Quantum Mechanics

Adapted from: http://www.chalkbored.com/lessons/chemistry-12/quantum-mechanics.ppt

Classical Mechanics v. Quantum Mechanics

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What are Quantum Numbers?

• An electron’s unique “fingerprint” that describes it position and behavior

• Quantum Mechanics = explains the behavior of very SMALL, FAST moving objects

Quantum Mechanics overview• We will see: electrons have discrete energies,

not because they are in shells but because they can only have certain wavelengths

• Line spectra are not due to electrons jumping from shell to shell (as in Bohr’s model)…

• Instead they’re due to electrons transforming from one wavelength (waveform) to another

• Each electron is a wave that can be described by a series of “quantum numbers”

• There are four quantum numbers: n, l, ml, ms

• The combination of the first 3 defines an “orbital”

Quantum Mechanics

I don't like it, and I'm sorry I ever had anything to do with it. -- Erwin Schrodinger talking about Quantum Physics

Classifying electron waves• The waves of electrons are similarly class-

ified according to certain variables (n, l, ml)

• The rationale for the numbers is not always clear. These numbers come from some pretty advanced math. You don’t have to know why we use certain formulas for determining quantum numbers.

• You do have to know what the formulas are, when to use them, and what the resulting quantum numbers represent.

Quantum Numbers

There are four numbers that come into the theory of electron clouds as waves called quantum numbers.

The first quantum number, n, is the principle energy level. This is the 1 in 1s2. It can have the values 1, 2, 3, …

The second quantum number, l, is the sublevel. The nth principle energy level has n sublevels. We refer to these sublevels by letters: s, p, d, f, g, h, i, j, k, … Sometimes numbers are used too: 0, 1, 2, 3, …(n-1)

l : The secondary quantum number• Each value of l is associated with a letter:• 0 = s, 1 = p, 2 = d, 3 = f• after 3, the associated letters go

alphabetically from f up, so 4 = g, 5 = h, etc.• Normally, we don’t talk about electrons

beyond l = 3 (the f subshell)• Whereas n represents size and energy, l

tells us of the shape (a.k.a. sublevel) of an electron (more detail later).

• We often identify electrons by shell and subshell: e.g. 1s, 3d, 2s, and 5d subshell

l : The secondary quantum number• If n can be thought of as shells, l can be

thought of as “subshells” dividing each shell into subsections … (l = 0 n - 1)

n = 1l = 0 (s)

n = 2l = 0 (s)l = 1 (p)

n = 3l = 0 (s)l = 1 (p)l = 2 (d)

Use QN WS as study tool

Quantum Numbers

The third quantum number, ml, is the orbital. Every sublevel has one or more orbitals. The s sublevel has 1 orbital, the p sublevel has 3 orbitals, the d sublevel has 5 orbitals, etc. These orbital can be indicated by the number ml = l, l-1, …0, -1, … -l

The fourth quantum number, ms, is the spin of the electron. Electrons can be either spin up or spin down. ms can be either +½ or -½

Spintronics: This is a new type of electronics which is based on the spin of the electrons.It is possible to filter electrons which have different spins using very thin magnetic films.

A. Principal QN (n = 1, 2, 3, . . .)

1. Related to size of the atomic orbital (distance from the nucleus).

2. Larger n value indicates higher energy

3. Larger n value means electrons are less strongly bound to nucleus

B. Angular Momentum (sublevel) QN (l = 0 to n 1)

1. Relates to shape of the atomic orbital.

2. Each l number is assigned a letter

3. n = 3, l = 0, 1, 2 (s, p, and d orbitals in the third shell)

C. Magnetic QN (ml = l to l)

1. Relates to orientation of the orbital in space relative to other orbitals.

2. 2. For l = 2, ml = -2, -1, 0, 1, 2 (Five d-orbitals)

D.Electron Spin QN

(ms = +1/2, 1/2)

• Relates to the spin states of the electrons.

2. Electrons are –1 charged and are spinning

3. Spinning charge creates a magnetic field

4. You can tell the direction of the spin by which way the magnetic moment lines up in an external magnetic field

5. The two possible spin directions are called +½ and –½

E. Pauli Exclusion Principle

1. In a given atom, no two electrons can have the same set of four quantum numbers (n, l, ml, ms).

2. Therefore, an orbital can hold only two electrons, and they must have opposite spins.

3. Electrons can have the same n, l, and ml values

a) n = 3, l = 2 (d-orbital), ml = -2 (a single d-orbital)

b) That single d-orbital can only hold 2 e-, one with ms = +1/2, and one with ms = 1/2

http://newsbureau.upmc.com/TX/Nanotubes04.htm http://www.news.utoronto.ca/bin6/050110-832.asp http://www.vega.org.uk/series/lectures/feynman/ http://informationweek.com/story/showArticle.jhtml?articleID=59300089

II. Orbital Shapes and EnergiesA. Atomic orbital shapes are surfaces that surround

90% of the total probability of where its electrons are1. Look at l = 0, the s-orbitals

2. Basic shape of an s-orbital is spherical

3. centered on the nucleus

3. Basic shape is same for same l values

4. Nodes = areas of zero probability

5. Number of nodes changes for larger n

6. We will usually just use outer surface

7. to describe the shape of an orbital

B. p-orbitals1. There are no 1p orbitals (n = 1, l = 0 only)

2. 2p orbitals (n = 2, l = 1) have 2 lobes with a node at the nucleus

3. There are three different p-orbitals (l = 1, ml = -1, 0, 1)a. 2px lies along the x-axis

b. 2py lies along the y-axis

c. 2pz lies along the z-axis

4. All three 2p orbitals have the same energy = degenerate

5. 3p, 4p, 5p, etc… have the same shape and number, just larger

C.d-orbitals1. There are no 1d or 2d orbitals (d needs l = 2, so

n = 3)

2. 5 degenerate d-orbitals (ml = -2, -1, 0, 1, 2)

3. 4 of the d-orbitals have 4 lobes which lie in

planes on or between the xyz axes: 3dxy, 3dxz, 3dyz, 3dx2-y2

4. 1 is composed of 2 lobes and a torus-shaped

area: 3dz2

5. The 4d orbitals etc…are the same shape, only larger

D. f-orbitals1. n = 4, l = 3, ml = -3, -2, -1, 0, 1, 2, 3

2. 7 f-orbitals in the fourth shell are degenerate

3. The f-orbital are only used for the lanthanides and actinides and are complex shapes. We won’t use them.

E. Orbital Energies1. Orbital energies are largely determined by

2. the n value: 3 > 2 > 1 for H atom (s = p)

2. But, for polyelectron atoms, the different

• l values are not all degenerate (s ≠ p)

• a. 2s is larger than 2p orbital

• b. 2s “penetrates” the 2p, so is lower energy

• c. Penetration effects help explain energy ordering

III. The History of the Periodic TableA. Patterns in element properties were recognized

1. Dobereiner (1780-1849) found “triads” of similar elements: Cl, Br, I

2. Newlands suggested in 1864 that elements should be arranged in “octaves” because similarities occurred every 8th element

B. The Modern Periodic Table1. The German Meyer (1830-1895) and Russian Mendeleev

(1834-1907) independently developed the current arrangement of elements

2. Mendeleev predicted the properties of “missing” elements

Hund’s Rule

Orbitals of equal energy are each occupied by ONE electron before any orbital is occupied by a SECOND electron

All electrons in a single occupied orbital must have the same spin.

Principal Quantum Number

Symbol = nRepresents the main energy level of the

electronRange = 1- 7Ex. = 3s

Principal Quantum number = 3

Angular Momentum Quantum Number

Symbol = l (small letter L)Represents the shape of the orbital (also

called sublevel)Range = 0 – n-1 (whole number)Shapes:0 = s (sphere) 1 = p (petal)

2 = d (double petal) 3 = f (flower)

Magnetic Quantum Number

Symbol = mRepresents the orientation of the orbital

around the nucleusEach line holds 2 electrons

___ = s 0___ ___ ___ = p

-1 0 +1

Magnetic Quantum Number (cont.)

___ ___ ___ ___ ___ = d

-2 -1 0 +1 +2

___ ___ ___ ___ ___ ___ ___ = f

-3 -2 -1 0 +1 +2 +3

So what is ms (or just “s”)?

• The spin (clockwise or counterclockwise) on the electron

• It describes which of the 2 possible electrons in any orbital is being described

• Values: +/- ½

Spin Quantum Number

2 Spin StatesClockwise spin = +1/2 (upward arrow)Counterclockwise spin = -1/2 (downward

arrow)

A Single orbital can hold two electrons, but they must have opposite spins

Unit 2 – Electrons and Periodic Behavior

Cartoon courtesy of NearingZero.net

Cartoon courtesy of NearingZero.net