value of n form a shell. Different orbital types within a shell are subshells.

6.5 Quantum Mechanics and Atomic Orbitals • Orbitals and Quantum Numbers For a one-electron hydrogen atom, orbitals on the same energy level have the same energy. That is, they are degenerate. Figure 6.17

6.5 Quantum Mechanics and Atomic Orbitals • Orbitals and Quantum Numbers

6.6 Representations of Orbitals • The s Orbitals Spherically symmetric There is only one s orbital for each value of n (l = 0, ml = 0) Radial probability function Nodes

6.6 Representations of Orbitals • The s Orbitals Figure 6.21 probability density in the s orbitals of hydrogen.

6.6 Representations of Orbitals • The p Orbitals The value of l for p orbitals is 1 and the orbital has three magnetic quantum number ml. They have two lobes with a node between them.

6.6 Representations of Orbitals • The d and f orbitals • The value of l for a d orbital is 2 and the orbital has five magnetic quantum number ml. • Four of the five d orbitals have 4 lobes; the other resembles a p orbital with a doughnut around the center.

6.7 Many-electron Atoms • Orbitals and Their Energies • As the number of electrons increases, though, so does the repulsion between them. • In many-electron atoms, orbitals on the same energy level are no longer degenerate • For a given value of n, the energy of an orbital increases with increasing value of l.

In the 1920s, it was discovered that a beam of neutral atoms is separated into two groups when passing them through a nonhomogeneous magnetic field • Electron has two equivalent magnetic field

6.7 Many-electron Atoms • Electron Spin • Observation of closely spaced double lines in spectrum of many-electron atoms • Electrons have spin. • Spin magnetic quantum number, ms • The spin quantum number has only 2 allowed values: +1/2 and −1/2

6.7 Many-electron Atoms • Pauli Exclusion Principle • No two electrons in the same atom can have identical sets of quantum numbers. • An orbital can hold a maximum of two electrons and they must have opposite spins

6.8 Electron Configuration • Electron distribution in orbitals of an atom. • The ground state: the most stable electron configuration of an atom • The orbitals are filled in order of increasing energy with no more than two electrons per orbital • Consider the lithium atom • How many electrons? • Electron configuration? paired unpaired

6.8 Electron Configuration • Hund’s Rule • For degenerate orbitals, the lowest energy is attained when the number of electrons with the same spin is maximized - Hund’s rule

6.8 Electron Configuration • Hund’s Rule

6.8 Electron Configuration (b) How many unpaired electrons exist?

6.8 Electron Configuration • Drawbacks to using x-rays for medical imaging • MRI overcomes the drawbacks

6.8 Electron Configuration • Transition Metals • The condensed electron configuration includes the electron configuration of the nearest noble-gas element of lower atomic number • Note that the 19th electron of potassium is occupied in 4s (not 3d) • Transition metals: elements in the d-block of the periodic table • Lanthanides and actinides

6.9 Electron Configurations and the Periodic Table • The structure of the periodic table reflects the orbital structure

6.9 Electron Configurations and the Periodic Table • The periodic table is your best guide to the order in which orbitals are filled

6.9 Electron Configurations and the Periodic Table

6.1 The Wave Nature of Light