Chapter Six (1)

Chapter 6 Slide 1 of 82

Chapter Six (1)

Atomic Structure


Introduction

The chapter consists of three parts:

–The discovery of the electron and the determination of its properties

–Understanding the nature of light

–The behavior of electrons in atoms


Cathode Rays

Electric Discharge Tube


Cathode Rays• Cathode rays are the carriers of electric current from

cathode to anode inside a vacuumed tube• Cathode rays have the following characteristics:

– Emit from the cathode when electricity is passed through an evacuated tube

– Emit in a direction perpendicular to the cathode surface

– Travel in straight lines– Cause glass and other materials to fluoresce– Deflect in a magnetic field similarly to negatively

charged particles


Cathode rays deflect in a magnetic field

magnet

N

S


Thomson’s Mass-to-Charge Ratio Apparatus

me /e = –5.686 X 10-12 kg/C


Millikan’s Oil Drop Experiment

e = -1.602 X 10-19 C


Determined Values

• J.J. Thomson devises an experiment telling the ratio of the cathode ray particle’s mass to the charge expressed:

me /e = –5.686 X 10-12 kg/C• George Stoney names the particle an electron• Robert Millikan then determines a value for

the chargee = -1.602 X 10-19 C

• From these two values the mass of an electron:me = 9.109 X 10-31 kg/electron


J.J. Thomson’s Model• J.J. Thomson proposed an atom with a positively

charged sphere containing equally spaced electrons inside– Proposed for a Hydrogen atom that there was one

electron at the exact center of the sphere– Proposed for a Helium atom, that two electrons

existed along a straight line through the center, with each electron being halfway between the center and the outer surface of the sphere

– Applied this analysis to atoms with up to 100 electrons


Thomson’sRaisin Pudding Model


Rutherford’s Alpha Scattering Experiment


Rutherford’s Model

• Ernest Rutherford discovered the positive charge of an atom is concentrated in the center of an atom,the nucleus

– An atom, can be visualized as a giant indoor football stadium

– The nucleus can be represented by a pea in the center of the stadium,

– The electrons are a few bees buzzing throughout. The roof of the stadium prevents the bees from leaving.


The Wave Nature of Light

• Electromagnetic waves originate from the movement of electric charges– The movement produces fluctuations in electric

and magnetic fields– Electromagnetic waves require no medium

• Electromagnetic radiation is characterized by its wavelength, frequency, and amplitude.


An Electromagnetic Wave


Wavelength And Frequency

• Wavelength (λ ) is the distance between any two identical points in consecutive cycles, and is measured in nanometers and angstroms

• Frequency (v ) of a wave is the number of cycles of the wave that pass through a point in a unit of time, and is measured in hertz (s-1)

• Amplitude (I) of a wave is its height: the distance from a line of no disturbance through the center of the wave peak.


The Electromagnetic Spectrum


Continuous and Line Spectra• White light passed through a prism produces a spectrum of

rainbow colors in continuous form. The different colors of light correspond to different wavelengths and frequencies.

• When light is produced through an element, a discontinuousspectrum is displayed.

• The pattern of lines produced by the light emitted by excited atoms of an element is call a line spectrum.

• Emission Spectroscopy is the analysis of light emitted from a strongly heated or energized element

• A photograph or other record of the emitted light is called emission spectrum.


Continuous Spectra

Line Spectra


Emission Spectrum of Hydrogenin Visible Light Region


Emission Spectrum of Helium


Line Spectra of Some Elements


Planck’s Constant

• Planck’s quantum hypothesis states that energy can be absorbed or emitted only as a quantum or as whole multiples of a quantum, thereby making variations discontinuous, changes can only occur in discrete amounts.

• The smallest amount of energy, a quantum, is given by:

E = hvas Planck’s constant, h = 6.626 X 10-34 J s.


The Photoelectric Effect• Albert Einstein considered electromagnetic energy

to be bundled in to little packets called photons.Energy of photon = E = hv

– Photons of light hit surface electrons and transfer their energy hv = B.E. + K.E.

– The energized electrons overcome their attraction and escape from the surface

hv e- (K.E.)


Photoelectric Effect Illustrated


Bohr’s Hydrogen AtomPostulations• Rutherford’s nuclei model• The energy of an electron in a H atom is quantized• Planck & Einstein’s photon theory

E = hv• Electron travels in a circle• Classical electromagnetic theory is not applied

ve-rZ • Orbit

(1) Classical physics

centripetal force = Coulombic attractionmv2/r = Ze2/r2

(2) Total energy

E = 1/2 mv2 - Ze2/r

(3) Quantizing the angular momentum

mvr = n (h/2π )

E = - ( 2π2 mZ2e4)/(n2 h2)

when n =1, E(1)= - (2π2 mZ2e4)/( h2) E = E (1) /n2

r = (n2 h2)/ (4π2 mZe2)

Quantum number


Bohr’s Hydrogen Atom

• Niels Bohr found that the electron energy (En) was quantized, that is, that it can have only certain specified values.

• Each specified energy value is called an energy level of the atom

En = - B/n2

– n is an integer, and B is a constant which equals 2.179 x 10-18 J

– The energy is zero when the electron is located infinitely far from nucleus

– The negative sign represents the forces of attraction


The Bohr Model


Bohr Explains Line Spectra

• Bohr’s equation is most useful in determining the energy change (∆Elevel) that accompanies the leap of an electron from one energy level to another

• For the final and initial levels:Ef = -B / nf

2 Ei = -B / ni2

The energy difference between nf and ni is:

∆Elevel = Ef - Ei

= ( -B / nf2 ) – (-B / nf

2 )= B(1/ni

2 – 1/nf2)


Energy Levels and Spectral Lines for Hydrogen


Ground States and Excited States

• When an atom has its electrons in their lowest possible energy levels, it is in its ground state

• When an electron has been promoted to a higher level, it is in an excited state– Electrons are promoted through an electric

discharge, heat, or some other source of energy– An atom in an excited state eventually emits

photons as the electron drops back down to the ground state


λ= ? nm


Problems of Bohr’s Model of Atom

• The energy levels of Bohr’s H atom cannotbe applied to other atoms.

• The orbit of electrons cannot of defined.


De Broglie’s Equation• Louis de Broglie speculated that matter can behave

as both particles and waves, just like light• He proposed that a particle move with a mass m

moving at a speed v will have a wave nature consistent with a wavelength given by the equation:

p = mcE = mc2 = pc = hνp = hν /c = h/ λ

• De Broglie’s prediction of matter waves led to the development of the electron microscope

λ = h/p = h/mv


Experimental Determinationof Crystal Structures

Bragg’s Law2 d sinθ = n λ


X-Ray Diffraction Image & Pattern

Single crystal Powder


What is the speed of an electron to have a wavelength of X-ray?

Wavelength of X-ray ~ 0.10 nm = 1.0 x 10-10 mMass of electron = 9.11 x 10-31 kgPlanck constant h = 6.626 x 10-34 Js (kg m2/s)

<Answer>λ= h/p = h/mvv = h/m λ = (6.626 x10-34)/[(9.11 x10-31)(1.0 x 10-10)]

= 7.3 x106 m/s


How to achieve the speed of an electron of 7.3 x106 m/s?

V

<Answer>E= eV= ½ mv2

V = ½ mv2 /e = ½ (9.11 x10-31)(7.3 x106)2 /(1.6022 x 10-19)V = 150 V(1 eV = 1.6022 x 10-19 J)


Electron diffraction

In 1927, Davisson and Germer of the Bell Laboratories investigated the scattering of electrons from various surfaces.


The Uncertainty Principle

• Werner Heisenberg’s uncertainty principle states that we can’t simultaneously know exactly where a tiny particle like an electron is and exactly how it is moving

(∆Px) (∆x) > h/4π, Px = mvx

• The act of measuring the particle actually interferes with the particle

• In light of the uncertainty principle, Bohr’s model of the hydrogen atom fails, in part, because it tells more than we can know with certainty.


Uncertainty Principle Illustrated


Wave Functions

• Quantum mechanics, or wave mechanics, is the treatment of atomic structure through the wavelike properties of the electron

• Wave mechanics provides a probability of where an electron will be in certain regions of an atom


• Erwin Schrödinger developed a wave equation to describe the hydrogen atom

• An acceptable solution to Schrödinger’s wave equation is called a wave function

• A wave function (ψ) represents an energy state of the atom

2),,( zyxψ : the probability of finding an electron

at (x, y, z) position in an atom

1),,( 2 =∫+∞

∞−τψ dzyx The probability of finding

an electron in the universe is equal to 1.

y Traveling wavey(x, t) = y0sin(kx−ωt)y: amplitude of the wave

If y is a function of x only,then, y(x) = y0sinkx

x

n=1Standing wavey(x) = y0sinkx

Boundary condition:y = 0, when x = 0y = 0, when x = L

kL = nπ, k = nπ/Ln = integers (quantum number)

y(x) = y0sin[(nπ/L) x]

n=2

L

L x0n=3


Standing Waves & Quantum Number

Wave MechanicsSchrödinger equation

• ψ : amplitude of the wave• The probability that a particle will be detected is

proportional to .

( )[ ] 082

2

2

2

=−+ ψπψ xUEh

mdxd

one-dimensional

three-dimensional

( )[ ] 0,,82

2

2

2

2

2

2

2

=−+∂∂

+∂∂

+∂∂ ψπψψψ zyxUE

hm

zyx

2ψ

• (normalization)• ψ(x,y,z) is a single valued function w.r.t. the

coordinates• ψ(x,y,z) is a continuous function

• ψ(x,y,z) is a finite function

1),,( 2 =∫+∞

∞−τψ dzyx

Boundary condition for solving ψ in Schrödinger eq. :

For H atom

re

rZeU

22 −=

−=

To solve the equation more easily,Cartesian coordinates x, y, z are

transformed to polar coordinates r, θ, φ.

( )( ) ( ) ( )( ) ( )φθ

φθφθψψ

,

,,),,(

Υ=ΦΘ=

=

rRrRrzyx

Wavefunctions of Hydrogen Atom


( )224 nlRrπ


Quantum Numbers and Atomic Orbitals

• The wave functions for the hydrogen atom contain three parameters that must have specific integral values called quantum numbers.

• A wave function with a given set of these three quantum numbers is called an atomic orbital.

• These orbitals allow us to visualize the region in which there is a probability of find an electron.


Quantum Numbers

The principal quantum number (n)– Can only be a positive integer

– The size of an orbital and its electron energy depend on the n number

– Orbitals with the same value of n are said to be in the same principle shell

n = 1, 2, 3 · · · ·

When values are given to quantum numbers, a specific atomic orbital is defined

Quantum Numbers (continued)

• The orbital angular momentum quantum number (l)– Can have positive integral values

0≤ l ≤ n-1– Determines the shape of the orbital– All orbitals having the same value of n and the

same value of l are said to be in the same subshell– Orbitals and subshells are also designated by a

letter:Value of l 0 1 2 3

Orbital or subshell s p d f


Quantum Numbers (continued)

• The magnetic quantum number (ml):– Can be any integer from -l to +l

-l ≤ ml ≤ l– Determines the orientation in space of the

orbitals of any given type in a subshell– The number of possible value for ml = 2l + 1,

and this determines the number of orbitals in a subshell

The relationship between quantum numbers0≤ l ≤ n-1For example: n= 1, l= 0

n= 2, l= 1, 0n= 3, l= 2, 1, 0

-l ≤ ml ≤ lFor example: l= 1, ml = -1,0,1

l= 2, ml = -2,-1,0,1,2

1s- orbital2p, 2s- orbitals

3d, 3p, 3s- orbitals

px, py, pz- orbitals

dxy, dyz, dzx, dz2, dx2-y2

orbitals


Quantum Numbers Summary


The 1s Orbital

( ) ⎟⎠⎞

⎜⎝⎛ −⎟⎟

⎠

⎞⎜⎜⎝

⎛=

2exp2

2/3

010

ρaZrR pma

aZr 92.52 2

00

==ρ

Υ0, 0(θ,φ) = 1/2π1/2

• The 1s orbital has spherical symmetry.• The electrons are more concentrated near the center

(+)

(-)0

r

0

The 2s Orbital

Υ0,0(θ,φ) = 1/2π1/2

( ) ( ) ⎟⎠⎞

⎜⎝⎛ −−⎟

⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛=

2exp22

21

2/3

020

ρρaZrR

node

0


The 2s Orbital• The 2s orbital has two regions of high electron

probability, both being spherical• The region near the nucleus is separated from the outer

region by a spherical node- a spherical shell in which the electron probability is zero

node


The Three p Orbitals


The Five d Orbital Shapes


The Seven f Orbital Shapes


Electron Spin – the 4th Quantum Number

• The electron spin quantum number (ms) explains some of the finer features of atomic emission spectra– The number can have two values: +1/2 and –1/2

( ms= ½ , -½ )– The spin refers to a magnetic field induced by the

moving electric charge of the electron as it spins– The magnetic fields of two electrons with opposite

spins cancel one another; there is no net magnetic field for the pair.


The Stern-Gerlach Experiment

Hydrogen atom and Schrödinger equation• energy is quantized (n)• magnitude of angular momentum is

quantized (l)• the orientation of angular momentum is

quantized (ml)

Electron has spin (ms)


hydrogen atom

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−= 2

2

2

422nZ

hmeE π


light-emitting diode (LED)

• A light-emitting diode (LED) is a semiconductor device that emits incoherent narrow-spectrum light when electrically biased in the forward direction. • This effect is a form of electroluminescence. • The color of the emitted light depends on the chemical composition of the semiconducting material used.

AlGaAs - red and IRAlGaP - green AlGaInP - high-brightness orange-red, orange, yellowyellow,

and greenGaAsP - red, orange-red, orange, and yellowyellowGaN - green, and blueInGaN - near UV, bluish-green and blueAlN, AlGaN - near to far UV


Organic Light-Emitting Diodes (OLEDs)

• vs. inorganic LEDsFlexibilitySimple and easy thin film fabrication and micronscale patterning (vs. wire-bonded epitaxial AlGaAs or group III nitride discrete semiconductor LEDs)

• vs. liquid crystal display, LCDWide viewing angleVery bright and highly contrastNo back-lighting needed (low energy consumption)Fast switching times (video-rate display)Multicolor emission (RGB)Thin and light weightFoldable, very thin screen possible


比一比看誰炫比一比看誰炫

有機發光二極體(

ＯＬＥＤ)

液晶顯示器(

ＬＣＤ)


Summary

• Cathode rays are negatively charged fundamental particles of matter, now called electrons.

• An electron bears one fundamental unit of negative electric charge.

• A nucleus of an atom consists of protons and neutrons and contains practically all the mass of an atom.

• Mass spectrometry establishes atomic masses and relative abundances of the isotopes of an element

• Electromagnetic radiation is an energy transmission in the form of oscillating electric and magnetic fields.


Summary (continued)

• The oscillations produce waves that are characterized by their frequencies (v), wavelengths (λ), and velocity (c).

• The complete span of possibilities for frequency and wavelength is described as the electromagnetic spectrum.

• Planck’s explanation of quantums gave us E = hv• The photoelectric effect is explained by thinking of quanta

of energy as concentrated into particles of light called photons.

• Wave functions require the assignment of three quantum numbers: principal quantum number, n, orbital angular momentum quantum number, l, and magnetic quantum number, ml.


Summary (continued)

• Wave functions with acceptable values of the three quantum numbers are called atomic orbitals.

• Orbitals describe regions in an atom that have a high probability of containing an electron or a high electronic charge density.

• Shapes associated with orbitals depend on the value of l. Thus, an s orbital (l = 0) is spherical and a p orbital (l = 1) is dumbbell-shaped.

• A fourth quantum number is also required to characterize an electron in an orbital - the spin quantum number, ms.

Chapter Six (1)

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