Quantum Mechanics

Atomic Structure Electron Cloud Model Electron Cloud Model “Cloud” is a probability distribution. “Cloud” is a probability distribution. Probability is the most information that we can possibly know. Probability is the most information that we can possibly know.

Schrödinger Wave Equation Allows you to calculate the probability of an electron being in a given place and energy. Allows you to calculate the probability of an electron being in a given place and energy. Gives us a good idea of where the electrons are in an atom. Gives us a good idea of where the electrons are in an atom.

Quantum Numbers Address of an electron in an atom Address of an electron in an atom The collection of 4 quantum numbers gives you as much information as you can possibly get for a particular electron. The collection of 4 quantum numbers gives you as much information as you can possibly get for a particular electron. Describes a hierarchical structure Describes a hierarchical structure Shell Shell Subshell Subshell Orbitals Orbitals Electron Spin States Electron Spin States

Principle Quantum Number How far from the nucleus an electron is. How far from the nucleus an electron is. Also describes most of the energy. Also describes most of the energy. Will be integers from 1 to 7 for our purposes. Will be integers from 1 to 7 for our purposes. Call this the “shell” the electron is in. Call this the “shell” the electron is in. n = 1 (the first shell) is shell closest to nucleus with lowest energy n = 1 (the first shell) is shell closest to nucleus with lowest energy n

Angular Momentum Quantum Number Angular momentum describes movement in circles. Angular momentum describes movement in circles. Describes the shape of the area an electron is in. Describes the shape of the area an electron is in. Can be integers from 0 through 3 Can be integers from 0 through 3 Usually called the “sub-shell” Usually called the “sub-shell”ℓ A cursive L

Angular Momentum Quantum Number Specific Values have specific names Specific Values have specific names ℓ = 0 is called an s subshell ℓ = 0 is called an s subshell ℓ = 1 is called a p subshell ℓ = 1 is called a p subshell ℓ = 2 is called a d subshell ℓ = 2 is called a d subshell ℓ = 3 is called a f subshell ℓ = 3 is called a f subshell

s Orbitals within an s Subshell Have a spherical shape Have a spherical shape

p Orbitals within a p Subshell Have a dumbbell shape Have a dumbbell shape

d Orbitals within a d Subshell Have a four leaf clover shape Have a four leaf clover shape

f Orbitals within an f Subshell Look like two d orbitals pushed together in the middle Look like two d orbitals pushed together in the middle

Magnetic Quantum Number Describe which direction the shapes point in space. Describe which direction the shapes point in space. Refers to specific orbitals Refers to specific orbitals Orbital – area of space analogous to an orbit but taking probability into account Orbital – area of space analogous to an orbit but taking probability into account Each orbital holds up to two electrons. Each orbital holds up to two electrons. mℓmℓmℓmℓ m with a subscript L

Magnetic Quantum Number Only one direction in space for a sphere Only one direction in space for a sphere Only one s orbital Only one s orbital

Magnetic Quantum Number There are three directions for p orbitals to point There are three directions for p orbitals to point There are three p orbitals There are three p orbitals

Magnetic Quantum Number There are five solutions for the d orbitals There are five solutions for the d orbitals

Magnetic Quantum Number Seven f orbitals Seven f orbitals

Magnetic Quantum Number n = 1 is where the s subshells start n = 1 is where the s subshells start n = 2 is where the p subshells start n = 2 is where the p subshells start n = 3 is where the d subshells start n = 3 is where the d subshells start n = 4 is where the f subshells start n = 4 is where the f subshells start

Spin Quantum Number Describes the spin of an electron Describes the spin of an electron Imagine a top spinning on its axis Imagine a top spinning on its axis Can either be +½ or -½ Can either be +½ or -½ msmsmsms m with a subscript s

For the Math People For a given electron For a given electron n = [1, 2, 3, 4, 5, 6, or 7] n = [1, 2, 3, 4, 5, 6, or 7] ℓ = [0, … n – 1] ℓ = [0, … n – 1] m ℓ = [- ℓ … +ℓ] m ℓ = [- ℓ … +ℓ] m s = [-½, +½] m s = [-½, +½]

How do we apply all this to electrons? Aufbau Principle – fill up electrons in the lowest energy first Aufbau Principle – fill up electrons in the lowest energy first Use the diagonal line diagram. Use the diagonal line diagram.

Hydrogen Start with the simplest atom. Start with the simplest atom. Has one electron Has one electron

Orbital Filling Diagrams Use horizontal lines to represent an orbital Use horizontal lines to represent an orbital Use half arrows to represent electrons Use half arrows to represent electrons Arrow up means spin +½ Arrow up means spin +½ Arrow down means spin -½ Arrow down means spin -½

Helium Has two electrons Has two electrons Where do you put the extra electron? Where do you put the extra electron? Pauli Exclusion Principle – No two electrons in the same atom can have the same set of four quantum numbers Pauli Exclusion Principle – No two electrons in the same atom can have the same set of four quantum numbers Practically this means two electrons in the same orbital must have opposite spins. Practically this means two electrons in the same orbital must have opposite spins.

More Elements Lithium Lithium Three electrons Three electrons Beryllium Beryllium Four electrons Four electrons

Boron 5 electrons 5 electrons Does it matter which p-orbital you put the electron in? Does it matter which p-orbital you put the electron in? No because they are all equivalent No because they are all equivalent Degenerate – have the same energy Degenerate – have the same energy

Carbon Six electrons Six electrons Where does the sixth electron go? Where does the sixth electron go? Hund’s Rule – Electrons stay unpaired as long as possible. Hund’s Rule – Electrons stay unpaired as long as possible.

More Elements Nitrogen Nitrogen 7 electrons 7 electrons Oxygen Oxygen 8 electrons 8 electrons Fluorine Fluorine 9 electrons 9 electrons Neon Neon 10 electrons 10 electrons

More Elements Chromium Chromium Krypton Krypton

More Elements Tin Tin Promethium Promethium

Even More Elements Tungsten Tungsten Lead Lead

Even More Elements Plutonium Plutonium Bohrium Bohrium