Chem6A_F11 Hol - Chapter 7 Lecture Slides

Handout #7 (Fall 2011) Chem 6A - Prof. John E. Crowell

Chem 6A - Chapter 7 1

Quantum Theory and Atomic Structure

7.1 The Nature of Light

7.2 Atomic Spectra

7.3 The Wave-Particle Duality of Matter and Energy

7.4 The Quantum-Mechanical Model of the Atom

Particle Nature of Light: Blackbody Radiation §  Three phenomena could not be explained by classical physics: blackbody radiation, photoelectric effect, and atomic spectra. A new picture of energy was required.

§  Blackbody Radiation: Heat a solid object: at 1000 K it begins to emit soft red light; at 1500 K it begins to glow orange and is brighter; at 2000 K it is still brighter and white. How do we explain this?

§  Planck: proposed that only certain quantities of energy could be emitted or absorbed:

§  These packets of energy are called quanta, and n is a quantum number. Since it was (correctly) assumed that most transitions occur between adjacent states, Δn = 1 and the change in energy is:



Amplitude (intensity) of a Wave Light is Electromagnetic Radiation that is visible to the human eye, and consists of waves. The amplitude of a wave refers to the maximum amount of displacement of a particle from its rest position.

The wavelength of a wave is simply the length of one complete wave cycle.

The basic unit of light or electromagnetic radiation is a photon. A photon is a discrete bundle of electromagnetic energy.

Energy of a

Photon:

c = 2.998 x 108 m/sec

Frequency (nu) = cycles per

second

Wavelength (lambda) = distance (meters) per cycle

h = Planck’s constant: h = 6.626 x 10-34 J-sec



Formation of a Standing Wave

If at each time step, we add these two waves together at every point in space,

we obtain a wave that doesn't move. Even though the amplitude changes in time, the location of those amplitude

shifts do not. It is a standing wave. A standing wave is thus distinguished by

waves in which the position of the nodes remains fixed at all times. Standing waves arise if the space in which the

waves can propagate is bounded.

Traveling waves are distinguished

by the whole wave, including nodes, propagating at a given velocity.

Shown is a traveling wave

moving to the right & one traveling to

the left. . .

The Electromagnetic

Spectrum Decreasing Energy

Increasing Wavelength

Electromagnetic Radiation consists of visible light, ultraviolet radiation, infrared radiation, gamma rays, X-rays, radio waves, etc. This type of wave involves periodic oscillations in electrical and magnetic fields.



Electric and Magnetic Fields of a Light Wave

Interaction of an electric field with an oscillating dipole moment: The two interact if the frequency and phase are the same.

Spectral Range of Various Spectroscopies

Increasing Frequency

Decreasing Wavelength



Diffraction by a Single Slit

When parallel rays of light of wavelength λ are incident on a slit of width a, diffraction occurs due to constructive & destructive interference. The extent of interference

depends on the angle that the light waves emerge at.

In (a) above, constructive interference is maximize, while in (b), destructive interference occurs, resulting in a minimum. In (c), every other wave is out of phase,

resulting in primarily destructively interference, and a near minimum.

Constructive Interference

Destructive Interference

Single Slit Diffraction

When parallel rays of light of wavelength λ are incident on a slit of width a, diffraction occurs due to constructive & destructive interference. The extent of interference depends on the angle that the light waves emerge at. Destructive interference is maximized by the equation given above. If no interference (diffraction) occurs, a

simple square wave intensity profile would be observed.

Interference minima obey:



Two Slit Experiment

Up Close

Constructive and destructive interference upon light waves

coming from two point sources (left), or equivalently, upon

emerging from two slits (above).

Constructive interference results in maxima (red or bright spots to left) and destructive interference results in minima (black or dark

spots to left).

single slit

results

double slit

results

Incident particle

(e.g., electron,

atom)

A particle going through one slit displays wave-like behavior, as a diffraction pattern is observed. Particle behavior is observed in that exactly half of the electrons strike

each slit when one is closed. Having both slits open changes the measurement, and a diffraction pattern is observed, indicating each particle goes through BOTH SLITS. Behavior as both a wave and a particle is a fundamental aspect of the microscopic

world described by quantum mechanics.

Particle Diffraction by a Double Slit



Photoelectric Effect:

Observations Hertz (1887): showed that when a metal is irradiated with UV light, an electron may be ejected.

l  Experimental Observations: Ø  No electrons were ejected below a certain (metal-specific) cut-off or

threshold frequency (ν0). Ø  For light of frequency ν > ν0, the kinetic energy of the ejected electron was

directly proportional to the frequency ν but independent of the light intensity.

Ø  The number of ejected electrons depends on the light intensity. Ø  Even at arbitrarily low light intensity, electrons are ejected immediately if

the frequency ν exceeds the threshold frequency (ν0).

Photoelectric Effect: Explanation l  Einstein’s Explanation (1905):

Ø  Light of frequency ν is composed of particles (now referred to as “photons”), each of energy E = hν.

Ø  The intensity of the light is proportional to the number of photons.

Ø  Each metal has a characteristic minimum threshold energy (work function, Φ) required to remove an electron (determined by bonding in the metal, but similar to an ionization energy).

Ø  The kinetic energy of the ejected electron is the photon energy minus the work function (threshold energy).

KE = 1

2mv2 = hν −Φ= hν − hν

0

Slope = h

Threshold = νo

Independent determination of Planck’s constant



Photoelectric Effect: Consequences

Photons - a spatially localized packet of light.

Each electron

ejected by one photon

KE = 1

2mv2 = hν −Φ= hν − hν

0

Einstein showed that the energy of light is proportion to its frequency and that h is a universal constant. The photoelectric effect demonstrates that light

(photons) displays particle-like behavior

No electrons are ejected

when hν < Φ.

When hν > Φ,

electrons with

energy mv2/2 are ejected.

De Broglie Waves: Wave – Particle Duality De Broglie’s Postulate (1924): Building upon Einsteins’ result that light could display particle-like behavior, de Broglie hypothesized that one can associate a wave of

wavelength λ with a particle moving at momentum p according to the de Broglie Relationship:

Wave-like behavior of particles is only observed for very small

masses and low velocities (e.g. an electron in an atom) where the

wavelength λ is sufficiently large to be observable. It is irrelevant for

macroscopic objects.

Davisson – Germer Experiment (1925): Scattering of an electron beam from a NiO crystal surface

demonstrated a variation of scattering intensity characteristic

of a diffraction experiment in which waves interfere

constructively and destructively, depending upon direction.



Comparing Diffraction Patterns of X-rays and Electrons

X-ray Diffraction of Aluminum

Photons behave like waves, exhibiting constructive and

destructive interference.

Electron Diffraction of Aluminum

Electrons show wave-like behavior too.

Wave – Particle Duality

De Broglie Wavelength & Wave – Particle Duality

Particles (matter) behaves as if it moves in a wave. These “matter waves” depend on size (mass) & speed:

λbaseball =h

mv=

6.626x10−34 kg m2s−1

(142x10-3 kg)(90mph)

⎛

⎝⎜⎞

⎠⎟mile

5280 ft⎛⎝⎜

⎞⎠⎟

ft12 in

⎛⎝⎜

⎞⎠⎟

in2.54cm

⎛⎝⎜

⎞⎠⎟

100cmm

⎛⎝⎜

⎞⎠⎟

3600shr

⎛⎝⎜

⎞⎠⎟

λbaseball @90mph = 1.16x10−34 m wavelength



Emission from Highly Excited Atoms

Light emitted or absorbed by atoms or molecules appears at certain discrete wavelengths. Rydberg fit emission spectra for H atoms to the following: Rydberg constant: RH=109,677.581 cm-1.

Evidence of Quantization of Energy

Emission Spectra for Several Elements



The Bohr explanation of the series of spectral lines for H

RH = Rydberg constant:

Energy Quantization: The Quantum Staircase The energy levels for the

hydrogen atom are quantized.

An electron bound to hydrogen can absorb a

photon if the energy (wavelength) is correct and move to a higher

energy level.

Conversely, an electron can emit a photon and move to a lower energy

level.

However, the electron cannot be bound to

hydrogen and in a state other than the allowed

quantized energy states (i.e. the stationary states shown for n = 1, 2, 3, …).



SOLUTION:

Determining ∆E and λ of an Electron Transition

PROBLEM: A hydrogen atom absorbs a photon of visible light and its electron enters the n = 4 energy level. Calculate (a) the change in energy of the atom and (b) the wavelength (in nm) of the photon.

PLAN: (a) The H atom absorbs visible light, so the electron is going from n = 2 to n = 4. Calculate ΔE. (b) Use ΔE = hν = hc / λ to find λ.

(a)

(b)

Summary of Historical Perspective

De Broglie Postulate / Wavelength: Particles show wave-like behavior too.

Planck’s Quantum Hypothesis: Radiators (energy sources) exist only in certain allowed energy levels, and radiate only in certain integer multiples of a basic energy

unit ( a quanta of hν, with n = 0, 1, 2, 3, …).

Einstein’s Quantum Hypothesis: Photons are spatially localized packets of light whose energy

is proportion to its frequency.

Line Spectra of Atoms: Atoms & Molecules have discrete energy

levels, and emit or absorb energy at discrete wavelengths.

Quantization of Energy Wave – Particle Duality



Wave motion in restricted systems:

A. Waves on a string are similar to a particle confined to a small region (the so called “one dimensional particle in a box”).

B.  Circular waves are similar to a confined “particle on a ring”, and mimics molecular rotational motion (in 2-D) or electron motion around the nucleus.

Standing Waves: Quantized States

ml = ± integer

ml ≠ ± integer



Heisenberg Uncertainty Principle l  Classical mechanics: particles have trajectories and

can precisely be described according to location (x) and linear momentum (p)

l  In quantum mechanics we talk about probability: waves are spread out as they oscillate

l  Wave-Particle Duality denies the ability of knowing the trajectory of particles (complementarity)

l  Heisenberg Uncertainty principle:

l  Important for wave equations

Precisely determined momentum:

A sine wave of wavelength λ implies that the momentum p is precisely known (using the de Broglie relationship). But the wavefunction and the probability density Ψ*Ψ is spread over all space. Hence p is exact (precisely known), and the position x is unknown.

Adding together several waves of different wavelength (a linear combination) will produce an interference pattern (a superposition) which begins to localize the wave!

However, this localization in position results in a spreading of the momentum values since each wave added has a different momentum.

Each different wavelength has a different momentum:

Hence, there is an inherent increase in the uncertainty in Δp as the position becomes more localized (i.e. Δx decreases).

Heisenberg Uncertainty Principle



A quantum mechanical system is completely specified by a wavefunction Ψ = Ψ(x,y,z,t). The wavefunction contains all the information there is to know about the system. An operator "operates" upon a function and produces another function. For every operator, there is a collection of functions which, when operated on by the operator, produces THE SAME function, modified only by a constant factor. Such a function is called an eigenfunction of that operator and the constant factor is called the eigenvalue of that eigenfunction.

Quantum mechanics involve eigenvalue equations. Any observable property of a system (energy, momentum, angular momentum, dipole moment, etc.) has its own operator. The eigenfunctions are the wavefunctions of the system and the eigenvalues are the values of the property - the number that we measure. The Hamiltonian is the energy operator for the system and the Schrödinger equation is an eigenvalue equation.

Quantum Mechanics or Quantum Chemistry

− 2

2md 2

dx2 +V (x)⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪ψ

n(x)= nE ψ

n(x)

H ψn(x)= nE ψ

n(x)

Schrödinger Equation

Particle Confinement: The Particle in a 1-D Box l  We can exactly solve the

Schrodinger Equation for simple cases.

l  The particle in a box solution:

l  Solving the Schrodinger Equation yields:

and, since ΔE = hν and ΔE = En+1 – En

L = length of box n = 1, 2, 3, … x = distance between 0 and L

Since n is an integer, E is quantized with discrete values of E with E > 0

x = 0 x = L



ψ n(x) = 2

Lsin nπ x

L; n = 1,2,…

Eigenfunctions Probability Density

ψ n

*(x)ψ n(x) = 2L

sin2 nπ xL

n = 4

n = 3

n = 2

n = 1

Particle in a

1-D Box

E/(h

2 /8m

L2)

E =

2k 2

2m=

2n2π 2

2mL2 = h2

(2π )2

n2π 2

2mL2 = h2n2

8mL2

Particle in a 1-D Box: Energy Levels

The allowed energy levels for a particle in a box. Note that the energy levels increase as n2, and that their separation increases as the quantum number increases.



The lowest energy state has all quantum numbers equal to one. This is called the ground-state energy or zero-point energy:

Particle in a 3-D Box

The energy is quantized with 3 quantum numbers: nx, ny, nz.

ψ nx ,ny ,nz

(x, y, z) = 8Lx Ly Lz

sin(nxπ x

Lx

)sin(nyπ y

Ly

)sin(nzπ y

Lz

)

For the case of a cubic box (Lx = Ly= Lz), the energy is:

Where V(x,y,z) = 0 inside the box, and V(x,y,z) = ∞ outside the box. Outside the 3-D box, Ψ(x,y,z) = 0, and inside the box:

Electron Probability Density in the ground-state H atom

Panel A & B: Plots of the Probability Density (Ψ*Ψ) for the 1s Orbital. It is greatest at the nucleus and spherically symmetric, decreasing as one moves out from the nucleus. Panel C & D: Plots of the Radial Distribution Function for the 1s Orbital. This is the probability that the electron will be found anywhere in a shell of radius r. It is equal to zero at the nucleus, maximizes at the Bohr radius, and decreases as one moves out further from the center. It too is spherically symmetric. Panel E: A contour plot of the 1s Orbital showing the volume where the electron spends 90% of its time.



Radial Probability Density of the Hydrogen Atom A constant-volume electron-sensitive detector (the small cube) gives its greatest reading at the nucleus, and a smaller reading

elsewhere. The same reading is obtained anywhere on a circle of given radius: the s orbital is spherically symmetrical.

1s Orbital

2s Orbital

Radial Probability Density of the Hydrogen Atom The radial distribution function Pnl gives the probability that the electron will be

found anywhere in a shell of radius r. For a 1s electron in hydrogen, Pnl is a

maximum when r is equal to the Bohr radius a0. The value of Pnl is equivalent to the reading that a detector shaped like a spherical shell would give as its radius is

varied. The probability of finding an electron between r and r + dr is:



Quantum Numbers and Atomic Orbitals An atomic orbital is specified by three quantum numbers:

n the principal quantum number - a positive integer ( 1, 2, 3, ……)

l the angular momentum quantum number - an integer from 0 to (n - 1)

ml the magnetic moment quantum number - an integer from –l to +l

The organization of orbitals into shells (characterized by n) and subshells (characterized by I).

The energy levels of the hydrogen atom showing the subshells and (in square brackets) the numbers

of orbitals in each subshell. In hydrogenic atoms, all orbitals of a given shell have the same energy.



Shells, Subshells, and Orbitals

Energy Levels for the Hydrogen Atom



The Hydrogen Atom Hamiltonian Hydrogen Atom

Hamiltonian

Kinetic energy

of electron

Potential energy

(nucleus-electron

interaction)

Kinetic energy

of nucleus

We can change the variables to the center of mass coordinates:

Center of mass coordinates:

Particle separation

(relative coordinates):

We seek solutions then of the form:

Eigenfunctions for the Hydrogen Atom



(3) It is further required that the quantum number l in the total solution satisfies l < n-1. Thus, we have the following form for the wavefunction, and three quantum numbers:

Eigenfunctions for the Hydrogen Atom

Principle quantum number:

Angular momentum quantum number:

Magnetic quantum number:

radial part

angular part

Normalization constant Associated Laguerre

Polynomials

Decaying exponential

in r Term to make all wavefunctions go to

zero at the nucleus except for l=0

The radial wave-function has the following form:

Hydrogen Wavefunctions

a0 is the Bohr radius

R decreases exponentially as r increases

Angle independent



1s orbital

2s orbital

3s orbital

2p orbitals



The 3d orbitals

One of the seven

possible 4f orbitals



SOLUTION:

PLAN:

Determining Quantum Numbers for an Energy Level

PROBLEM: What values of the angular momentum (l) and magnetic (ml) quantum numbers are allowed for a principal quantum number (n) of 3? How many orbitals exist for n = 3?

Follow the rules for allowable quantum numbers found in the text.

l values can be integers from 0 to (n – 1); ml can be integers from -l through 0 to +l.

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

For l = 0, ml = 0

For l = 1, ml = -1, 0, +1

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

There are 9 ml values and therefore, 9 orbitals with n = 3.

SOLUTION:

PLAN:

Determining Sublevel Names and Orbital Quantum Numbers

PROBLEM: Give the name, magnetic quantum numbers, and number of orbitals for each sublevel with the following quantum numbers:

(a) n = 3, l = 2 (b) n = 2, l = 0 (c) n = 5, l = 1 (d) n = 4, l = 3

Combine the n value and l designation to name the sublevel. Knowing l, we can find ml and the number of orbitals.

n l sublevel name

possible ml values # of orbitals

(a)

(b)

(c)

(d)

3

2

5

4

2

0

1

3

3d

2s

5p

4f

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

0

-1, 0, +1

-3, -2, -1, 0, +1, +2, +3

5

1

3

7

Chem6A_F11 Hol - Chapter 7 Lecture Slides

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