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Chapter 7 Electronic Structure. Waves. Waves are periodic disturbances – they repeat at regular intervals of time and distance. Properties of Waves. Wavelength ( l ) is the distance between one peak and the next . Frequency ( n ) is the number of waves that pass a fixed point each second.
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Waves • Waves are periodic disturbances – they repeat at regular intervals of time and distance.
Properties of Waves • Wavelength(l) is the distance between one peak and the next. • Frequency(n) is the number of waves that pass a fixed point each second.
Electromagnetic Radiation • Light or electromagnetic radiation consists of oscillating electric and magnetic fields.
Speed of Light • All electromagnetic waves travel at the same speed in a vacuum, 3.00×108 m/s. • The speed of a wave is the product of its frequency and wavelength, so for light: • So, if either the wavelength or frequency is known, the other can be calculated.
Example: Electromagnetic Radiation • An FM radio station broadcasts at a frequency of 100.3 MHz (1 Hz = 1 s-1). Calculate the wavelength of this electromagnetic radiation.
Kinds of Electromagnetic Radiation • Visible light is only a very small portion of the electromagnetic spectrum. • Other names for regions are gamma rays, x rays, ultraviolet, infrared, microwaves, radar, and radio waves.
Quantization of Energy • In 1900, Max Planck proposed that there is a smallest unit of energy, called a quantum. The energy of a quantum is where h is Planck’s constant, 6.626×10-34 J·s.
The Photoelectric Effect • The photoelectric effect: the process in which electrons are ejected from a metal when it is exposed to light. • No electrons are ejected by light with a frequency lower than a threshold frequency, n0. • At frequencies higher than n0, kinetic energy of ejected electron is hn – hn0.
Photoelectric Effect (cont.) • Einstein suggested an explanation by assuming light is a stream of particles called photons. • The energy of each photon is given by Planck’s equation, E = hn. • The minimum energy needed to free an electron is hn0. • Law of conservation of energy means that the kinetic energy of ejected electron is hn – hn0.
Dual Nature of Light? • Is light a particle, or is it a wave? • Light has both particle and wave properties, depending on the property. • Particle behavior, wave behavior no longer considered to be exclusive from each other.
Spectra • A spectrum is a graph of light intensity as a function of wavelength or frequency. • The light emitted by heated objects is a continuous spectrum; light of all wavelengths is present. • Gaseous atoms produce a line spectrum – one that contains light only at specific wavelengths and not at others.
The Rydberg Equation • Study of the spectrum of hydrogen, the simplest element, show that the wavelengths of lines of light can be calculated using the Rydberg equation: • n1 and n2 are whole numbers and RH = 1.097×107 m-1.
Example: Rydberg Equation • Calculate the wavelength (in nm) of the line in the hydrogen atom spectrum for which n1 = 2 and n2 = 3.
The Bohr Model of Hydrogen • Bohr assumed: • that the electron followed a circular orbit about the nucleus; and • that the angular momentum of the electron was quantized. • Using these assumptions, he found that the energy of the electron was quantized:
Bohr Model and the Rydberg Equation • Assume that when one electron transfers from one orbit to another, energy must be added or removed by a single photon with energy hn. • This assumption leads directly to the Rydberg equation.
Matter as Waves • Louis de Broglie proposed that matter might be viewed as waves as well as particles. • de Broglie suggested that the wavelength of matter is given by where h is Planck’s constant, p is momentum, m is mass, and v is velocity.
Example: de Broglie Wavelength • At room temperature, the average speed of an electron is 1.3×105 m/s. The mass of the electron is about 9.11×10-31 kg. Calculate the wavelength of the electron under these conditions. • What is the wavelength of a marathon runner moving at a speed of 5 m/s? (mass of the runner is 52 kg)
h 4 (x) (mv) Uncertainty • Heisenberg showed that the more precisely the momentum of a particle is known, the less precisely is its position known: • Cannot know precisely where and with what momentum an electron is. • New ideas for determining this information based on probability • Quantum Mechanics was born
Standing Waves • The vibration of a string is restricted to certain wavelengths because the ends of the string cannot move.
de Broglie Waves in the H Atom • The de Broglie wave of an electron in a hydrogen atom must be a standing wave, restricting its wavelength to values of l = 2pr/n, with n being an integer. • This leads directly to quantized angular momentum, one of Bohr’s assumptions.
Schrödinger Wave Equation • The wave function (Y) gives the amplitude of the electron wave at any point in space. • Y2 gives the probability of finding the electron at any point in space. • There are many acceptable wave functions for the electron in a hydrogen (or any other) atom. • The energy of each wave function can be calculated, and these are identical to the energies from the Bohr model of hydrogen.
Quantum Numbers in the H Atom • The solution of the Schrödinger equation produces quantum numbers that describe the characteristics of the electron wave. • Three quantum numbers, represented by n, l, and ml, describe the distribution of the electron in three dimensional space. • An atomic orbital is a wave function of the electron for specific values of n, l, and ml.
The Principal Quantum Number, n • The principal quantum number, n, provides information about the energy and the distance of the electron from the nucleus. • Allowed value of n are 1, 2, 3, 4, … • The larger the value of n, the greater the average distance of the electron from the nucleus. • The term principal shell (or just shell) refers to all atomic orbitals that have the same value of n.
Angular Momentum Quantum Number, l • The angular momentum quantum number, l, is associated with the shape of the orbital. • Allowed values: 0 and all positive integers up to n-1. • The l quantum number can never equal or exceed the value of n. • A subshell is all possible orbitals that have the same values of both n and l.
Notations for Subshells • To identify a subshell, values for both n and l must be assigned, in that order. • The value of l is represented by a letter: l 0 1 2 3 4 5 etc. letter s p d f g h etc. • Thus, a 3p subshell has n = 3, l = 1. • A 2s subshell has n = 2, l = 0.
Magnetic Quantum Number, ml • The magnetic quantum number, ml, indicates the orientation of the atomic orbital in space. • Allowed values: all whole numbers from –l to l, including 0. • A wave function described by all three quantum numbers (n, l, ml) is called an orbital.
Example: Quantum Numbers • Give the notation for each of the following orbitals if it is allowed. If it is not allowed, explain why. (a) n = 4, l = 1, ml = 0 (b) n = 2, l = 2, ml = -1 (c) n = 5, l = 3, ml = +3
Test Your Skill • For each of the following subshells, give the value of the n and the l quantum numbers. (a) 2s (b) 3d (c) 4p
Electron Spin • An electron behaves as a small magnet that is visualized as coming from the electron spinning. • The electron spin quantum number, ms, has two allowed values: +1/2 and -1/2.
Electron Density Diagrams • Different densities of dots or colors are used to represent the probability of finding the electron in space.
Contour Diagrams • In a contour diagram, a surface is drawn that encloses some fraction of the electron probability (usually 90%).
Shapes of p Orbitals • p orbitals (l = 1) have two lobes of electron density on opposite sides of the nucleus.
Orientation of the p Orbitals • There are three p orbitals in each principal shell with an n of 2 or greater, one for each value of ml. • They are mutually perpendicular, with one each directed along the x, y, and z axes.
Shapes of the d Orbitals • The d orbitals have four lobes where the electron density is high. • The dz2 orbital is mathematically equivalent to the other d orbitals, in spite of its different appearance.
Energies of Hydrogen Atom Orbitals • The energies of the hydrogen atom orbitals depend only on the value of the n quantum number. • The s, p, d, and f orbitals in any principal shell have the same energies.
Other One-Electron Systems • The energy of a one-electron species also depends on the value of n, and are given by the equation where Z is the charge on the nucleus. • This equation applies to all one-electron species (H, He+, Li2+, etc.).
Effective Nuclear Charge • In multielectron atoms, the energy dependence on nuclear charge must be modified to account for interelectronic repulsions. • The effective nuclear charge is a weighted average of the nuclear charge that affects an electron in the atom, after correction for the shielding by inner electrons and interelectronic repulsions.
Effective Nuclear Charge • Electron shielding is the result of the influence of inner electrons on the effective nuclear charge. • The effective nuclear charge that affects the outer electron in a lithium atom is considerably less than the full nuclear charge of 3+.
Energy Dependence on l • The 2s electron penetrates the electron density of the 1s electrons more than the 2p electrons, giving it a higher effective nuclear charge and a lower energy.
Multielectron Energy Level Diagram • Within any principal shell, the energy increases in the order of the l quantum number: 4s < 4p < 4d < 4f.
Increasing Energy Order • Based on experimental observations, subshells are usually occupied in the order 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s < 5f < 6d
Electrons in Multielectron Atoms • Each electron in a multielectron atom can be described by hydrogen-like wave functions by assigning values to the four quantum numbers n, l, ml, and ms. • These wavefunctions differ from those in the hydrogen atom because of interelectronic repulsions. • The energy of these wave functions depends on both n and l.
Pauli Exclusion Principle • The Pauli Exclusion Principle: no two electrons in the same atom can have the same set of four quantum numbers. • A difference in only one of the four quantum numbers means that the sets are different.
The Aufbau Principle • The aufbau principle: as electrons are added to an atom one at a time, they are assigned the quantum numbers of the lowest energy orbital that is available. • The resulting atom is in its lowest energy state, called the ground state.
Orbital Diagrams • An orbital diagram represents each orbital with a box, with orbitals in the same subshell in connected boxes; electrons are shown as arrows in the boxes, pointing up or down to indicate their spins. • Two electrons in the same orbital must have opposite spins.
Electron Configuration • An electron configuration lists the occupied subshells using the usual notation (1s, 2p, etc.). Each subshell is followed by a superscripted number giving the number of electrons present in that subshell. • Two electrons in the 2s subshell would be 2s2 (spoken as “two-ess-two”). • Four electrons in the 3p subshell would be 3p4 (“three-pea-four”).
