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# Chapter 7 Electronic Structure - PowerPoint PPT Presentation

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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### Chapter 7Electronic Structure

• Waves are periodic disturbances – they repeat at regular intervals of time and distance.

• 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.

• Light or electromagnetic radiation consists of oscillating electric and magnetic fields.

• 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.

• An FM radio station broadcasts at a frequency of 100.3 MHz (1 Hz = 1 s-1). Calculate the wavelength of this 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.

• 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 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.

• 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.

• 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.

• 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.

• 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.

• Calculate the wavelength (in nm) of the line in the hydrogen atom spectrum for which n1 = 2 and n2 = 3.

• 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:

• 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.

• 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.

• 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)

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

• The vibration of a string is restricted to certain wavelengths because the ends of the string cannot move.

• 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.

• 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, 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.

• 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.

• 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.

• 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.

• 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

• For each of the following subshells, give the value of the n and the l quantum numbers.

(a) 2s

(b) 3d

(c) 4p

• 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.

• Different densities of dots or colors are used to represent the probability of finding the electron in space.

• 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.

• 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.

• 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.).

• 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.

• 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+.

• 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.

• Within any principal shell, the energy increases in the order of the l quantum number: 4s < 4p < 4d < 4f.

• 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

• 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.

• 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: 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.

• 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.

• 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”).

• Hydrogen contains one electron in the 1s subshell.

1s1

• Helium has two electrons in the 1s subshell.

1s2

• Lithium has three electrons.

1s2 2s1

• Beryllium has four electrons.

1s2 2s2

• Boron has five electrons.

1s2 2s2 2p1

• Carbon, with six electrons, has the electron configuration of 1s2 2s2 2p2.

• The lowest energy arrangement of electrons in degenerate (same-energy) orbitals is given by Hund’s rule: one electron occupies each degenerate orbital with the same spin before a second electron is placed in an orbital.

• N 1s2 2s2 2p3

• O 1s2 2s2 2p4

• F 1s2 2s2 2p5

• Ne 1s2 2s2 2p6

• Heavier atoms follow aufbau principle in organization of electrons.

• Because their electron configurations can get long, larger atoms can use an abbreviated electron configuration, using a noble gas to represent core electrons.

Fe: 1s2 2s2 2p6 3s2 3p6 4s2 3d6→ [Ar] 4s2 3d6

Ar

• The electron configurations for some atoms do not strictly follow the aufbau principle; they are anomalous.

• Cannot predict which ones will be anomalous.

• Example: Ag predicted to be

[Kr] 5s2 4d9; instead, it is

[Kr] 5s1 4d10.