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  • 1.Chapter 7Quantum Theory and Atomic Structure7-1

2. Goals & Objectives See the following Learning Objectives on pages 283 and 322. Understand these Concepts: 7.1-7, 9-12; 8.1-8. Master these Skills: 7.1-2, 5; 8.1-2. 7-2 3. The Wave Nature of Light Visible light is a type of electromagnetic radiation. The wave properties of electromagnetic radiation are described by three variables: - frequency ( , cycles per second - wavelength ( ), the distance a wave travels in one cycle - amplitude, the height of a wave crest or depth of a trough.The speed of light is a constant: c= x = 3.00 x 108 m/s in a vacuum 7-3 4. Figure 7.1 The reciprocal relationship of frequency and wavelength.7-4 5. Figure 7.27-5Differing amplitude (brightness, or intensity) of a wave. 6. Figure 7.37-6Regions of the electromagnetic spectrum. 7. Sample Problem 7.1Interconverting Wavelength and FrequencyPROBLEM: A dental hygienist uses x-rays ( = 1.00) to take a series of dental radiographs while the patient listens to a radio station ( = 325 cm) and looks out the window at the blue sky ( = 473 nm). What is the frequency (in s-1) of the electromagnetic radiation from each source? (Assume that the radiation travels at the speed of light, 3.00x108 m/s.) PLAN:Use the equation c = to convert wavelength to frequency. Wavelengths need to be in meters because c has units of m/s. wavelength in units given use conversion factors 1 = 10-10 mwavelength in m = cfrequency (s-1 or Hz)7-7 8. Sample Problem 7.1 SOLUTION: For the x-rays:-10 = 1.00 x 10 m = 1.00 x 10-10 m 1c = 3.00 x 108 m/s = 1.00 x 10-10 mFor the radio signal:For the blue sky:7-8= c == c =3.00 x 108 m/s 10-2 m 325 cm x 1 cm 3.00 x 108 m/s 10-9 m 473 nm x 1 cm= 3.00 x 1018 s-1= 9.23 x 107 s-1= 6.34 x 1014 s-1 9. Figure 7.47-9Different behaviors of waves and particles. 10. Figure 7.57-10Formation of a diffraction pattern. 11. Energy and frequency A solid object emits visible light when it is heated to about 1000 K. This is called blackbody radiation. The color (and the intensity ) of the light changes as the temperature changes. Color is related to wavelength and frequency, while temperature is related to energy. Energy is therefore related to frequency and wavelength:E = nh7-11E = energy n is a positive integer h is Plancks constant 12. Figure 7.6Familiar examples of light emission related to blackbody radiation.Smoldering coal7-12Electric heating elementLightbulb filament 13. 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.E = nh where n can only be a whole number.7-13 14. Figure 7.77-14The photoelectric effect. 15. Sample Problem 7.2Calculating the Energy of Radiation from Its WavelengthPROBLEM: A cook uses a microwave oven to heat a meal. The wavelength of the radiation is 1.20 cm. What is the energy of one photon of this microwave radiation? PLAN:We know in cm, so we convert to m and find the frequency using the speed of light. We then find the energy of one photon using E = h .SOLUTION: E=h =7-15hc=(6.626 x 10-34) Js)(3.00 x 108 m/s) 10-2 m (1.20 cm)( ) 1 cm= 1.66 x 10-23 J 16. Figure 7.8A7-16The line spectrum of hydrogen. 17. Figure 7.8B7-17The line spectra of Hg and Sr. 18. Figure 7.9 Three series of spectral lines of atomic hydrogen.Rydberg equation1=R1 2 n11 n22R is the Rydberg constant = 1.096776x107 m-1 for the visible series, n1 = 2 and n2 = 3, 4, 5, ...7-18 19. The Bohr Model of the Hydrogen Atom 1913 Bohrs atomic model postulated the following: 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.7-19 20. 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. The energy of the photon (h ) equals the difference between the energies of the two energy states. When the E electron is in any orbit higher than n = 1, the atom is in an excited state.7-20 21. Figure 7.107-21A quantum staircase as an analogy for atomic energy levels. 22. Figure 7.117-22The Bohr explanation of three series of spectral lines emitted by the H atom. 23. Tools of the Laboratory Figure B7.1Flame tests and fireworks.strontium 38Srcopper 29CuThe flame color is due to the emission of light of a particular wavelength by each element.7-23Fireworks display emissions similar to those seen in flame tests. 24. Tools of the Laboratory Figure B7.27-24Emission and absorption spectra of sodium atoms. 25. Light Waves Atomic Emission Spectra Light in discrete lines; individual wavelengths, not continuum 26. Tools of the Laboratory Figure B7.37-26Components of a typical spectrometer. 27. Tools of the Laboratory Figure B7.4 Measuring chlorophyll a concentration in leaf extract.7-27 28. 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:h = mu 7-28m = massu = speed in m/s 29. Figure 7.127-29Wave motion in restricted systems. 30. Table 7.1The de Broglie Wavelengths of Several ObjectsSubstanceMass (g)Speed (m/s)slow electron9x10-281.0fast electron9x10-285.9x1061x10-10alpha particle6.6x10-241.5x1077x10-15 7x10-29one-gram mass1.00.01baseball14225.0Earth7-306.0x10273.0x104(m)7x10-42x10-34 4x10-63 31. Figure 7.15CLASSICAL THEORY Matter particulate, massiveEnergy continuous, wavelikeMajor observations and theories leading from classical theory to quantum theorySince matter is discontinuous and particulate, perhaps energy is discontinuous and particulate.Observation Blackbody radiationTheoryPhotoelectric effectEnergy is quantized; only certain values allowed Einstein: Light has particulate behavior (photons)Atomic line spectra7-31Planck:Bohr:Energy of atoms is quantized; photon emitted when electron changes orbit. 32. Figure 7.15 continued Since energy is wavelike, perhaps matter is wavelike. Observation Davisson/Germer: Electron beam is diffracted by metal crystalTheory deBroglie: All matter travels in waves; energy of atom is quantized due to wave motion of electrons Since matter has mass, perhaps energy has massObservation Compton: Photons wavelength increases (momentum decreases) after colliding with electronTheory Einstein/deBroglie: Mass and energy are equivalent; particles have wavelength and photons have momentum. QUANTUM THEORY7-32Energy and Matter particulate, massive, wavelike 33. Heisenbergs Uncertainty Principle Heisenbergs Uncertainty Principle states that it is not possible to know both the position and momentum of a moving particle at the same time. xmu h 4x = position u = speedThe more accurately we know the speed, the less accurately we know the position, and vice versa.7-33 34. Sample Problem 7.5Applying the Uncertainty PrinciplePROBLEM: An electron moving near an atomic nucleus has a speed 6x106 m/s 1%. What is the uncertainty in its position ( x)? The uncertainty in the speed ( u) is given as 1% (0.01) of 6x106 m/s. We multiply u by 0.01 and substitute this value into Equation 7.6 to solve for x.PLAN:SOLUTION:u = (0.01)(6x106 m/s) = 6x104 m/s xm u h 4x7-34h 4 m u6.626x10-34 kgm2/s4 (9.11x10-31 kg)(6x104 m/s) 1x10-9 m 35. The Quantum-Mechanical Model of the Atom The matter-wave of the electron occupies the space near the nucleus and is continuously influenced by it. The Schrdinger wave equation allows us to solve for the energy states associated with a particular atomic orbital. The square of the wave function gives the probability density, a measure of the probability of finding an electron of a particular energy in a particular region of the atom.7-35 36. The Shrodinger Atom (1926) Quantum Mechanical Model Wave equation is solved to obtain Wave function (orbital) Square of wave function 2 gives probability of finding an electron in a certain space7-36 37. The Schrdinger Equation H=Ewave mass of function d2 electron d2 d2 8 2m + + + 2 2 2 dx dy dz h2 how changes in space7-37potential energy at x,y,z (E-V(x,y,z) (x,y,z) = 0total quantized energy of the atomic system 38. Figure 7.16 Electron probability density in the ground-state H atom.7-38 39. The Shrdinger Atom (1926) Wave function for each electron in an atom contains four quantum numbers n l ml ms7-39Principle Angular momentum Magnetic Spin 40. Quantum Numbers and Atomic Orbitals An atomic orbital is specified by three quantum numbers. The principal quantum number (n) is a positive integer. The value of n indicates the relative size of the orbital and therefore its relative distance from the nucleus.The angular momentum quantum number (l) is an integer from 0 to (n -1). The value of l indicates the shape of the orbital.The magnetic quantum number (ml) is an integer with values from l to +l The value of ml indicates the spatial orientation of the orbital.7-40 41. Quantum Numbers and The Exclusion Principle Each electron in any atom is described completely by a set of four quantum numbers. The first three quantum numbers describe the orbital, while the fourth quantum number describes electron spin.Paulis exclusion principle states that no two electrons in the same atom can have the same four quantum numbers. An atomic orbital can hold a maximum of two electrons and they must have o