Quantum Numbers: specify the properties of atomic orbitals and their electrons.

Four Quantum Numbers Principal Quantum Number Orbital Quantum Number Magnetic Quantum Number Spin Quantum Number

Principal Quantum Number, n Indicates main energy levels n = 1, 2, 3, 4… Each main energy level has sub-levels

The maximum number of electrons in a principal energy level is given by: Max # electrons = 2n2 n= the principal quantum number

Orbital Quantum Number, ℓ (Angular Momentum Quantum Number) Indicates shape of orbital sublevels ℓ = n-1 ℓ sublevel 0 s 1 p 2 d 3 f 4 g

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 = 1 l = 0 (s) n = 3 l = 0 (s) l = 1 (p) l = 2 (d) n = 2 l = 0 (s) l = 1 (p) Use QN WS as study tool

Atomic Orbital s 2s

II. Orbital Shapes and Energies Atomic orbital shapes are surfaces that surround 90% of the total probability of where its electrons are Look at l = 0, the s-orbitals Basic shape of an s-orbital is spherical centered on the nucleus Basic shape is same for same l values Nodes = areas of zero probability Number of nodes changes for larger n We will usually just use outer surface to describe the shape of an orbital

Degenerate Orbitals

The 3 p orbitals http://www.rmutphysics.com/CHARUD/scibook/crystal-structure/porbital.gif

p-orbitals There are no 1p orbitals (n = 1, l = 0 only) 2p orbitals (n = 2, l = 1) have 2 lobes with a node at the nucleus There are three different p-orbitals (l = 1, ml = -1, 0, 1) 2px lies along the x-axis 2py lies along the y-axis 2pz lies along the z-axis All three 2p orbitals have the same energy = degenerate 3p, 4p, 5p, etc… have the same shape and number, just larger

The d orbitals

d-orbitals There are no 1d or 2d orbitals (d needs l = 2, so n = 3) 5 degenerate d-orbitals (ml = -2, -1, 0, 1, 2) 4 of the d-orbitals have 4 lobes which lie in planes on or between the xyz axes: 3dxy, 3dxz, 3dyz, 3dx2-y2 1 is composed of 2 lobes and a torus- shaped area: 3dz2 The 4d orbitals etc…are the same shape, only larger

f orbitals

f-orbitals n = 4, l = 3, ml = -3, -2, -1, 0, 1, 2, 3 7 f-orbitals in the fourth shell are degenerate The f-orbital are only used for the lanthanides and actinides and are complex shapes. We won’t use them.

Magnetic Quantum Number, ml Indicates the orientation of the orbital in space. Values of ml : integers -l to l The number of values represents the number of orbitals. Example: for l= 2, ml = -2, -1, 0, +1, +2 Which sublevel does this represent? Answer: d

Electron Spin QN (ms = +1/2, 1/2) Relates to the spin states of the electrons. Electrons are –1 charged and are spinning Spinning charge creates a magnetic field You can tell the direction of the spin by which way the magnetic moment lines up in an external magnetic field The two possible spin directions are called +½ and –½

Electron Spin Quantum Number, (ms or s) Indicates the spin of the electron (clockwise or counterclockwise). Values of ms: +1/2, -1/2

Example: List the values of the four quantum numbers for orbitals in the 3d sublevel. Answer: n=3 l = 2 ml = -2,-1, 0, +1, +2 ms = +1/2, -1/2 for each pair of electrons

The Electron Cloud The electron cloud represents positions where there is probability of finding an electron.

The Electron Cloud The higher the electron density, the higher the probability that an electron may be found in that region. http://www.chemeng.uiuc.edu/~alkgrp/mo/gk12/quantum/H_S_orbital.jpg

The Electron Cloud for Hydrogen 90% probability of finding the electron within this space

Probability Curve for Hydrogen

Quantum Mechanical Model Electrons are located in specific energy levels. There is no exact path around the nucleus. The model estimates the probability of finding an electron in a certain position.

Let's Summarize Quantum Numbers

Principal Quantum Number Symbol = n Represents the main energy level of the electron Range = 1- 7 Ex. = 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 = m Represents the orientation of the orbital around the nucleus Each line holds 2 electrons ___ = s ___ ___ ___ = 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 States Clockwise 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