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Chapter 7 The Quantum-Mechanical Model of the

Atom

- Outline
- Intro to Quantum Mechanics
- The Nature of Light
- Atomic Spectroscopy and the Bohr Model
- The Wave of Nature of Matter
- Quantum Mechanics and the Atom
- The Shapes of Atomic Orbitals

The Behavior of the Very Small

- electrons are incredibly small

- electron behavior determines much of the behavior

of atoms

- directly observing electrons in the atom is

impossible

Intro to QM

A Theory that Explains Electron Behavior

- the quantum-mechanical model explains the manner

electrons exist and behave in atoms

- helps us understand and predict the properties of

atoms that are directly related to the behavior

of the electrons

- why some elements are metals while others are

nonmetals - why some elements are very reactive while others

are practically inert

Intro to QM

The Nature of Light

Before we introduce quantum mechanics we must

first understand a few things about light.

- light is a form of electromagnetic radiation
- composed of perpendicular oscillating waves, one

for the electric field and one for the magnetic

field

- all electromagnetic waves move through space at

the same constant speed

Light

Electromagnetic Radiation

EM radiation can be described as a wave composed

of oscillating electric and magnetic fields.

Light

Waves

What do we mean by waves ?

Light

Characterizing Waves

Maximum height above centre line (or the maximum

depth below the centre line) is the amplitude

Distance between successive peaks is called the

wavelength (?, lambda)

Number of peaks (or troughs) that pass through a

given point in a unit of time is the frequency,

(?, nu).

The difference in time between successive

occurrences of the same displacement is the

period. (t)

Light

Relating Wavelength and Frequency

- for waves traveling at the same speed, the

shorter the wavelength, the more frequently they

pass

- this means that the wavelength and frequency of

electromagnetic waves are inversely proportional

- since the speed of light is constant, if we know

wavelength we can find the frequency, and vice

versa

Light

Wavelength x Frequency speed of light

Speed of light is known and equal to 3.00 x 108

m s-1

This equation applies to all the forms of

electromagnetic radiation (not just visible

light).

Light

Types of Electromagnetic Radiation

There are many forms of EM radiation that you may

already be familiar with

Light

Light

A Closer Look at Visible Light

The colour of visible light depends on its

wavelength.

Visible light wavelengths are on the order of

100s of nm.

Light

Example Questions

1. Many cordless phones operate on signals at 600

MHz. What is the equivalent wavelength ?

2. HeliumNeon lasers (the light used to scan

your groceries at the checkout) produce light at

633 nm. What is the frequency of the lasers

light ?

Light

Properties of EM Radiation

Interference

When two sets of waves (for example water waves)

intersect, there are places where the waves

disappear and other places where the waves

persist.

Light

When the waves are in-step, (called being

in-phase) the waves add together to give the

highest crests and the deepest troughs.

When the waves are out-of-step, (called being

out-of-phase) the waves cancel each other out.

Light

Properties of EM Radiation

Diffraction

When a wave encounters an obstacle or a slit that

is comparable in size to its wavelength, it bends

around. This phenomenon is called diffraction.

Light

Light

So Light is a Wave

The Photoelectric Effect

When light strikes the surface of certain metals,

electrons are detected. This was first observed

by Heinrich Hertz in 1888 (12 years before

Plancks quantum theory).

- Electron emission only occurs when the _________

of the light exceeds a threshold value. - The number of electrons emitted depends on the

________ of the light but - The kinetic energy of the emitted electrons

depends on the __________of the light.

Light

Light

Einstein Explains the PEE

- Einstein proposed that the light energy was

delivered to the atoms in packets, called quanta

or photons

- the energy of a photon of light was directly

proportional to its frequency - __________ proportional to it wavelength
- the proportionality constant is called Plancks

Constant, (h) and has the value 6.626 x 10-34 Js

Light

- 1 photon at the threshold frequency has just

enough energy for an electron to escape the atom

- for higher frequencies, the electron absorbs more

energy than is necessary to escape

- this excess energy becomes kinetic energy of the

ejected electron

Light

Atomic Spectra

The atoms of group 1 give a characteristic colour

when placed in a flame.

Atomic Spectroscopy

Exciting Gas Atoms to Emit Light with Electrical

Energy

Atomic Spectroscopy

Emission Spectra

Atomic Spectroscopy

The Atomic Spectrum of Hydrogen

Atomic Spectroscopy

Example Question

- Use the Rydberg equation for n 3. Does it agree

with the experimental atomic spectra for hydrogen

? - Repeat the exercise for n 7. Can you explain

why this line is not observed by the human eye ?

Bohrs Model

- the electrons traveled in orbits that were a

fixed distance from the nucleus - therefore the _______of the electron was

proportional to the distance the orbital was from

the nucleus

- Niels Bohr proposed that the electrons could only

have very specific amounts of energy

- electrons emitted radiation when they jumped

from an orbit with higher energy down to an orbit

with lower energy

Bohrs Model of the Atom

Bohrs Model of the Atom

The Wave Nature of Matter

if electrons behave like particles, there should

only be two bright spots on the target

Wave Nature of Matter

- de Broglie proposed that ____particles could have

wave-like character - Incredibly, electrons which we were thought of as

negatively charged _______also exhibit ________

properties - because it is so small, the wave character of

electrons is significant - de Broglie predicted that the wavelength of a

particle was _________ proportional to its

momentum

Wave Nature of Matter

Examples

1. Calculate the de Broglie length of an electron

travelling at one-tenth the speed of light.

2. In last nights ALCS, a fastball was clocked

at 97 miles an hour (43 m/s). Given that a

baseball weighs 145 g. Calculate the de Broglie

length of his fastball and comment on whether

that was a feasible reason why the batters

couldnt hit the pitches.

Uncertainty Principle

- Heisenberg stated that the product of the

uncertainties in both the position and speed of a

particle was inversely proportional to its mass - x position,
- v velocity,
- m mass
- the means that the more accurately you know the

position of a small particle, like an electron,

the less you know about its speed - and vice-versa

Wave Nature of Matter

Quantum Mechanics

Standing waves are waves where the magnitude of

the oscillation is different from point to point

along the wave. Points that undergo no

displacement are called nodes.

Consider a plucked guitar string of length, l.

n Number of nodes

1

2

3

Particle in a Box (PIAB)

Schrödinger suggested that if an electron in an

atom has wave-like properties then it should be

describable using a mathematical equation called

a wavefunction (?). The wavefunction must be a

solution to Schrödingers equation The

wavefunction should correspond to a standing wave

within the boundary of the system being

described..

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1. The energy of the particle in a 1D PIAB is

quantized.

- The minimum energy of the particle in a 1D PIAB

is never zero

3. n is called the __________ quantum number

4. The square of the wavefunction, ?2, at a given

point in space represents the __________ of

finding the particle there.

Exercise

Use de Broglies equation for matter waves, the

fact that the kinetic energy of a particle is

given by the following expression

and the equation for the wavelength of a standing

wave

to derive the equation for the energy of a 1D

PIAB.

Quantum Mechanics

- The energy of an electron dictates the properties

of an element. For example, bonding.

- However, if we very accurately know the energy of

an electron, Heisenberg says we cant precisely

know its position.

- for an electron with a given energy, the best we

can do is describe a region in the atom of high

probability of finding it

- To determine the energy of an electron the

Schrödinger equation must be solved.

Quantum Mechanics

Wave Function, y

- A wavefunction, ?, is just a mathematical

function that is a solution to the Schrödinger

equation.

- The square of the wavefunction, ?2 gives a

probability map of finding the electron in a

region of space.

- calculations show that the size, shape and

orientation in space of an orbital are determined

be three integer terms in the wave function

- these integers are called quantum numbers
- __________quantum number, n
- __________ momentum quantum number, l
- __________ quantum number, ml

Quantum Mechanics

Principal Quantum Number, n

- characterizes the energy of the electron in a

particular orbital

- n can be any integer ³ 1
- the larger the value of n,

- energies are defined as being negative
- the larger the value of n, the larger the orbital

Quantum Mechanics

Angular Momentum Quantum Number, l

- The angular quantum number is an integer that

determines the shape of the orbital (see later).

- Possible values for l are 0,1,2,,(n-1).

Value of l Letter Designation

0

1

2

3

Quantum Mechanics

Magnetic Quantum Number, ml

- The magnetic quantum number is an integer that

determines the orientation of the orbital (see

later).

- Possible values for ml are l, (l-1), (l-2)-l.

- Each specific combination of n,l,ml specifies one

atomic orbital.

- Orbitals with the same principal quantum number

are said to be in the same principal level

(shell).

- Orbitals with the same value of n and m are said

to be in the same sublevel (subshell).

Quantum Mechanics

Example

Levels and Sublevels

Quantum Mechanics

Orbital energies for a hydrogen atom depend only

on the principal quantum number n. This means

that all the subshells within a principal shell

have the same energy. Orbitals at the same energy

level are said to be __________.

Electronic Orbitals of Hydrogen

Quantum Mechanics

Example

Question 32 from Tro (Chapter 7 End of Chapter

Problems)

List all the orbitals in each of the following

principal levels. Specify the three quantum

numbers for each orbital.

- n 1
- n 2
- n 3
- n 4

Quantum Mechanics

Principal Energy Levels in Hydrogen

Quantum Mechanics

The Hydrogen Spectrum Explained !

- both the Bohr and Quantum Mechanical Models can

predict these lines very accurately

Quantum Mechanics

Quantum Mechanics

The Shapes of Atomic Orbitals

Recall that ?2 gives the probability density

The Shapes of Atomic Orbitals

The Radial Distribution Function

Function Meaning

Prob. density Probability of finding e- at a _______ r

RDF Probability of finding e- at a _______r

The Shapes of Atomic Orbitals

A node is a point where both ? and ?2 all equal

zero.

The ns orbitals (n gt 1) are spherically symmetric

like the 1s orbital. They are just bigger and

have nodes.

The Shapes of Atomic Orbitals

The Shapes of Atomic Orbitals

p Orbitals (l 1)

There are three types of p orbitals. Each

corresponds to a different ml quantum number.

The Shapes of Atomic Orbitals

d Orbitals (l 2)

The Shapes of Atomic Orbitals

Example

Write an orbital designation corresponding to the

quantum numbers n 4, l 2, ml 0.

Write an orbital designation corresponding to the

quantum numbers n 3, l 1, ml 1.