1 Chapter 7 QUANTUM THEORY & ATOMIC …streaming.missioncollege.org/atran/media/CHEM_001A_38367/...the Quantum Theory of Energy - Any object (including atoms) can emit or absorb only

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© 2006 Brooks/Cole - Thomson

Chapter 7 QUANTUM THEORY & ATOMIC

STRUCTURE

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7.1 The Nature of Light

• Most subatomic particles behave as PARTICLES and obey the physics of waves.

• Light is a type of electromagnetic radiation

• Light consists of energy particles called photons that travel as waves.

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© 2006 Brooks/Cole - Thomson 3

Wavelength (λ) The distance a wave travels in one cycle The distance between two corresponding points

on a wave Units are meters (m) or commonly nanometers

(nm = 10-9 m)

The Wave Nature of Light

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Frequency (ν) The number of wave cycles that move through a

point in space in 1 sescond Units are hertz (Hz) which are the same as

inverse seconds (1/s)


Long wavelength --> low

frequency

Short wavelength --> high

frequency

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Amplitude - The height of the crest (or depth of a trough) - Indication of the light intensity or brightness

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Long wavelength --> low freq.

Short wavelength --> high freq.


wavelength

wavelength Node

Speed The distance the wave moves per unit time (m/s) The product of its frequency (cycles per second)

and wavelength (m/s)

• = c

In a vacuum, speed of light: c = 3.00 x 108 m/sec

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Electromagnetic Spectrum arranges wavelength from shortest to longest

arranges frequency from highest to lowest

shows visible light with wavelengths from 400–700 nm

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© 2006 Brooks/Cole - Thomson p. 273

Wavelength (), Frequency () & Energy (Ephoton)

= C C = 3.00 x 108 m/s (speed of light)

h = 6.626 × 10-34 J.s (Planck’s constant)

Photons are packets of light energy

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Red light has = 700. nm; calculate the frequency and energy of a photon.

Freq = 3.00 x 108 m/s

7.00 x 10-7 m 4.29 x 1014 sec-1

700 nm • 1 x 10 -9 m

1 nm = 7.00 x 10 -7 m

Ephoton = h•

= (6.63 x 10-34 J•s)(4.29 x 1014 s-1)

= 2.85 x 10-19 J per photon

Problem

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Figure 7.6 Familiar examples of light emission related to

blackbody radiation.

Lightbulb filament Smoldering coal Electric heating element

The Particle Nature of Light Black body radiation

A solid object emits visible light when it is heated to

about 1000 K. The intensity ) of the light changes as

the temperature changes.

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The Particle Nature of Light Black body radiation &

the Quantum Theory of Energy

- Any object (including atoms) can emit or absorb

only certain quantities of energy.

- Energy is quantized; it occurs in fixed quantities,

rather than being continuous. Each fixed quantity of

energy is called a quantum.

- An atom changes its energy state by emitting or

absorbing one or more quanta of energy.

DE = n • h • n can only be a whole number

h = 6.6262 x 10-34 J•s (the Planck’s constant) Max Planck

(1858-1947)

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Problem

Ephoton = h•

= (6.63 x 10-34 J•s)(4.29 x 1014 s-1)

= 2.85 x 10-19 J per photon

E per mol =

(2.85 x 10-19 J/ph)(6.02 x 1023 photons/mol)

E = 171.6 kJ/mol

Calculate the energy of 1.00 mol of photons of red

light (wavelength 700. nm, frequency 4.29 x 1014

sec-1).

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7.2 Atomic Spectra

•Excited atoms can emit light characteristic of that element.

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Spectrum of White Light

White light (sunlight, or light from regular light

bulbs) that passes through a prism

is separated into all colors that together are

called a continuous spectrum gives the colors of a rainbow

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Line Spectra of Excited Atoms & the Rhydberg Equation

• Excited atoms emit light of only certain wavelengths

Line Spectrum of Excited Hydrogen Gas

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Line Spectra of Other Elements in the Gas Phase

Line Spectra of Excited Atoms & the Rhydberg Equation

The wavelengths of emitted light are specific for

the element.

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Three series of spectral lines of atomic hydrogen.

The Rydberg Equation

is wavelength of the line

R is Rydberg’s constant: 1.0967776 x 107 m-1

n1 and n2 are positive integers with n2 > n1

for the visible series, n1 = 2 and n2 = 3, 4, 5, ...

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The Bohr Model of Hydrogen Atom

The H atom has only certain energy

levels, which Bohr called stationary states.

- Each state is associated with a fixed circular orbit of the electron around the nucleus.

- The higher the energy level, the farther the orbit is from the nucleus.

- When the H electron is in the first orbit, the atom is in its lowest energy state, called the ground state.

Bohr’s atomic model postulated the following:

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The atom does not radiate energy while in one of its stationary states.

The atom changes to another stationary state only by absorbing or emitting a photon.

ΔE = Efinal Einitial

- The energy of the photon (h) equals the difference between the energies of the two energy states.

Ephoton = ΔE = h

When the E electron is in any orbit higher than n = 1, the atom is in an

excited state.

The Bohr Model of Hydrogen Atom

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Features of the Bohr Model Quantum numbers are integers: n = 1, 2, 3, …

Radius of electron orbit directly relates to the

electron’s energy: the lower the n value, the

smaller the electron orbit, and the lower the

energy level.

If electrons are in quantized

energy states, then ∆E of

states can have only

certain values. This

explains sharp line spectra.

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Features of the Bohr Model

Ground state: when the electron is in the lowest

possible orbit, which is closest to the

nucleus.

Excited state: when the electron is any orblt

farther from the nucleus.

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Absorption when H atom absorbs a photon whose energy equals the difference between the lower and higher energy levels, then the electron moves to the higher orbit. Emission when H atom in a higher energy level returns to a lower energy level, then the atom emits a photon whose energy equals the difference between the two levels.

Features of the Bohr Model

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The Bohr explanation of three series of

spectral lines emitted by the H atom

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The Energy Levels of the Hydrogen Atom

For an energy level n:

For a transition between two energy levels

The Bohr Equation:

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Problem

Solution:

Energy of transition:

∆E = Efinal - Einitial = -2.18 x 10 -18 J [(1/12) - (1/2)2]

= -1.635 x 10-18 J ==> EMISSION PROCESS

Energy of emitted light

Ephoton = hC/ = 1.635 x 10-18 J

Wavelength of this light: = 121.6 nm

This is exactly in agreement with experiment!

What is the wavelength corresponding to the

transition of electron from n =2 to n = 1?

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7.3 The Wave-Particle Duality of

Matter and Energy Matter and Energy are alternate forms of the same entity.

E = mc2

All matter exhibits properties of both particles and waves.

Electrons have wave-like motion and therefore have only

certain allowable frequencies and energies.

Matter behaves as though it moves in a wave, and the

de Broglie wavelength for any particle is given by:

m = mass in kg

u = speed in m/s muhλ

L. de Broglie

(1892-1987)

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Problem Determine the de Broglie wavelength of

(a) an electron with a speed of 1.00 x 106 m/s.

[electron mass = 9.11 x 10-11 kg].

(b) a lithium atom moving at 2.5 x 105 m/s

Solution:

(a)

(b)

mxJ

smkg

smxkgx

sJx

mv

h 1022

631

34

1027.71

/.1

)/1000.1)(1011.9(

.10626.6

mass of a lithium atom:

gxatomsLix

mol

molLi

g 23

2310152.1

10023.6

1

1

941.6

mxJ

smkg

smxkgx

sJx

mv

h 1322

526

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103.21

/.1

)/105.2)(10152.1(

.10626.6

This wavelength is within the range of g-rays (10-12 – 10-17 m)

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7.4 The Quantum-Mechanical Model

of the Atom - Electron behaves simultaneously as a wave and a

particle. The matter-wave of the electron occupies the space

near the nucleus and is continuously influenced by it.

- The Schrödinger wave equation allows us to solve for the

energy states associated with a particular atomic orbital. The

wave function (or atomic orbital) is a mathematical

description of the electron’s matter-wave in 3-D — the region of

3D-space within which an electron is most likely to be found,

and NOT the path the electron follows.

- The square of the wave function 2 gives the probability

density, a measure of the probability of finding an electron of

a particular energy in a particular region of the atom. It

describes the shape of an orbital.

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Figure 7.16

Electron probability density in

the ground-state H atom.

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Quantum Numbers of an Atomic Orbital

An atomic orbital is specified by three quantum numbers:

n (principal): 1, 2, 3, ….

l (angular momentum): 0 to n-1

ml (magnetic): -l to 0 to +l

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Quantum Numbers

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Quantum Numbers & Energy Levels The energy states and orbital of an atom are

associated with one or more quantum numbers:

Level (or shell) given by the n values. Each designates

the energy (size) of the electron. The lower the n value means the greater probability that the electron is closer to the nucleus.

Sublevel (or subshell) given by the l values. Each

designates the orbital shape with a letter:

l = 0 is an s sublevel l = 1 is a p sublevel

l = 2 is a d sublevel l = 3 is an f sublevel

Name a subshell by its n value with the designated subshell letter.

Orbital given by the ml values. Each designates the orbital orientation.

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Problems

a) What are the n, l, ml values for the 3d and 5f sublevels?

b) Identify the incorrect quantum numbers:

n l ml Sublevel

name

a) 2 1 +1 2p

b) 1 0 -1 1s

c) 5 2 0 5d

c) Fill in the quantum numbers or sublevel names:

n l ml Sublevel

name

a) 4 1 +1 ?

b) ? 3 -2 6f

c) 5 0 ? 5s

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Shapes of Atomic Orbitals

s orbital l=0, no node

p orbital l=1, 1 nodal

plane

d orbital l=2, 2 nodal

planes

f orbital l=3, 3 nodal

planes

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Size and Shape of Atomic Orbitals

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Figure 7.21 Energy levels of the H atom.

The Special Case: Energy Level in Hydrogen Atom

Hydrogen is

the only atom

whose energy

state depends

completely on

the principal

quantum

number n.

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1 Chapter 7 QUANTUM THEORY & ATOMIC …streaming.missioncollege.org/atran/media/CHEM_001A_38367/...the Quantum Theory of Energy - Any object (including atoms) can emit or absorb only