Quantum Mechanical Model of the Atom

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• Many scientists contributed to the development of the quantum mechanical model of the atom.– Bohr– Planck– DeBroglie– Heisenberg– Schrodinger– Pauli

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What was already known..

• Early 1900’s…believed that– Energy is quantized– Electrons have both wave and matter

properties– Electrons can be at a variety of specific

energy levels in an atom• Energy levels are called orbits

– Proposed that electron had both wave and matter properties

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Schrodinger & deBroglie

• S & deB pictured the electron bound to the atom in a standing wave– Standing vs. traveling waves

• See page 301

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Schrodinger

• Sch.. Proposed that electrons move around the nucleus in standing waves– Each orbit represents some whole number

multiple of a wavelength– Schrodinger analyzed the hydrogen data

based on the assumption that the electrons behaved as standing waves.

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Schrodinger

– Schrodinger’s equation takes into account:• The position of the electron in 3D space (its x,y,z

coordinates)• Potential energy of the atom due to the attraction

between electrons and protons• Kinetic energy of the electron

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Schrodinger

• Schrodinger’s equation has many solutions– Each solution is called a wave function ()

and is correlated to a specific amount of energy

• Each wave function is more commonly called an orbital.

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Orbitals

Each solution to Schrodinger’s equation describes a specific wave function () /orbital– The square of a wave function, ()2,

generates a probability distribution for an electron in that orbital

• Also called an electron density map for a given orbital

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Orbitals

• Orbitals are regions in space where an electron is likely to be found– 90% of the time the electron is within the

boundaries described by the electron density map

– The exact path of an electron in a given orbital is not known!

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Heisenberg

• Heisenberg uncertainty principle states that we cannot know both the position and the momentum of an electron at the same time.– Therefore, we do not know the exact path of

the electron in an orbital.

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Orbitals

• Schrodinger’s equation and the associated electron density maps, allow us to describe the energy, size, and shape of orbitals.– The lowest energy solution to Sch..’s equation

for an electron in a hydrogen atom describes what is known as the 1s orbital.

• See pages 306/307

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Describing Orbitals

• Use quantum numbers to describe orbitals. A given orbital can be described by a set of 3 quantum numbers:

1. Principal quantum number (n)

2. Angular momentum quantum number (l) 3. Magnetic quantum number (ml)

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Principal Quantum Number (n)

• (n) describes the size and energy of the oribital– Possible values: whole number integer

• 1, 2, 3, …

– As “n” increases so does the size and energy of the orbital

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Angular momentum quantum number (l)

• (l) is related to the shape of the orbital– Possible values: (l) is an integer between 0

and n-1– Each (l) value is also assigned a letter

designation

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Angular momentum quantum number (l)

(l) Value Letter Designation

0 s

1 p

2 d

3 f

n Possiblel values

Designation

1 0 1s

2 01

2s2p

3 012

3s3p3d

4 0123

4s4p4d4f

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Magnetic quantum number (ml)

• (ml) is related to the orientation of the orbital in 3-D space– Possible values: - l to + l

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Magnetic quantum number (ml)

• Consider the p orbital…it has an l value of 1 and thus the possible ml values are -1, 0, +1– These 3 ml values correspond to the 3

possible orientations of the p orbital

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Ml and Orbitals

l ml # orbitals

0 (s) 0 1

1 (p) -1, 0, 1 3

2 (d) -2, -1, 0, 1, 2 5

3 (f) -3, -2, -1, 0, 1, 2, 3 7

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Quantum Number Summary

• See page 304 and board.– A set of 3 quantum numbers describes a

specific orbital• Energy and size - n• Shape - l

• Orientation – ml

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4th Quantum Number!

• A 4th quantum number was added to describe the spin on a given electron.– Called the electron spin quantum number - ms

• Possible values: +1/2 and -1/2

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More on electron spin.

• Each orbital can hold a maximum of 2 electrons of opposite spin.

• Pauli exclusion principle states that no two electrons in an atom can have the same set of 4 quantum numbers

Summary

• Three quantum numbers describe a specific orbital– Energy and size, shape, and orientation

• Four quantum numbers describe a specific electron in an atom

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7.9 Polyelectronic atoms

• The Schrodinger model was based on H and works in principle for atoms with more than one electron.– The shapes and possible orientations of the

hydrogen based orbitals holds true for polyelectronic atoms.

– However, the size and energy of the orbitals in polyelectronic atoms differ from those calculated for hydrogen.

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Polyelectronic Atoms

• In general, find that in a given principal quantum number (n) – S is lower energy than p, which is lower

energy than d…..• s < p < d < f

– Already know that 1s < 2s < 3s… and 2p < 3p < 4p…. (in terms of size and energy)

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7.11 The Aufbau Principle

• Putting electrons in to orbitals…– Aufbau means “building up” in German– Electrons always enter the lowest energy

orbital with room

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Hund’s Rule

• The orbitals of a given sublevel (e.g. p, or d, or f) are degenerate (of the same energy).

• The lowest energy state occurs with the maximum number of unpaired electrons.– Meaning…..electrons enter an empty orbital

of a given sublevel before pairing up.

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Goals

• To be able to write for any atom:– Electron configuration– Box/energy diagram– Lewis dot symbol

• State the quantum numbers for each electron in an atom.

• To relate the electron configuration of an atom to its location on the periodic table and its properties.

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Goals Elaborated

• Electron configuration – shows the number of electrons in each sublevel– Format: 1s22s22p4 or [He] 2s22p4

• Box/energy diagram – shows electrons as arrows and each orbital as a box. Electrons of opposite spin are indicated by up and down arrows.– Format:

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Goals Elaborated

• Lewis Dot Symbol – shows valence electrons as dots around the symbol for the atom– Maximum of 2 electrons per side of the

symbol– Valence electrons are all of the electrons in

the highest occupied principle quantum level (n)

– Format:

1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s…

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