Quantum Numbers

A Short Tutorial

Bohr Model of Hydrogen Atom• an e-’s is found in specific

energy levels.• These levels represent a

fixed distance from the nucleus.

• A quantum of energy is the amount of energy needed to move one electron.

• Neils Bohr was also a GK for the Danish National Team !

The Bohr Model• Like the rungs of this ladder, the

energy levels in an atom are not equally spaced.

• The higher the energy level of an e-, the less energy it takes to move the e- to the next higher energy level.

• Moving requires a quantum leap.

5.1

Modern View of Atoms

1. Energy of e-’s is quantized.2. e-’s sometimes act like

waves.3. Position of e- is based on

probability.• Modern view is the

quantum-mechanical model of the atom.

Location of Electrons Based on Probability

• e-’s are mathematically predicted to exist in electron clouds.

• 90% chance that e-’s exist in this shape.

• e- clouds have high electron density.

Schrodinger:• An Atom contains

4 quantum numbers

• Schrodinger was always very serious about Quantum Numbers.

Principal Quantum Number• Symbol is n.• Distance from nucleus.• Values are n = 1, 2, 3, 4, 5, 6, 7, ….

n = 1

n = 2n = 3

Clearly, the further the distance from the nucleus, the bigger the volume of the energy level is.

Orbital Quantum Number

Symbol is l. It corresponds to the type of orbital,

either s, p, d or f. It has values of 0, 1, 2, 3, ….Check the next several slides for a

look at the different orbitals.

The s - orbital

The s-orbital is shaped like a sphere with its center at the origin of a 3 dimensional grid. The axes on the grid are the x-axis, the y-axis, and the z-axis.

The p - orbitalThe p – orbitals are dumbbell shaped and lie across each axis, so they are 90° from each other. There are always 3 p – orbitals for each energy level.

The p - orbitalAnother look at p – orbitals and how they orientate themselves to each other.

The d - orbitals5.1

The d – orbitals are also dumbbell shaped and generally lie between the axes. There are always 5 d – orbitals.

The f - orbitalsHere is a list of the 7 f – orbitals. fxyz, fx3, fy3, fz3, fz(x2-y2), fy(z2-x2), fx(z2-y2)

They have complex shapes.

Magnetic Quantum Number

Symbol is m though sometimes lm. It corresponds to the orientation of the

orbital around the axis. It has values of - l, … 0, …. + lYou have seen these on earlier slides.

Check the next slide in the presentation to look at the p – orbitals again.

Magnetic Quantum Numbers

• Notice the x, y, and z part of the orbital notation below ?

• This is telling us how the orbital is orientated around the axes.

• This is the magnetic quantum number.

Magnetic Quantum Number

• There are as many orientations as there are orbitals.

• Check the table: Orbital type # of orbitals Magnetic QN s 1 0 p 3 -1 0 + 1 d 5 -2 -1 0 +1 +2 f 7 -3 -2 -1 0 +1 +2 +3

Magnetic Quantum Number• Now let’s apply this to the p – orbitals.• Recall: There are three p – orbitals and they are the

px , py , and pz .• From the slide above we see that the magnetic

quantum numbers for the p – orbitals are as follows: -1 , 0, +1.

• Therefore the px corresponds to -1, the py corresponds to 0, and the pz corresponds to + 1.

• The other orbitals work in the same way – there are as many orbitals as magnetic quantum numbers.

Spin Quantum Number (ms) • Symbol is s, though

sometimes ms • It tells us that the two

electrons in the orbital have opposite spins.

• It’s values are either + ½ or – ½ .

• Remember that there can be no more than 2 electrons per orbital.

For a given energy level n,there are n2 orbitals

for n = 1 : 1orbitalfor n = 2 : 4 orbitals (one s and three p orbitals)for n = 3 : 9 orbitals (one s, three p, and five d

orbitals)

Is there a pattern here?

Since there can be 2 electrons per orbital, the number of electrons in any energy level is described as 2 n2

Lets put all the info together

Lets put all the info together

Lets put all the info together

Summary• Remember that the quantum numbers are the

solutions to the Schrodinger equations.• They are actually numbers but it would be

confusing to have 4 numbers right next to each other so some are given letter designations to make it easier to read.

• For example: For orbital quantum numbers the s – orbital = 0. The p – orbital = 1. The d – orbital = 2 etc.

Summary• Let’s look at a certain electron in an Iron atom. We

will look at the last electron to be filled in its’ electron configuration which would be 3d6

• Principal QN = 3 It is in the 3rd energy level.• Orbital QN = 2 It is a d – orbital• Magnetic QN = -2 It is filling up the d – orbital• Spin QN = - ½ It has an opposite spin from

the electron already there.