Electronic structure of atoms - PDF Free Download

Chapter 1 Electronic structure of atoms light photons spectra Heisenberg s uncertainty principle atomic orbitals electron configurations the periodic table

1.1 The wave nature of light Much of our understanding of the electronic structure of atoms comes from analyzing light emitted or absorbed by substances The electromagnetic spectrum Radiation carries energy through space The visible region accounts for only a small portion near the middle of the electromagnetic spectrum Electromagnetic radiation is characterized by its wave nature

Electromagnetic radiation can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. waves have three characteristics: peak Wavelength λ Frequency v amplitude trough Amplitude is the height of the wave maximum from the centre. Wavelength is the distance between successive wave peaks. wavelength, λ (lambda) has units of distance, m The number of waves passing a given point per unit of time is the frequency frequency, ν (nu) has units of Hertz, s 1 For waves traveling at the same velocity, the longer the wavelength, the smaller the frequency. ALL electromagnetic radiation travels at the same velocity: the speed of light: c = 299,792,458 m/s ~3 10 8 m s -1

The wavelength and frequency of light are therefore related in a straightforward way: c = λν violet light infrared radiation ν = 7.50 10 14 s 1 ν = 3.75 10 14 s 1 What we perceive as different kinds of electromagnetic energy are waves with different wavelengths and frequencies. Interference: constructive destructive

Wave nature of light successfully explains a range of different phenomena: Examples of interference: stones and ripples, and reflections from a DVD Wave model of light useful BUT has some limitations 1.2 Quantized Energy and Photons Some phenomena cannot be explained using a wave model of light: 1. Blackbody radiation emission of light from hot objects 2. The photoelectric effect 3. Emission spectra emission of electrons from metal surfaces on which light shines emission of light from electronically excited gas atoms

Hot Objects and the Quantization of Energy Heated solids emit radiation (blackbody radiation): wavelength distribution depends on temperature In 1900, Max Planck investigated black body radiation, and he proposed that energy can only be absorbed or released from atoms in certain amounts called quanta A quantum is the smallest amount of energy that can be emitted or absorbed as electromagnetic radiation The relationship between energy, E, and frequency is: classical theory = the ultraviolet catastrophe T = 7000 K T = 5000 K Light emission by molten iron E = hν where h is the Planck constant = 6.626 10-34 joule-seconds (J s)

The Photoelectric Effect and Photons The photoelectric effect (right) provides evidence for the particle nature of light and for quantization. Light shining on the surface of a metal can cause electrons to be ejected from the metal. light electrons The electrons will only be ejected if the photons have sufficient energy Below a threshold frequency no electrons are ejected. Above the threshold frequency, the excess energy appears as the kinetic energy of the ejected electrons. Einstein proposed that light could have particle-like properties, which he called photons. Energy of one photon = E = hν = h c/λ Light has wave-like AND particle-like properties

1.3 Line Spectra and the Bohr Model Line spectra e.g. laser light Radiation composed of only one wavelength is called monochromatic. a whole array of different wavelengths is called continuous radiation When radiation from a light source, such as a light bulb, is separated into its different wavelength components, a spectrum is produced. White light passed through a prism provides a continuous spectrum: Not true of light emitted from excited gaseous elements; instead, get a line spectrum White light spectrum Helium spectrum Discontinuous lines of different colours

Hydrogen lamp Prism separates component wavelengths detector Hydrogen spectrum slit Lines are characteristic of the element a fingerprint Rutherford s model of the atom Rutherford assumed that electrons orbited the nucleus analogous to planets orbiting the sun; however, a charged particle moving in a circular path should lose energy! the atom should be unstable according to Rutherford s theory!

Bohr s model of the atom Niels Bohr noted the line spectra of certain elements and assumed that electrons were confined to specific energy states. These he called orbits. Bohr s model is based on three postulates: 1. Only orbits of specific radii are permitted for electrons in an atom these correspond to certain definite energies 2. An electron in a permitted orbit has a specific energy an "allowed" energy state 3. Energy is only emitted or absorbed by an electron as it moves from one allowed energy state to another energy is gained or lost as a photon

The Energy States of the Hydrogen Atom Colours from excited gases arise because electrons move between energy states in the atom. since the energy states are quantized, so is the light emitted from excited atoms appear as line spectra Bohr showed mathematically that E = (hcr H ) 1 n 2 = ( 2.18 10-18 J) 1 n 2 where n is the principal quantum number (i.e., n = 1, 2, 3 ) and R H is the Rydberg constant. R H = 1.097 10 7 m The first orbit in the Bohr model has n = 1 and is closest to the nucleus. The furthest orbit in the Bohr model has n = and corresponds to E = 0.

Electrons in the Bohr model can only move between orbits by absorbing and emitting energy in quanta (E = hn). The ground state = the lowest energy state An electron in a higher energy state is said to be in an excited state The amount of energy absorbed or emitted by moving between states is given by E = E f E i = hν = hc λ Limitations of the Bohr Model The Bohr Model has several limitations: 1 = 2.18 10-18 J 2 n f 1 n i 2 It cannot explain the spectra of atoms other than hydrogen Electrons do not move about the nucleus in circular orbits However, the model introduces two important ideas: 1. the energy of an electron is quantized: electrons exist only in certain energy levels described by quantum numbers 2. energy gain or loss is involved in moving an electron from one energy level to another.

1.4 The wave behavior of matter Louis de Broglie suggested that if light can have material properties, matter should exhibit wave properties confirmed by the diffraction of a expected: bright spots beam of electrons particle beam actual: interference pattern electron microscopes use electrons instead of a beam of light de Broglie proposed that the characteristic wavelength of the electron or of any other particle depends on its mass, m, and on its velocity, v λ = h / mv electrons The momentum, mv, is a particle property, whereas λ is a wave property Matter waves is the term used to describe wave characteristics of material particles. Applicable to all matter but for objects of ordinary size, the calculated wavelength is unimaginably short and has no sensible interpretation.

The Uncertainty Principle Werner Heisenberg: we cannot determine the exact position, direction of motion, and speed of subatomic particles simultaneously Wave-like AND particle-like The dual nature of matter sets a limit on how precisely we can know the location and momentum of an object. The measuring process interferes with what is being measured Heisenberg imagined a gamma ray microscope to explain his uncertainty principle. - photons are used to illuminate an electron - the position of the electron can be determined from the scattering of the photons electrons photons - BUT the photon scatters in a random direction and transfers a large and unknown amount of momentum to the electron Heisenberg related the uncertainty of the position, Δx, and the uncertainty in momentum Δ(mv) to a quantity involving Planck s constant: Δx Δ(mv) h / 4π detector

1.5 Quantum Mechanics and Atomic Orbitals Erwin Schrödinger proposed an equation containing both wave and particle terms. The solution of the equation is known as a wave function, Ψ (psi). describes the behavior of a quantum mechanical object, like an electron Ψ 2 is the probability density Ψ 2 gives the electron density for the atom A region of high electron density = high probability of finding an electron Schrödinger treated electrons in the hydrogen atom as standing waves Higher energy standing waves have nodes

Orbitals and quantum numbers If we solve the Schrödinger equation we get wave functions and corresponding energies. These wave functions are called orbitals The probability density (or electron density) described by an orbital has a characteristic energy and shape. The energy and shape of orbitals are described by three quantum numbers. These arise from the mathematics of solving the Schrödinger equation. the principal quantum number, n must be a positive integer n = 1,2,3,4, the angular momentum quantum number, l maximum value is (n-1), i.e. l = 0,1,2,3 (n-1) use letters for l (s, p, d and f for l = 0, 1, 2, and 3). describes the main energy level; specifies electron shell describes the shape; specifies subshell the magnetic quantum number, m l maximum value depends on l, can take integral values from l to + l m l = (-l),,0,,(+ l) designates specific orbital; specifies orientation

Orbitals can be ranked in terms of energy. Orbital energy levels in the hydrogen atom as n increases energy level spacing becomes smaller each cluster of boxes = one subshell each row = one shell n = 3 shell has three subshells composed of nine orbitals n = 2 shell has two subshells composed of four orbitals n = 1 shell has one orbital each box = one orbital

1.6 Representations of Orbitals node = probability of finding an electron is 0 The s orbitals For all orbitals the number of nodes is given by n 1 All s orbitals are spherical As n increases, the s orbitals get larger As n increases, the number of nodes increases height of graph indicates electron probability at a single point at distance r height of graph indicates electron probability density per volume of a shell

The p orbitals p orbitals are dumbell-shaped 3 values of m l two lobes and a nodal plane at the nucleus 3 different orientations p z p x p y 3 representations of electron probability for 2p z orbital

The d orbitals d orbitals have two nodal planes (well, most do. Some nodes are not planes.) 5 values of m l so 5 different orbitals with different orientations xy plane 3 d orbitals have lobes that lie between the x-, y-, and z-axes 2 d orbitals have lobes that lie aligned along the x-, y-, or z-axes 4 of the d orbitals have 4 lobes each 1 d orbital has 2 lobes and a donut

1.7 Many-Electron Atoms Orbitals and Their Energies In a many-electron atom, for a given value of n, the energy of an orbital increases with increasing value of l Orbitals of the same energy are said to be degenerate For n 2, the s and p orbitals are no longer degenerate Therefore, the energy-level diagram looks slightly different for many-electron systems Electron Spin and the Pauli Exclusion Principle Line spectra of many-electron atoms show each line as a closely spaced pair of lines. Wolfgang Pauli postulated that electron must exist in two states

Stern and Gerlach designed an experiment to determine why. A beam of atoms was passed through a slit and into a non-homogeneous magnetic field and the atoms were Beam detected: Magnet collector Two spots were found, plate corresponding to silver atoms with electrons in different states. (Note that Slit silver atoms have one unpaired electron each.) Beam of atoms Beam splits in two we define a 4 th quantum number m s = spin magnetic quantum number = ±½ electron spin is quantized

How do we show these two states? m s = + ½ (spin up) m s = ½ (spin down) Pauli s exclusion principle states that: - no two electrons can have the same set of four quantum numbers - therefore, two electrons in the same orbital must have opposite spins i.e. m s =+1/2 and m s =-1/2 1.8 Electron Configurations Electron configurations tell us how the electrons are distributed among the various orbitals (energy levels) of an atom. The most stable configuration, or ground state, is that in which the electrons are in the lowest possible energy state When writing ground-state electronic configurations: - electrons fill orbitals in order of increasing energy - no two electrons can fill one orbital with the same spin (Pauli)

H 1s 1 He 1s 2 Li 1s 2 2s 1 Hund s Rule Quantum numbers are like postal codes For degenerate orbitals, the lowest energy is attained when the number of electrons with the same spin is maximized - electrons fill each orbital singly with their spins parallel before any orbital gets a second electron - electron-electron repulsions are minimized Condensed Electron Configurations Electron configurations may be written using a shorthand notation: 1. Write the core electrons corresponding to the noble gas in square brackets = electrons in the inner shells; generally not involved in bonding.

2. Write the valence electrons explicitly Transition Metals = electrons in the outer shell; those gained or lost in reactions After Ca the 3d orbitals begin to fill The block of the periodic table in which the d orbitals are filling represents the transition metals. The Lanthanides and Actinides Elements 57-70 are the lanthanides ( rare earths ) in which the 7 degenerate 4f orbitals fill: Actinides are 5f elements 89-102 usually put under the rest of the PT

1.9 Electron Configurations and the Periodic Table The periodic table can be used as a guide for electron configurations. the period number is the value of n s-block alkali and alkaline earth metals d-block transition metals p-block main group elements f-block lanthanides and actinides Blocks of elements in periodic table relate to which orbital is being filled - the 4s orbital fills before the 3d orbitals - the 6s orbital fills before the 4f orbitals (which fill before the 5d orbitals)

Anomalous Electron Configurations There are elements that appear to violate the electron configuration guidelines: e.g. Cr [Ar] 3d 5 4s 1 not [Ar] 3d 4 4s 2 Cu [Ar] 3d 10 4s 1 not [Ar] 3d 9 4s 2 - due to special stability of half-filled and filled shell configurations When atomic number > 40, energy differences are small and other anomalies often occur. These usually act to reduce electron repulsions. Summary of rules We can collect these rules for adding electrons to orbitals under the useful umbrella known as the Aufbau Principle. We apply them in the following order: 1. Lower energy orbitals fill with electrons first Aufbau - German for building up 2. Any orbital can hold up to 2 electrons. If there are two electrons in the same orbital, they have opposite spins (Pauli Exclusion Principle). 3. If two or more degenerate orbitals are available, one electron goes into each until all are half-filled, then they start pairing up (Hund s rule). 4. A particularly stable configuration is one in which a set of p or d orbitals is either filled or half-filled (e.g. Cr, Cu)