Quantum Mechanical Theory

Learning Goal: I will understand the quantum mechanical model of the atom, be able to determine an atom’s set of quantum numbers, electron configuration and orbital diagram

Dalton’s Atomic Theory (1805)Experimental Work

Theoretical Explanation

Atomic Theory

Law of definite proportions: elements combine in a characteristic mass ratio.

Each atom has a particular combining capacity.

Matter is composed of indestructible, indivisible atoms, which are identical for one element, but different from other elements.

Law of multiple proportions: there may be more than one mass ratio.

Some atoms have more than one combining capacity.

Law of conservation of mass: total mass remains constant

Atoms are neither created nor destroyed in a chemical reaction.

Thomson Atomic Theory (1897)Experimental Work Theoretical

ExplanationAtomic Theory

Arrhenius: the electrical nature of chemical solutions.

Atoms may gain or lose electrons to form ions in solution.

Matter is composed of atoms that contain electrons (negative particles) embedded in a positive material. The kind of element is characterised by the number of electrons in the atom.“The cookie model for the atom.”

Faraday: quantitative work with electricity & solutions.

Particular atom and ions gain or lose a specific number of electrons.

Crookes: qualitative studies with the cathode ray.

Electricity is composed of negatively charged particles.

Thomson: quantitative studies with the cathode ray.

Electrons are a component of all matter.

Millikan: charged oil drop experiment

Electrons have a specific fixed electric charge.

Rutherford Atomic Theory (1911)Experimental Work

Theoretical Explanation

Atomic Theory

Rutherford: A few positive alpha particles are deflected at large angles when fired at gold foil.

The positive charge in the atom must be concentrated in a very small volume of the atom.

An atom is comprised of a very tiny nucleus which contains positive charges and most of the mass of the atom. Very small negative electrons occupy most of the volume of the atom.

Most materials are very stable and do not self destruct or disintegrate.

A very strong force holds the positive charges together within the nucleus. (Strong nuclear force)

Rutherford: Most alpha particles pass straight through gold foil.

Most of the atom is empty space.

The Limitations of Rutherford’s Atomic Model

Based on the understanding of physics at the time, for an electron in motion around a central core:

• If an atom absorbed radiation (light, UV, x-rays, etc), radiation must be emitted, so it was expected that a continuous spectrum of light energy was being given off

• because of radiation, the electron would lose energy and its orbit would decrease until it spiraled into the nucleus, destroying the atom

Rethinking Atomic Structure Based on the Nature of Energy

Light is one form of electromagnetic radiation, whichtravels through space as waves

Electromagnetic waves:• have frequency, wavelength, and amplitude• interact with matter in discrete particles called photons

When atoms are excited due to absorption of energy, they emit light as they lose energy and return to a non-excited state.

Atomic Spectra

Each element has a characteristic line spectrum.

Atoms of each element emit light of particular wavelengths called a line spectrum or emission spectrum.

The Bohr Model of the Hydrogen Atom

• are in circular orbits• can only exist in certain “allowed” orbits or energy levels (energy of

electrons is quantized)• do not radiate energy while in one orbit• can jump between orbits by gaining or losing a specific amount of energy

Niels Bohr set out to explain the stability of the nuclear model of the atom. In this model, electrons

Bohr’s Atomic Model

Hydrogen’s

Energy Level Diagram

n = 1

n = 2

n = 3

e-

Ground State

Energy is added to the gas sample.

e-

A quantum of energy is

absorbed by the electron and it

undergoes “transition”

Excited State

A quantum of energy, equal to

the absorbed quantity, is

released as a “photon” (EMR)

when the electron returns

to its ground state.

Bohr’s explanation of Spectra

Hydrogen’s

Energy Level Diagram

n = 1

n = 2

n = 3 e-Emission Line

Spectrum

IR R O Y G B V UV

Bohr’s Atomic Model Explains the Line Spectrum of Hydrogen

• Calculated wavelengths of the possible energies of photons that could be emitted from an excited hydrogen atom (transitions from n = 6, 5, 4, and 3 to n = 2) corresponded with hydrogen’s visible line spectrum

Today’s quantum mechanical model of the atom incorporates the wave properties of electrons.

The Quantum Mechanical Model of the Atom

An electron density diagram represents an atomic orbital.

Wave functions, initially described by Erwin Schrodinger, represent a region in space around a nucleus where an electron will be found. This region of space is called an atomic orbital

The Quantum Mechanical Model of the Atom

The circle does not represent a real boundary.

Atomic orbitals can be visualized as “fuzzy clouds” • The higher the density of the “cloud,” the higher the probability of finding

an electron at that point.• The cloud has no definite boundary.• The region where an electron will spend 90 percent of its time is depicted by

drawing a circle.

Heisenberg’s Uncertainty principle

•Electrons behave as both waves and particles

• Heisenberg determined that it is impossible to know BOTH the exact position and velocity of an electron.

W. Heisenberg1901 - 1976

Quantum Mechanics

• Wave function – mathematical description of an orbital in an atom where an electron of a certain energy is likely to be found.

• Electron probability density – indicates regions around the nucleus with the greatest probability of finding an electron.

▫ 3D shapes of the atomic orbitals▫ “cloud of electron density” E.

Schrodinger1887 - 1961

Quantum Numbers Describe Orbitals

Electrons in the quantum mechanical model of the atom are described using quantum numbers.

Three quantum numbers describe the distribution of electrons in the atom and a fourth describes the behaviour of each electron.

Symbols for the four quantum numbers:

n l ml ms

The Principle Quantum Number, n

• Is the first quantum number• Describes the energy level, or shell, of an orbital • All orbitals with the same n value are in the same shell• The larger the n value, the larger the size of the shell• Values can range from n = 1 to n = ∞

n = 1 first shelln = 2 second shelln = 3 third shelln = 4 fourth shell

The Orbital-Shape Quantum Number, l• Is the second quantum number• Describes the shape of an orbital• Refers to energy sublevels, or subshells• Values depend on the value of n. They are positive integers from 0 to (n – 1)• Each value is identified by a letter

l = 0 orbital sl = 1 orbital pl = 2 orbital dl = 3 orbital f

An energy sublevel is identified by combining n with the orbital letter. For example, n = 2, l = 1: 2p sublevel

Shapes of Orbitals• The act of measuring or identifying an electron requires

us to interfere with its motion and energy.▫This interference may result in an inaccuracy in identifying

the very characteristic we intended to measure. • The act of measuring the location of an electron requires

the absorption or release of its energy, which in turn affects its speed and maybe location. ▫Heisenburg’s uncertainty principle indicates that it is

impossible to know the exact position and speed of a particle at the same time.

• Such an understanding reinforces a sense of probability in determining the location of an electron.

•Schrodinger’s wave equations can be used to predict the likelihood of “finding” and electron at a specific location. These probabilities can be used to plot an electron probability density.▫The second quantum number, l, indicates the variety in

“shape” of the orbital. (s – sharp, p – principal, d – diffuse, f – fundamental) Recent theory suggests that there may be g-orbitals!?!

•An orbital is associated with a size, three dimensional shape and orientation around the nucleus.▫Together the size, shape and position represent the

probability of finding a specific electron at that location.

Shape of orbitals

•The probability density plot can be assessed in multiple dimensions to generate a 3-d density cloud or “shape” for the orbital.

Shape of orbitals

p-orbital

s-orbital

d-orbital

Shape of orbitals

• The culmination of all of the electrons produces a combined effect from all involved orbitals to generate a unique “orbital shape” for each atom. For example, the structure of Ne . . .

Shape of orbitals•The diagram we used to represent

oxygen is;

8 Proton

s-

-

-

-

-

-

-

-

8

16O

Shape of orbitals The diagram we might currently use to

represent oxygen is;

Types of orbitals: s

•When l = 0, the orbital is called s

Types of Orbitals (l)

•When l = 1, the orbital is called p

p orbitals• The p sublevel has 3 orbitals• The three p orbitals lie 90 degrees

apart in space• They are designated px, py, pz for the

axis

d orbitals• When l = 2, the orbital is called d•d sublevel has 5 orbitals

f orbitals• What are the values for ml when l = 3?• When l = 3, the orbital is called f

The Magnetic Quantum Number, ml

• Is the third quantum number • Indicates the orientation of the

orbital in space• Named because in a magnetic

field, these different orientations have different energies

• For a given l there are (2l +1) values for ml

• How do we know p orbitals have 3 sublevels?▫ Magnetic number represents the

sublevels

▫ ml = -l to +l

▫ So for the p orbital l = 1, possible sublevels are -1, 0, 1

s, p, and d orbitals have characteristic shapes.

The Spin Quantum Number, ms

• Is the fourth quantum number • Specifies the orientation of the axis of electron spin• Two possible values: +½ or –½• e- spin in their orbits and as they spin they generate a magnetic field• e- can spin either clockwise or counterclockwise

To summarize:

Identifying Electrons Using Sets of Quantum Numbers

According to the Pauli exclusion principle:• an orbital can have a maximum of two electrons• two electrons in an orbital must have opposite spins

No two electrons of an atom have the same set of four quantum numbers.

Summary of Quantum Numbers

Principal quantum number, n: the main electron energy level or shell (n )

Secondary quantum number, l: the electron sublevels or subshells (0 to n-1)

Magnetic quantum number, ml: the orientation of the sublevel (-l to +l)

Spin quantum number, ms: the electron spin (-1/2 to +1/2)

Energy shell Orbital shape Orbital orientation

Electron Spin

1 0 0 +1/2,-1/2

201

0-1,0,+1

+1/2,-1/2+1/2,-1/2

3012

0-1,0,+1

-2,-1,0,+1,+2

+1/2,-1/2+1/2,-1/2+1/2,-1/2

Try it!

•Write a set of quantum numbers for the 2 electrons in a 1s orbtial

1 s

Tells us the principal quantum number (n) is 1

Tells us the shape of the orbital (l) is s

n = 1l = 0ml = 0ms = +1/2

n = 1l = 0ml = 0ms = -1/2

Try some more!•Write a set of quantum numbers

for an electron in a 3p orbital3 p

Tells us the principal quantum number (n) is 3

Tells us the shape of the orbital (l) is p

n = 3l = 1ml = -1ms = +1/2 n = 3l = 1ml = -1ms = -1/2

n = 3l = 1ml = 0ms = +1/2 n = 3l = 1ml = 0ms = -1/2

n = 3l = 1ml = +1ms = +1/2 n = 3l = 1ml = +1ms = -1/2

Representing Electrons: Electron Configurations and Orbital Diagrams

The electron configuration for hydrogen

The electron configuration for an atom shows the number and arrangement of its electrons, in the ground state.

An orbital diagram uses boxes or lines to represent orbitals at each n and shows electron spin.

Orbital diagrams often accompany electron configurations.

Identify the atom.

Three rules are used to build Energy Level Diagrams:

▫Aufbau principle

▫Pauli Exclusion Principle

▫Hund’s Rule

Aufbau Principle (German for “building up’)

•Electrons occupy orbitals of lower energy first.

-Pauli Exclusion Principle(Wolfgang Pauli, Austria, 1900-1958)

•An orbital can hold only two electrons and they must have opposite spin.

•Electron Spin Quantum Number (ms):

+1/2, -1/2

Hund’s Rule

In a set of orbitals, the electrons will fill the orbitals in a way that would give the maximum number of parallel spins (maximum number of unpaired electrons).one electron is placed in each orbital at the same energy level before the second electron is placed

Analogy: Students could fill each seat of a school bus, one person at a time, before doubling up.

Conventions for Creating Energy-Level Diagrams

•Circles or squares are used to represent the orbitals•Arrows are used to represent the electrons

▫up represents one electron rotation (clockwise) while down the other (counter clockwise)

▫there is no convention as to which one you must start with

Conventions for Creating Energy-Level Diagrams

1s 2s 2p 3s 3p

O (z = 8)

1s 2s 2p 3s 3p

P (z = 15)

1s 2s 2p 3s 3p

Ar (z = 18)

Conventions for Creating Energy-Level Diagrams

•Energy-level diagrams may be written in a vertical manner to exemplify the energy level subtleties.

Ord

er

of

fill

ing

orb

itals

1s

2s

3s

4s

5s

6s

2p

3p

4p

6p5d

4f

5p4d

3d

2e-

2e-

6e-

2e-

6e-

2e-10e

-

6e-

2e-10e

-

6e-

2e-14e

-

10e-

6e-

2e-

8e-

8e-

18e-

18e-

32e-

•As the number of energy levels and orbitals increase, so too does the complexity of the energy-level diagram.

•The diagram indicates nicely the order in which the orbitals are filled

Energy-Level Diagrams for ions

▫For anions – add the proper number of electrons using regular conventions

▫For cations – remove the correct number of ions in the proper manner

1s 2s 2p 3s 3p

S 2- (z = 16)

1s 2s 2p 3s 3p

Al 3+

(z = 13)

Electron configurations• The energy-level diagram is the best way to

visualize the energy relationships between electrons. However it may prove cumbersome.

• Electron configurations show the same information in a more concise manner.

Electron configurations

becomes . . .Cl: 1s2 2s2 2p6 3s2 3p5

In the shorthand form of electron configurations, the previous noble gas structure is used to reflect the lower energy level electrons. So . . .

Cl: 1s2 2s2 2p6 3s2 3p5

becomes . . . Cl: [Ne] 3s2 3p5

1s 2s 2p 3s 3p

Cl (z = 17)

•Completely filled orbitals are more stable than half filled orbitals which are more stable than partially filled orbitals.

> >

•As the principle energy level increases the difference between each orbital’s energy level comes to the point where the d-orbitals energies are very close to the s-orbital’s.

•In situations like this the “s-” and “d-orbitals” may be treated as a similar energy level and the application of Hund’s rule may be applied over the range of orbitals rather than the single second quantum orbital.

•The stability of having each orbital containing one electron is greater than having a partially filled group of orbitals and a filled orbital of only minimally less energy.

Anomalies in Electron Configurations

Anomalies in Electron Configurations •Vanadium has the electron configuration of –

V (z=23) [Ar] 4s2 3d3

•Chromium has one extra electron and conventions would predict it to be added to the next open d-orbital

Cr (z=24) [Ar] 4s2 3d4

•However, the stability of having one electron in each orbital of similar takes president and the true electron configuration becomes

Cr (z=24) [Ar] 4s1 3d5

n=3

n=4

s p d

Explaining the multivalent ions •Many transitional metals have the ability to have multiple charges

as ions and the explanation for this behaviour has yet to be explained.

•As orbitals are being filled there are varying levels of stability due to the interaction of forces or electrostatic repulsion between electrons and force associates with the magnetic field due to the electron’s spin.

•Filled orbitals are most stable because the electrostatic repulsion is balanced against the magnetic attraction.

e- e-Electrostatic repulsion

e-South

Magnetic field

Magnetic attractione-North

Magnetic field

Explaining the multivalent ions •Orbitals that are completely filled, like Noble gases, have the most

stable structure due to the balanced forces between electrostatic repulsion and magnetic attraction.

•Transitional metals often have partially filled d-orbitals and complete s-orbitals with similar energy levels. Electrons are lost to achieve the best combination of stability.

• Best stability – completely filled orbitals (2 electrons/orbital)Electrostatic repulsion and Magnetic attraction

• Next best stability – half filled orbitals (1 electron/orbital)Electrostatic repulsion with minimal crowding

Co: [Ar] 4s2 3d7

Co: [Ar]

Co2+: [Ar]

Co3+: [Ar]

- Filled 4s and half filled 3d

- Half filled 4s & 3d less stable than Co2+ so less common ion

Explaining Magnetism •Some materials exhibit strong magnetic properties naturally

these are referred to as being ferromagnetic.

•As moving charges have magnetic fields so too do spinning electrons. Those spinning in one direction would have one magnetic polarity as those spinning in the opposite direction the opposing magnetic pole.

•For example – clockwise – south pole and counter clockwise –north pole

•Those atoms that have a number of similarly spinning electrons would have similar magnetic fields.

•If these “magnetic atoms” were free to align themselves with neighbouring atoms of similar characteristics they would create regions of magnetism in the material called domains.

•The arrangement of these domains in a material results in a magnet.

Atoms themselves have magnetic properties due to the spin of the atom’s electrons.

These areas of atoms are called “domains”

Groups of atoms join so that their magnetic fields are all going in the same direction

When an unmagnetized substance is placed in a magneticfield, the substance can become magnetized.

This happens when the spinning electrons line up in thesame direction.

Explaining Magnetism Iron, nickel and cobalt are such naturally occurring atoms.

Iron, nickel and cobalt are small enough atoms that they can realign themselves due to the magnetic properties of their surrounding and thereby create domains.

There are other such atoms that have similar electron configurations but limited ability to migrate. Hence, they are reduced in their magnetic properties.

Explaining Magnetism Ferromagnetic – materials with strong magnetic properties. Their presence increases a magnetic field substantially.

Paramagnetic – materials with weak magnetic properties. Their presence only slightly strengthens a magnetic field.

Diamagnetic – materials that have reduced magnetic properties. Their presence weakens a magnetic field.