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7. Atomic structure and periodicity

Electromagnetic radiation

electromagnetic radiation: light; radiant energy that has wavelike behavior and travels at the speed of light in a vacuum

e.g. light from sun, microwave, X rays, radiant heat from fireplace

waves have three primary characteristics

wavelength (λ, lambda): distance between two consecutive peaks or troughs in a wave

frequency (ν, nu): number of waves (cycles) per second that pass a given point in space

shortest wavelength has highest frequency

longest wavelength has lowest frequency

speed

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λ: wavelength (m)

ν: frequency (cycles/s)

c: speed of light (2.9979 × 10⁸ m/s)

hertz (Hz): 1 cycle per second (1/s)

cycles are implied

Nature of matter

initial definitions of matter and energy

end of 19th century: matter and energy were distinct

matter: particles

had mass

position in space could be specified

energy: waves

massless

delocalized; position in space could not be specified

beginning of 20th century

German Max Planck: studied radiation profiles emitted by solid bodies heated to incandescence

results could not be explained in terms of the present physics (that matter could absorb or emit any quantity of energy)

Planck’s constant:

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energy can be gained or lost only in whole-number multiples of hν

change in energy for a system:

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n: integer

h: Planck’s constant

ν: frequency fo electromagnetic radiation absorbed or emitted

quantization: energy transfer can only occur in discrete quanta (packets)

photon: quantum of electromagnetic radiation

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h: Planck’s constant

ν: frequency of radiation

λ: wavelength of radiation

Photoelectric effect

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photoelectric effect: electrons emitted from the surface of a metal when light strikes it

no electrons are emitted by a given metal below a specific threshold frequency (ν₀)

for light with frequency lower than the threshold frequency, no electrons are emitted regardless of the intensity of the light

for light with frequency greater than the threshold frequency, the number of electrons emitted increases with the intensity of the light

for light with frequency greater than the threshold frequency, the kinetic energy of the emitted electrons increases linearly with the frequency of the light

explanation

assume:

electromagnetic radiation is quantized (consists of photons)

threshold frequency represents the minimum energy required to remove the electron from the metal’s surface

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m: mass of electron

v: velocity of electron

hν: energy of incident proton

hν₀: energy required to remove electron from metal’s surface

intensity of light is a measure of the number of protons present in a given part of the beam, so a greater intensity means that more photons are available to release electrons (as long as ν > ν₀)

Einstein’s special theory of relativity:

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m: mass

c: speed of light

energy has mass

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can calculate mass associated with given quantity of energy

calculate apparent mass of photon

energy of each photon:

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duel nature of light: electromagnetic radiation exhibits wave properties and certain characteristics of particulate matter as well

does particulate matter exhibit wave properties?

mass and wavelength

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particle with velocity

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de Broglie’s equation

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allows you to calculate wavelength for a particle

diffraction: light scattered from a regular array of points or lines

different colors result

various wavelengths of visible light are not all scattered in the same way

colors “separated” like light passing through a prism

diffraction pattern: caused by scattered radiation; bright spots and dark patterns

can only be explained in terms of waves, so proves that matter obeys de Broglie’s equation

all matter exhibits both particulate and wave properties

Atomic spectrum of hydrogen

a sample of hydrogen gas receives a high-energy spark → H₂ molecules absorb energy → some H—H bonds are broken → excited hydrogen atoms

contain excess energy that is released by emitting light of various wavelengths to produce emission spectrum

continuous spectrum: contains all the wavelengths of visible light

when light passes through a prism

line spectrum: a few lines of light

each line corresponds to a discrete wavelength

only certain energies are allowed for the electron in the atom (quantized)

Bohr model

quantum model: electrons move around the nucleus only in certain allowed circular orbits

energy levels available to electron in hydrogen atom

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n: integer (larger value → larger orbit radius)

Z: nuclear charge

negative: energy of electron bound to nucleus is lower than if at infinite distance, where there is no interaction and energy is 0

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ground state: lowest possible energy state

two important points about Bohr model

model correctly fits quantized energy levels of hydrogen atom and postulates only certain allowed circular orbits for the electron

as the electron becomes more tightly bound, its energy becomes more negative relative to the zero-energy reference state (infinite distance from the nucleus). As the electron is brought closer to the nucleus, energy is released from the system

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does not work when applied to atoms other than hydrogen; electrons do not move around the nucleus in circular orbits

Quantum mechanical model of atom

standing wave: stationary; waves do not travel

wave or quantum mechanics: electron in hydrogen atom imagined to be standing wave

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ψ: wave function

function of the coordinates (x, y, z) of the electron’s position in three-dimensional space

Ĥ: operator

set of mathematical intsructions

produces total energy of the atom when the mathematical terms are applied to the wave function

E: total energy of the atom

potential energy due to attraction between proton and electron

kinetic energy of moving electron

orbital: specific wave function, characterized by a particular value of E

quantum (wave) mechanical model

do not know the pathway of the orbitals

Heisenberg uncertainty principle: there is a fundamental limitation to just how precisely we can know both the position and momentum of a particle at a given time

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Δx: uncertainty in a particle’s position

Δ(mv): uncertainty in a particle’s momentum

h: Planck’s constant

more accurately we know a particle’s position, the less accurately we can know its momentum and vice versa

Physical meaning of a wave function

the square of the function indicates the probability of finding an electron near a particular point in space

e.g. relative probability of finding electron at positions 1 and 2

substitute x, y, z for the two positions, square function value, compute ratio

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N₁/N₂: ratio of probabilities of finding the electrons at positions 1 and 2

probability distribution: intensity of color used to indicate probability value near a given point in space

a.k.a. electron density map

electron density and electron probability mean the same thing

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radial probability distribution: plots total probability of finding the electron in each spherical shell versus the distance from the nucleus

maximum occurs because:

probability of finding an electron at particular position greatest near nucleus

volume of spherical shell increases with distance from the nucleus

when moving away from the nucleus, probability decreases but summing more positions

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describing size of hydrogen 1s orbital: radius of the sphere that encloses 90% of the total electron probability

orbital is a wave function, but most helpful to picture as three-dimensional electron density map

Quantum numbers

quantum numbers: describe various properties of the orbital

principal quantum number (n): integral values

related to size and energy of orbital

as n increases:

orbital becomes larger

electron spends more time farther from the nucleus

higher energy (less tightly bound to the nucleus)

energy less negative

angular momentum quantum number (ℓ): integral values from 0 to n - 1 for each value of n

related to shape of atomic orbitals

value of ℓ for particular orbital commonly assigned a letter (from early spectral studies)

ℓ = 0: s

ℓ = 1: p

ℓ = 2: d

ℓ = 3: f

ℓ = 4: g

magnetic quantum number (m_ℓ): integral values between ℓ and - ℓ, including 0

related to orientation of the orbital in space relative to the other orbitals in the atom

subshell: each set of orbitals with a given value of ℓ

designated by giving the value of n and the letter for l

e.g. n = 2, ℓ = 1 → 2p

number of orbitals per subshell

s = 1

p = 3

d = 5

f = 7

g = 9

Orbital shapes and energies

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node/nodal surface: areas of high probability separated by areas of zero probability

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atomic orbital functions have signs

s orbital functions: positive everywhere in three-dimensional space

p orbital functions: different signs in different regions of space

p_z orbital has positive sign in all regions of space in which z is positive and a negative sign when z is negative

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degenerate

energy of a particular orbital is determined by its value of n

orbitals with the same value of n have the same energy

hydrogen’s single electron can occupy any of its atomic orbitals

ground state (lowest energy): 1s orbital

excited state (higher energy): higher-energy orbital

summary of hydrogen atom

in the quantum (wave) mechanical model, the electron is viewed as a standing wave

series of wave functions (orbitals) that describe possible energies and spatial distributions available to the electorn

in agreement with the Heisenberg uncertainty principle, the model cannot specify the detailed electron motions

square of the wave function represents the probability distribution of the electron in that orbital

orbitals are pictured in terms of probability distributions or electron density maps

the size of an orbital is arbitrarily defined as the surface that contains 90% of the total electron probability

the hydrogen atom has many types of orbitals

ground state: single electron resides in 1s orbital

electron can be excited to higher-energy orbitals if energy is put into the atom

Electron spin and the Pauli principle

electron spin: electron can have two spin states, producing two oppositely directed magnetic moments

electron spin quantum number (mₛ): can be + 1/2 or - 1/2

electron can spin in one of two opposite directions

Pauli exclusion principle: in a given atom no two electrons can have the same set of four quantum numbers (n, ℓ, m_ℓ, mₛ)

Polyelectronic atoms

polyelectronic atoms: atoms with more than one electron (e.g. helium)

three energy contributions must be considered in describing helium atom

kinetic energy of the electrons as they move around the nucleus

potential energy of attraction between the nucleus and the electrons

potential energy of repulsion between the two electrons

electron correlation problem: because the electron pathways are unknown, the electron repulsions (Schrödinger equation) cannot be calculated exactly

approximation used: treat each electron as if it were moving in a field of charge that is the net result of the nuclear attraction and the average repulsions of all the other electrons

e.g. sodium’s outermost electron

attraction to nucleus

repulsion caused by other 10 electrons

net effect: not bound nearly as tightly to nucleus as it would be if the other electrons were not present (screened or shielded)

hydrogen-like orbitals: same general shapes as orbitals for hydrpogen but sizes and energies differ

for a given principal quantum level, the orbitals vary in energy

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“prefer” orbitals in order s, p, d, then f

reason (e.g. 2s vs 2p)

2s electron penetrates to the nucleus (comes very close) more than once in the 2p orbital

electron in a 2s orbital attracted to the nucleus more strongly than an electron in a 2p orbital

2s orbital is lower in energy than the 2p orbitals in a polyelectronic ion

the more effectively an orbital allows its electron to penetrate the shielding electrons to be close to the nuclear charge, the lower is the energy of that orbital

History of the periodic table

Johann Dobereiner: triads (several groups of three elements that have similar properties)

John Newlands: elements should be in octaves (certain properties repeated for every eighth element)

current periodic table: independently by Julius Lothar Meyer and Dmitri Ivanovich Mendeleev

Mendeleev emphasized how useful the table could be in predicting the existence and properties of unknown elements

corrected several values for atomic masses

almost universally adopted

Aufbau principle and the periodic table

aufbau principle: as protons are added one by one to the nucleus to build up the elements, electrons are similarly added to these hydrogen-like orbitals

orbital diagram

ground state

arrow represents direction

Hund’s rule: the lowest energy configuration for an atom is the one having the maximum number of unpaired electrons allowed by the Pauli principle in a particular set of degenerate orbitals

represented as having parallel spins (up)

e.g. carbon

could be written as 1s²2s²2p¹2p¹ to indicate that the electorns occupy separate 2p orbitals

usually given as 1s²2s²2p² and understood that the electrons are in different 2p orbitals

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valence electrons: electrons in the outermost principal quantum level of an atom

e.g. nitrogen: 2s and 2p electrons

e.g. sodium: 3s electron

core electrons: inner electrons

transition metals: configurations obtained by adding electrons to the five 3d orbitals

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note that:

(n + 1)s orbitals always fill before the nd orbitals

e.g. 5s orbitals fill in rubidium and strontium before the 4d orbitals fill in the second row of transition metals (yttrium through cadmium)

due to penetration effect

e.g. 4s orbital allows for more penetration to the vicinity of the nucleus that it becomes lower in energy than the 3d orbital, thus the 4s fills before the 3d

after lanthanum ([Xe]6s²5d¹) the lanthanide series/lanthanides occur

correspond to the filling of the 7 4f orbitals

sometimes an electron occupies a 5d orbital instead of a 4f orbital (because they have similar energies)

after actinium ([Rn]6s²6d¹) actinide series/actinides occur

correspond to the filling of the 5f orbitals

sometimes one or two electrons occupy the 6d orbitals instead of the 5f orbitals (because they have similar energies)

group labels for Groups 1A, 2A, 3A, 4A, 5A, 6A, 7A, 8A indicate total number of valence electrons

e.g. all elements in Group 5A have configuration ns²np³ (d electrons fill one period late and are usually not counted as valence)

groups labelled 1A, 2A, 3A, 4A, 5A, 6A, 7A, 8A often called main-group elements or representative elements (same valence electron configuration)

Groups 1-18: indicates number of s, p, and d electrons added since the last noble gas

similar chemistry of members of a given group is due to the fact that they all have the same valence electron configuration (only principal quantum number of valence orbitals changes)

Periodic trends in atomic properties

Ionization energy

energy required to remove an electron from a gaseous atom or ion (assumed to be in ground state)

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in stepwise ionization process, always the highest-energy electron (least-tightly bound) that is removed first

first ionization energy (I₁): energy required to remove the highest-energy electron of an atom

second ionization energy (I₂): energy required to remove the second-highest-energy electron of an atom (much larger than I₁)

across a period: first ionization energy increases

down a group: first ionization energy decreases

Electron affinity

electron affinity: energy change associated with the addition of an electron to a gaseous atom

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more negative energy → greater quantity of energy released

across a period: generally become more negative (but there are exceptions)

down a group: generally become more positive (but there are exceptions)

Atomic radius

atomic radius: obtained by measuring the distance between atoms in chemical compounds

covalent atomic radii: nonmetallic diatomic molecules

nonmetallic non-diatomic molecules: estimated from various covalent compounds

metallic atomic radii: from the distance in metal crystals

across a period: atomic radius decreases

increasing effective nuclear charge (decreasing shielding)

valence elctrons drawn closer to nucleus

down a group: atomic radius increases

increases in orbital sizes

Properties of a group: alkali metals

Information contained in the periodic table

groups of representative elements exhibit similar chemical properties that change in a regular way

it is the number and type of valence electrons that primarily determine an atom’s chemistry

electron configuration of any representative element

predicted configurations sometimes incorrect fro transition metals (memorize chromium and copper)

certain groups in the periodic table have special names

most basic division of elements: metals and nonmetals

metals

tendency to give up electron(s) to form a positive ion

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