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Multielectron Atoms, As We Will See in the Next Lecture.) 16 ORBITRON: SECOND BEST ROBOT in PS1 Chemical Bonding, Energy, 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-matter interactions: reflection, refraction, absorption Introduction to Quantum Mechanics White light vs. atomic line spectra Bohr model of hydrogen Photochemistry 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 SPECTRUM 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. electrons) 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 LIGHT-MATTER INTERACTIONS: ABSORPTION Materials can also absorb certain wavelengths of light. When a material absorbs a particular wavelength in the visible light range, its color is usually roughly to the wavelengths of light absorbed. (But this is not completely straightforward. Recall the Harvard Color Predictor.) Example: chlorophyll molecules in plant leaves 12 LIGHT VS. ATOMS Take out the diffraction grating you picked up! Look at the ceiling lights: Now let’s look at excited atoms: The spectrum is… The emission spectrum 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 photon proportional to the frequency of light. Niels Bohr proposed that the possible energies of an electron in a hydrogen atom are quantized. Electrons jump from one level to another by absorbing or emitting photons 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 hydrogen atom: Minus sign: Plus sign: The corresponding wavelength of the photon is: Ionization is the complete removal of an electron. We can calculate the ionization energy 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 ground state 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? λ = 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 0 L 0 L 0 L Position (x) Position (x) Position (x) 2 |ψ n (x) | 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 molecule. 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 ultraviolet light of 217 nm.
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