sign in sign up
<<
Home , Excited state

Chemical Bonding, , and Reactivity: PHYSICAL SCIENCES 1 An Introduction to the Physical Sciences LECTURE 2

SPRING 2014 Reading: 8.1-8.4 Alán Aspuru-Guzik Carlos Amador

MWF 10-11 AM Science Center D Office Hours: Wed. 1-3 PM, Science Center 309

1 OUTLINE

Electromagnetic Radiation Wavelength, frequency, speed Light-light interactions: interference Light- interactions: reflection, refraction, absorption

Introduction to White light vs. atomic line spectra Bohr model of 5 ELECTROMAGNETIC RADIATION A disturbance that transmits energy through space or through a medium is known as a

Direction of propagation If this energy is transmitted by perpendicular travelling electric and magnetic fields, the wave is known as

Magnetic Field

Electric Field 6 FREQUENCY, WAVELENGTH, AND SPEED Let’s look in more detail at the electric (or magnetic) field:

The distance from crest to crest (or trough to trough) is known as the

Units

The number of crests that pass through a The two quantities are related by: point per unit time is the

Units: 7 THE Electromagnetic radiation can be classified according to its wavelength (or frequency):

λν = c c c λ = v = v λ

10−6 m 10−9 m 10−10 m

10−12 m 8 INTERFERENCE Light interacts with itself via interference. Let’s have a look… double-slit experiment! 9 LIGHT-MATTER INTERACTIONS: REFLECTION The electric field of light can interact with charges (e.g. ) in matter according to:

Similarly, oscillating charges can generate electric fields.

We see these processes at work in reflection.

Metal (eg. Ag) 10 LIGHT-MATTER INTERACTIONS: REFRACTION

Light slows down when it passes through a material. This change in speed is accompanied by a frequency-dependent (color-dependent) change in direction.

Example: glass prism 11 to the wavelengths of light absorbed. absorbed. of light to the wavelengths is usually its color range, light the visible in wavelength a material When of light. also can Materials ABSORPTION LIGHT-MATTERINTERACTIONS: Example: the Recall straightforward. (But this is not completely

roughly chlorophyll Harvard Color Predictor Harvard absorbs absorb

in plant leaves leaves plant in molecules

a particular a particular certain wavelengths wavelengths certain .)

12 LIGHT VS. Take out the diffraction grating you picked up!

Look at the ceiling lights: Now let’s look at excited atoms:

The spectrum is…

The is… 13 EXPLAINING ATOMIC LINE SPECTRA: ENTER QUANTUM MECHANICS…

Max Planck Niels Bohr Max Planck proposed that matter absorbs and emits electromagnetic 1858-1947 1885-1962 radiation in discrete chunks of energy known as quanta, with the energy of the proportional to the frequency of light.

Niels Bohr proposed that the possible of an in a hydrogen are quantized.

Electrons jump from one level to another by absorbing or emitting of discrete quantized energies. 15 THE HYDROGEN LINE SPECTRUM Putting it all together, we can write the equation for the energies of light emitted and absorbed by a :

Minus sign:

Plus sign:

The corresponding wavelength of the photon is:

Ionization is the complete removal of an electron. We can calculate the of an electron in hydrogen by using: 16 PREDICTING THE HYDROGEN LINE SPECTRUM Problem 1(a): Calculate the energy of the photon emitted when the electron in the hydrogen atom undergoes a transition from the n = 3 to the n = 2 level.

Problem 1(b): What is the wavelength of this photon? What color is it? Did you see this color in the demo? 17 PHOTOCHEMISTRY: SETTING OFF A REACTION WITH LIGHT

Certain chemical reactions only occur when a certain amount of energy E is provided. Here we need enough energy to cleave the Cl-Cl bond to initiate the reaction.

hν H2 + Cl2 2 HCl

High energy, E = hc Low energy, Short wavelength λ Long wavelength

Question: Which color LED has enough energy to set off the above reaction? 20 PHOTOSYNTHESIS: A NATURAL EXAMPLE OF PHOTOCHEMISTRY hν 6 CO + 6 H O C6H12O6 + 6 O2 2 2

Chlorophyll energy levels

excited states Soret Qy

Energy Qy

Soret 21 SUMMARY OF KEY IDEAS

Electromagnetic waves satisfy: λν = c (c = 3.0 x 108 m/s) hc Electromagnetic waves carry energy: E = hν = (h = 6.626 x 10-34 J s) λ Electrons in the hydrogen atom RH -18 E = − (RH = 2.179 x 10 J) occupy discrete energy levels n: n n2

Electronic transitions occur via ⎛ 1 1 ⎞ ΔE = −R ⎜ − ⎟ absorption or emission of a photon: H ⎜ 2 2 ⎟ n f ni ⎝ ⎠ (negative for emission;

positive for absorption) 22 Chemical Bonding, Energy, and Reactivity: PHYSICAL SCIENCES 1 An Introduction to the Physical Sciences LECTURE 3

SPRING 2014 Reading: 8.5-8.6 Alán Aspuru-Guzik Carlos Amador

MWF 10-11 AM Science Center D Office Hours: Wed. 1-3 PM, Science Center 309 1 LIGHT AND QUANTUM MECHANICS

Electromagnetic waves satisfy: λν = c (c = 3.0 x 108 m/s) hc Electromagnetic waves carry energy: E = hν = (h = 6.626 x 10-34 J s) λ Electrons in the hydrogen atom RH -18 E = − (RH = 2.179 x 10 J) occupy discrete energy levels n: n n2

Electronic transitions occur via ⎛ 1 1 ⎞ ΔE = −R ⎜ − ⎟ absorption or emission of a photon: H ⎜ 2 2 ⎟ n f ni ⎝ ⎠ (negative for emission;

positive for absorption) 3 TODAY

Wave-particle duality of light and matter Calculating de Broglie wavelengths of matter

Quantum mechanics Describing matter with wavefunctions Particle-in-a-box Modeling conjugated molecules 4 LAST LECTURE Last time, we looked at the double-slit experiment and observed an interference pattern.

From this, we concluded that light is a 5 WELL, ACTUALLY… What if we crank the intensity of the light WAY DOWN?

500,000 frame 1000 frames 200 frames Single frame

Then light behaves as a called a but…

Source: A Weis, University of Fribourg, Switzerland 7 WHAT ABOUT MATTER? Surely matter (such as electrons) behave like particles… well… actually…

Diffraction pattern of x-ray beam Diffraction pattern of electron beam passing through aluminum foil passing through aluminum foil

It seems that matter also behaves like a 9 DESCRIBING MATTER AS A WAVE Just like light, matter can also be thought of as a wave characterized by a

Louis de Broglie Nobel Prize 1918

Problem 1: A typical pitcher throws a fastball at 100 mph (45 m/s). The baseball weighs 0.15 kg. Calculate the de Broglie wavelength of the baseball. Does the baseball have significant wavelike character?

8-5 10 h WHEN DO MATTER WAVES MATTER? = mv For the wave nature of matter to be important, the de Broglie wavelength must be close to the wavelength of light it is interacting with.

Substance Mass (kg) Speed (m/s) λ (m)

slow electron 9x10-31 1.0 7x10-4 fast electron 9x10-31 5.9x106 1x10-10 proton 1.67x10-27 1600 2.5x10-10 alpha particle 6.6x10-27 1.5x107 7x10-15 baseball 0.15 40.0 1x10-34 Earth 6.0x1024 3.0x104 4x10-63 11 RECAP SO FAR

Light and matter each exhibit both wave-like and particle-like properties! 12 A CLOSER LOOK AT WAVES

Traveling waves

Fixed-end boundary condition

Standing waves

We will use these as a basis for describing matter quantum mechanically! 13 QUANTUM MECHANICAL MATTER WAVES

How can we describe matter quantum mechanically?

Let’s take into account wave-particle duality: describe particle by a standing wave! Erwin Schrödinger

Wavefunction of a particle:

Probability of finding the particle at any point in space is related to square of the wavefunction: Max Born 14 THE SIMPLEST QUANTUM SYSTEM: PARTICLE IN A BOX Consider a particle (e.g. an electron) confined to a one-dimensional box. We want to find standing waves to describe this particle.

Potential energy V(x) V(x) =

Boundary Conditions:

(wavefunction vanishes outside the box) 0 L

Position (x) 15 WHICH STANDING WAVES ARE VALID FOR PARTICLE-IN-A-BOX? (a) (b)

0 0 Amplitude Amplitude 0 L 0 L Position (x) Position (x)

(c) (d)

0 0 Amplitude Amplitude 0 L 0 L Position (x) Position (x) 16 STANDING WAVES ARE CALLED EIGENFUNCTIONS First Eigenfunction Second Eigenfunction Third Eigenfunction n = 1 n = 2 n = 3

ψ n (x) Amplitude Amplitude Amplitude Amplitude Amplitude

0 L 0 L 0 L Position (x) Position (x) Position (x)

2 |ψ n (x) | Probability Probability Probability Probability Probability Probability

0 L 0 L

0 L 17 Position (x) Position (x) Position (x) For derivation, EACH EIGENFUNCTION COMES see 8-6 p. 320 WITH AN ASSOCIATED ENERGY

Eigenfunctions: n = 3 Energy

Energies:

Number of nodes: n = 2 (point of zero probability) (Not counting endpoints) n = 1

Conclusion: A particle in a box occupies discrete energy levels, much like an electron in a hydrogen atom. 0 L 18 ASIDE: WHAT ABOUT A PARTICLE WHIZZING BACK AND FORTH?

Traveling waves can be constructed from superpositions of standing waves (e.g. combination of n = 1 and n = 2).

These traveling wavefunctions are not eigenfunctions—they are combinations. They do not have well-defined energies.

0 L For now, we will stick to depicting particles by Position (x) standing waves (eigenfunctions) with well- defined energies. 19 MODELING CONJUGATED MOLECULES

The particle-in-a-box model is not just a theoretical fantasy!

Conjugated molecules have alternating single and double bonds.

Example: butadiene

The pi electrons are free to move about the entire . Hence they can be modeled as particles in a box.

We will use the model to compute the length of the butadiene molecule. 20 MODELING BUTADIENE

Problem 2: Butadiene has 4 pi electrons (2 per pi bond). The lowest-energy electronic transition occurs when butadiene absorbs light of 217 nm.

(a) Draw the electronic configuration of butadiene before and after absorption of light. Note that two pi electrons can fit in each particle-in-a-box .

n = 3 n = 3 Energy Energy Absorption

n = 2 n = 2

n = 1 n = 1 21 Chemical Bonding, Energy, and Reactivity: PHYSICAL SCIENCES 1 An Introduction to the Physical Sciences LECTURE 4

SPRING 2014 Reading: 8.7-8.8 Alán Aspuru-Guzik Carlos Amador

MWF 10-11 AM Science Center D Office Hours: Wed. 1-3 PM, Science Center 309

1 TODAY: GETTING CLOSER TO REAL LIFE

Particle in a Box Modeling conjugated molecules Particle in a 3D box

The Hydrogen Atom Quantum numbers Atomic Orbitals 3 SUMMARY OF PARTICLE IN A BOX Matter has wavelike properties which are h λ = characterized by the de Broglie wavelength: mv

Pi electrons in conjugated molecules can be modeled by a particle in a box (the simplest quantum mechanical system) :

Eigenfunctions: Eigenfunctions Probability 2 ⎛ nπx ⎞ ψ n (x) = sin⎜ ⎟ L ⎝ L ⎠ Energies: n2h2 E = n 8mL2

Electronic Transitions:

2 h 2 2 ΔE = 2 (n f − ni ) 8mL 4 MODELING BUTADIENE

Problem 1: Butadiene has 4 pi electrons (2 per pi bond). The lowest-energy electronic transition occurs when butadiene absorbs ultraviolet light of 217 nm.

(a) Draw the electronic configuration of butadiene before and after absorption of light. Note that two pi electrons can fit in each particle-in-a-box energy level.

n = 3 n = 3 Energy Energy Absorption

n = 2 n = 2

n = 1 n = 1 5 MODELING BUTADIENE

Problem 1: Butadiene has 4 pi electrons (2 per pi bond). The lowest-energy electronic transition occurs when butadiene absorbs ultraviolet light of 217 nm.

(b) What is the energy difference between the electronically excited state of butadiene and the ground state of butadiene? 6 MODELING BUTADIENE

(c) Calculate the length L of butadiene. The mass of an electron is m = 9.11 x 10-31 kg. Is your answer qualitatively reasonable? (A typical carbon-carbon bond length is roughly 1.5 Å.) 7 PARTICLE IN A 3D BOX Last time: Particle in a 1D Box

Potential energy V(x) h2n2 E = n 8mL2

0 L Position (x) Today: Particle in a 3D Box The energies are additive!

h2 ! n2 n2 n2 $ E = # x + y + z & nx ,ny,nz # 2 2 2 & 8m " Lx Ly Lz % 8 THE HYDROGEN ATOM Instead of considering an electron moving freely in a box, now let’s consider an electron bound to a proton in a hydrogen atom. r p+ e-

V(r) The potential energy is: r

V(r) =

-12 2 −1 −2 ε0 = permittivity of free space (8.85 x 10 C ·N ·m ) Z = number of protons in nucleus (for hydrogen, Z = 1) e = electron charge (9.11 x 10-31 C) We want to find wavefunctions for this potential: these are called atomic orbitals.

How many quantum numbers for each wavefunction/orbital? 9 QUANTUM NUMBERS FOR ELECTRONS IN HYDROGEN

Name Symbol Permitted Values Property principal orbital size (energy)

angular orbital shape momentum l orbital shape l orbital shape

magnetic orbital orientation 10 QUANTUM NUMBERS: PRACTICE PROBLEM

Problem 2(a): Can an orbital have the quantum numbers n = 2, l = 2, and ml = 2?

Problem 2(b): For an orbital with n = 3 and ml = -2, what is (are) the possible value(s) of l?

11 l = 0: s ORBITALS The s orbitals are spherical. As n increases, the s orbitals get larger and number of radial nodes increases.

Radial Probability 1s 2s 3s Distribution

(from squaring the r wavefunction) r r

“Cartoon”

Surface plot

n = l =

ml =

Number of radial nodes 12 l = 1: p ORBITALS The p orbitals are shaped like a dumbbell. How many p orbitals for a given principal n?

3D representations:

2px 2py 2pz

Each 2p orbital contains one angular node, which forms a plane.

(More on counting nodes later…) 13 l = 2: d ORBITALS The d orbitals have more complex shapes. How many d orbitals for a given n?

3D representations:

2px 2py 2pz 2 2 3d 2 x - y 3dxy 3dxz 3dyz 3dz

Each 3d orbital contains two angular nodes. 14 COUNTING NODES So far we have seen two types of nodes (regions of zero probability): radial nodes and angular nodes.

x How can we count them? radial node 2 2 4dx - y n =

angular nodes y l =

# of radial nodes = =

# of angular nodes = =

# of total nodes = = 15 ORBITAL ENERGIES AND DEGENERACIES We already saw that the energy of an electron in a hydrogen orbital depends only on its principal quantum number (n):

This leads to degeneracies, or multiple orbitals with the same energy, since orbitals differing only in

l and ml have the same energy:

Energy Degeneracy 3s 3p 3d

2s 2p

1s

(Note: These degeneracies do not Degeneracy of energy level n = hold for multielectron atoms, as we will see in the next lecture.) 16 ORBITRON: SECOND BEST ROBOT IN PS1

Click here for the most fun you will http://winter.group.shef.ac.uk/orbitron/ ever have in your life!

4f 5g The orbitron is a wonderful tool for visualizing orbitals in all sorts of ways.

For example, look at these f and g orbitals which your high school chem teacher refused to show you… 17 REVIEW: PARTICLE-IN-A-BOX VS. HYDROGEN ATOM h2n2 RH En = 2 E = − 8mL Energy n n2

3s 3p 3d Similarities: 2s 2p • Discrete energy levels • Describe electronic transitions

1s Differences:

Number of Quantum Numbers:

Wavefunctions:

Energies are:

Spacing between energy levels:

Degeneracies: 18 ATOMIC ORBITALS: PRACTICE PROBLEM

Problem 3: Consider an electron in a hydrogen 4py .

(a) List all possible quantum numbers for this electron.

(b) How many angular nodes does this orbital have? How many radial nodes? 19 Chemical Bonding, Energy, and Reactivity: PHYSICAL SCIENCES 1 An Introduction to the Physical Sciences LECTURE 5

SPRING 2014 Reading: Alán Aspuru-Guzik 8.9-8.12 9.6 Carlos Amador

MWF 10-11 AM Science Center D Office Hours: Wed. 1-3 PM, Science Center 309 1 TODAY

Finishing the hydrogen atom

A fourth quantum number: electron

Multielectron atoms Electron configurations Periodic table 3 ATOMIC ORBITALS: PRACTICE PROBLEM

Problem 3: Consider an electron in a hydrogen 4py atomic orbital. (a) List all possible quantum numbers for this electron.

(b) How many angular nodes does this orbital have? How many radial nodes? 4 ATOMIC ORBITALS: PRACTICE PROBLEM

Problem 3: Consider an electron in a hydrogen 4py atomic orbital. (c) Sketch the orbital and label all of the nodes.

Any similarities to this brochette are purely coincidental

(d) True or False: This electron is higher in energy than an electron in a 4s orbital. 5 SUMMARY OF THE HYDROGEN ATOM

2 2 2 2 A particle in a 3D box is characterized h ⎛ n ny n ⎞ E = ⎜ x + + z ⎟ by three quantum numbers: nx ,ny ,nz 8m ⎜ L2 L2 L2 ⎟ ⎝ x x x ⎠ Similarly, electrons in hydrogen occupy atomic orbital wavefunctions labeled by three quantum numbers: Name Symbol Permitted Values Property

principal n positive integers (1, 2, 3, …) orbital energy (size)

angular l integers from 0 to n – 1 orbital shape (The l values 0, 1, 2, momentum and 3 correspond to s, p, d, and f orbitals, respectively.)

magnetic m l integers from –l to 0 to +l orbital orientation R Number of radial nodes = n – l – 1 H En = − Depends only on n Number of angular nodes = l n2 Number of total nodes = n – 1 The nth energy level is n2 degenerate. 6 HOW DO ELECTRONS RESPOND TO AN EXTERNAL MAGNETIC FIELD?

External Electrons behave as tiny magnets when placed in a magnetic field. Magnetic Field B

Option 1: If they were tiny CLASSICAL magnets… Option 2: If they were tiny QUANTUM magnets… 7 STERN-GERLACH EXPERIMENT (1922) We can separate silver atoms (each with one unpaired electron) according to their unpaired electron’s spin by putting them in an inhomogeneous (spatially-varying) magnetic field:

Conclusion: 8 ELECTRON SPIN: A FOURTH QUANTUM NUMBER Conclusion: Electrons carry spin. This spin can be aligned or anti-aligned with an external magnetic field. We characterize this by the spin quantum number:

Name Symbol Permitted Values Property

principal n positive integers (1, 2, 3, …) orbital energy (size)

angular l integers from 0 to n – 1 orbital shape (The l values 0, 1, 2, momentum and 3 correspond to s, p, d, and f orbitals, respectively.)

magnetic m l integers from –l to 0 to +l orbital orientation

Example: What are the possible quantum numbers for a 1s electron? 9 RECAP SO FAR

Electrons in hydrogen have a fourth quantum number…

Spin: ms = +1/2, -1/2

That’s it: we are done with the hydrogen atom!

But the hydrogen atom was simple: it had only a single electron.

What about multielectron atoms? How does our simple picture get modified? Can we understand the periodic table? 10 WHY ARE MULTIELECTRON ATOMS SO HARD? Theoretical chemists (like me) spend their whole lives studying multielectronic systems.

They are notoriously hard to study because…

1.

2.

RAPIDLY

Z = atomic number = # of protons in the nucleus INCREASING COMPLEXITY! 11 MULTIELECTRON ATOMS For multielectron atoms, we can’t write down an exact formula for the energy of each orbital. The problem is just too darn hard! Still, we can make useful generalizations…

In multielectron atoms…

Orbitals fill in the order…

< < < < < < < … 12 SCREENING AND PENETRATION Why do orbitals fill in this order? For example, why is a 2s orbital lower in energy than a 2p orbital?

Screening:

Penetration:

(Note: As you can see, shielding is purely a multielectron effect. In hydrogen, 2s and 2p orbitals are degenerate.) 13 THE AUFBAU PRINCIPLE The aufbau principle states that orbitals fill in order of increasing energy. Here is a simple mnemonic:

l=0 l=1 l=2 l=3

n=1

n=2

n=3

n=4 Luckily, you don’t even have to memorize this mnemonic. On the exam, you get to use an n=5 AWESOME cheat sheet! That cheat sheet is called… n=6 14 THE PERIODIC TABLE

What is the for He?

What is the electron configuration for P? 15 ELECTRON CONFIGURATIONS: A FEW RULES

Pauli Exclusion Principle:

(Otherwise there would be no chemistry!)

Hund’s Rule:

(Allows electrons to better penetrate the nucleus!) Magnetism and matter:

A substance with any unpaired electrons is (attracted to external magnetic field).

A substance with all electrons paired is (unaffected by external magnetic field). 16