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Homework Problems

Homework Problems Chapter 6 Homework Problems: 1, 6, 8, 16, 18, 30, 34, 41, 48, 51, 54, 58, 66, 68, 69, 76, 86, 91, 96, 108, 130. CHAPTER 6 Quantum Theory and the Electronic Structure of Atoms. Pre-Quantum Physics

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Homework Problems

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  1. Homework Problems Chapter 6 Homework Problems: 1, 6, 8, 16, 18, 30, 34, 41, 48, 51, 54, 58, 66, 68, 69, 76, 86, 91, 96, 108, 130

  2. CHAPTER 6 Quantum Theory and the Electronic Structure of Atoms

  3. Pre-Quantum Physics Scientists have generally believed that the behavior of objects in the universe can be summarized in terms of a small number of fundamental physical laws. By 1900, most scientists believed that the major laws of physics had been found. Conservation laws (conservation of mass, conservation of energy) Thermodynamics (First law, second law, third law) Laws of motion (Newton’s laws) Newton’s theory of gravity Maxwell’s equations for electricity and magnetism. Atomic theory

  4. Light Light is a general term used for electromagnetic radiation. Although some scientists (like Newton) believed that light was a particle phenomenon, by 1900 most scientists were convinced that light was a wave phenomenon, for reasons discussed below.

  5. Properties of Waves The following general terms are used for waves. wavelength () – The distance between successive peaks of the wave (SI units are m)

  6. frequency ()– The number of peaks that pass a given point per unit time (SI units are s-1, sometimes called Hertz (Hz)).

  7. Wavelength, Frequency, and Wave Velocity There is a general relationship between the wavelength, frequency, and velocity (c, the speed) of a wave.  = c where c is the speed of the wave (SI unit = m/s)

  8. Light Experimentally it is found that the speed at which light travels in a vacuum is independent of the wavelength or frequency of the light. For light in vacuum, c = 2.998 x 108 m/s. If we know either the frequency or the wavelength, we can use the relationship  = c to find the missing quantity. Example: A sodium lamp emits yellow light at a wavelength  = 589.3 nm. What is the frequency of the light?

  9. Example: A sodium lamp emits yellow light at a wavelength  = 589.3 nm. What is the frequency of the light? Since  = c, it follows that  = c/, so  = (2.998 x 108 m/s) = 5.087 x 1014 s-1 589.3 x 10-9 m

  10. Wavelength and Intensity The color of light (for visible light) depends on the wavelength of the light, while the intensity (energy per unit time) of light depends on the amplitude of the light.

  11. Electromagnetic Spectrum All light is fundamentally the same, and we often use the word “light” to mean any electromagnetic radiation. However, it is convenient to divide the spectrum into regions based on wavelength or frequency.

  12. Wave Properties of Light Classically, light was considered a wave phenomena. This was based on experimental observations such as the interference pattern for light observed in the two slit experiment. Interference – The increase or decrease in amplitude that occurs when two waves of the same wavelength are combined together, due to constructive or destructive interference.

  13. Constructive and Destructive Interference When two waves with the same wavelength combine together they can be in phase (peaks line up with one another) or out of phase (peaks of one wave line up with troughs of the other wave), or somewhere between the two. When the waves are in phase, adding the waves together increases the amplitude. This is called constructive interference. When the waves are out of phase, adding the waves together decreases the amplitude. This is called destructive interference.

  14. Two Slit Interference In the two slit experiment monochromatic light passes through two small openings in a barrier. A diffraction pattern is observed. white spots - places where light has illuminated the film dark spots - places where no light has illuminated the film If light were a particle, then no diffraction pattern should occur. Instead, there should be only two spots on the film corresponding to the two slits.

  15. Particle Properties of Light While it was generally accepted that light behaved as a wave, experiments at the end of the 19th century gave evidence that under some conditions light behaved like a particle. 1) Blackbody radiation (explained by Max Planck in 1900). 2) Photoelectric effect (explained by Albert Einstein in 1905). monochromatic light - light of a single color (wavelength) critical wavelength (0) – If  > 0, no electrons are detected

  16. Einstein Theory For the Photoelectric Effect Einstein explained the photoelectric effect as follows: 1) Light consists of particles (now called photons) which have an energy given by the Planck expression Ephoton = h = hc/ h = 6.626 x 10-34 J.s 2) There is a minimum energy (W, the binding energy or work function for the metal) required for an electron to escape from the metal. 3) When an electron in the metal absorbs a photon, the energy of the photon is transferred to the electron. There are two possibilities a) If the energy is less than the work function for the metal, the electron does not have enough energy to escape. b) If the energy is greater than the work function for the metal, the electron can escape and be detected. 0 then represents the boundary between these two regions

  17. Predictions From Einstein’s Theory Several predictions follow from Einstein’s theory for the photoelectric effect. 1) The critical wavelength 0 is related to the binding energy W. Ephoton = h = hc/0 = W for minimum photon energy 0 = hc/W 2) If  < 0 then electrons are produced instantaneously, even for dim light. If  > 0 no photons are produced. 3) The maximum kinetic energy of the ejected electrons can be found from conservation of energy Ephoton = h = KEmax + W KEmax = maximum KE of electron KEmax = h - W Therefore, a plot of KEmax vs  should have a slope equal to h.

  18. The x-intercept in the above plot can be used to find the work function for the metal. Ephoton = h = KEmax + W The minimum value for KEmax is 0 h0 = W , the work function for the metal. CONCLUSION – Light has both wave and particle properties.

  19. Atomic Spectra (Experimental) When a sample of an element is heated to high temperature it will emit light. The observed light emission is called a line spectrum.

  20. Atomic Spectra (Observed Results) 1) Light emission occurs only at particular values of wavelength. 2) Different elements emit light at different wavelengths (and so this light emission can be used to identify the presence of an element in a sample, and even determine the concentration of element present). 3) An element will absorb light at the same wavelengths at which it emits light.

  21. Hydrogen Spectrum For most elements the pattern of wavelengths where light is emitted or absorbed is complicated. For hydrogen, however, the pattern fits a simple equation called the Rydberg formula 1 = R 1 - 1 R = 0.01097 nm-1 ni, nf = 1, 2, 3, …  nf2 ni2 ni > nf Example: At what wavelength (in nm) will light be emitted for the transition ni = 3 --> nf = 2?

  22. Example: At what wavelength (in nm) will light be emitted for the transition ni = 3 --> nf = 2? 1/ = (0.01097 nm-1) [ (1/22) - (1/32)] = 1.524 x 10-3 nm-1  = 1 (1.524 x 10-3 nm-1) = 656.3 nm ni = 6 5 4 3

  23. The concept of energy levels can be used to explain the pattern of light absorption and emission for atoms. For example, for hydrogen the energy levels are given by the equation: En = - 2.18 x 10-18 J n = 1, 2, … n2 Other atomic spectra can also be explained in terms of energy levels, though there is no simple formula for their location as there is for hydrogen. However, there is no expla-nation for the origin of these energy levels by classical physics.

  24. Wave Properties of Matter Since light has “particle-like” properties it is reasonable to consider the possibility that matter might, under some conditions, have “wave-like” properties. The first person to explore this idea was Louis de Broglie, in 1924. E = mc2 (Einstein) E = hc (Planck)  If we set these equal to one another, then mc2 = hc or  = hc = h  mc2 mc For a particle, replace c (speed of light) with v (speed of the particle), to get deBroglie = h the de Broglie wavelength mv

  25. Interpretation of de Broglie Wavelength What does the de Broglie wavelength mean? It is interpreted to mean the length scale at which particles can exhibit wave behavior. X-ray diffraction electron diffraction Aluminum foil

  26. Using the de Broglie Equation Example: What is the speed of an electron whose de Broglie wavelength is  = 0.100 nm (approximate spacing between particles in a crystal)?

  27. Example: What is the speed of an electron whose de Broglie wavelength is  = 0.100 nm (approximate spacing between particles in a crystal)? Since deBroglie = h mv then v = h = (6.626 x 10-34 J.s) m(deBroglie) (9.109 x 10-31 kg) (0.100 x 10-9 m) = 7.27 x 106 m/s This is approximately 2.4 % of the speed of light. Electron diffraction was first observed experimentally in 1927 by the American physicists Clinton Davisson and Lester Germer.

  28. Summary From the previous discussion we get the following important points: 1) Light has both wave-like and particle-like properties, which can be brought out by different experiments. 2) Matter also has both wave-like and particle-like properties, which can also be brought out in different experiments. 3) Many systems, such as atoms, behave as if they can only have certain values for energy (energy levels).

  29. Quantum Mechanics Quantum mechanics is the theory developed to account for the above observations. It is based on solving the Schrodinger equation. [ (- 2/2m) 2 + V(x) ] n(x) = Enn(x) Schrodinger Wavefunction Probability equation Energy By solving the Schrodinger equation for a system we can find the possible values for energy and the probability of finding the particles making up the system at a particular location in space.

  30. Uncertainty Relationship One general property of systems described by quantum mechan-ics is that they must satisfy an uncertainty principle (Heisenberg, 1927). (x) (p) = (x) (mv)  (h/4) p = mv = momentum What this means is that it is not possible to assign a definite position for a particle in a system. All that can be given is the probability of finding the particle at a particular location. This is why, for example, we describe the electrons in an atom as a “cloud” of charge surrounding the nucleus.

  31. Hydrogen Atom The solutions to the Schrodinger equation for the hydrogen atom are given in terms of quantum numbers, a series of numbers used to label the solutions and which describe the solutions. For an electron in a hydrogen atom there are four quantum numbers, each which gives information about the state the electron is in.

  32. Quantum Numbers The quantum numbers and their possible values are as follows. Principal quantum number. Determines the energy and average distance between the electron and the nucleus, and orbital size. Angular momentum quantum number. Determines the shape of the electron cloud (orbital shape). Magnetic quantum number. Determines the orientation of the orbital. Spin quantum number. Determines the orientation of the electron spin. n = 1, 2, 3, …  = 0, 1, …, (n-1) m = 0, 1, 2, …,   ms =  1/2

  33. Example If n = 2, what are the possible values for ? If  = 2, what are the possible values for m? Are these possible sets of quantum numbers for an electron? n = 2,  = 1, m = -1, ms = ½ n = 3,  = 0, m = 1, ms = ½

  34. Example If n = 2, what are the possible values for ? Answer: Since  ranges in value from  = 0 up to  = (n – 1), the possible values for  are 0 or 1. If  = 2, what are the possible values for m? Answer: Since m  = 0, ±1, …, ± , the possible values for m are m = 2, 1, 0, -1, and -2. Are these possible sets of quantum numbers for an electron? n = 2,  = 1, m = -1, ms = ½ yes n = 3,  = 0, m = 1, ms = ½ no

  35. Relationship Between n and Energy For the hydrogen atom the energy of the electron depends only on the value of the quantum number n. En = (- 2.18 x 10-18 J) n = 1, 2, 3, ... n2

  36. Angular Momentum (Orbital) Quantum Number We use letters to designate the various electron orbitals, which describe the shape of the region where the electron is most likely to be found. orbital m 0 s orbital (1) 0 1 p orbital (3) 1, 0, -1 2 d orbital (5) 2, 1, 0, -1, -2 3 f orbital (7) 3, 2, 1, 0, -1, -2, -3 Note that the total number of orbitals corresponding to a particular value of  is equal to the number of possible values of m, or (2 + 1). We usually label the orbitals by both their value for n and their value for . So, for example, if n = 3 and  = 1, we have the 3p orbitals.

  37. Orbital Shape (s and p Orbitals) For each value of  there is a set of orbitals with distinctive shapes. These orbitals represent the region of space where it is most likely to find the electron.  = 0 (s-orbital) (1)  = 1 (p-orbitals) (3)

  38. Orbital Shape (d Orbitals)  = 2 (d-orbital) (5)

  39. Orbitals With Different Values of n The value of n does not affect the shape of an orbital. Therefore a 1s and a 2s orbital have the same shape, and the 2p and 3p orbitals have the same shape. There are two effects that n has on the orbitals: 1) The larger the value of n, the larger the orbital. This leads to the electron being further away from the nucleus. 2) Orbitals have radial nodes, or distances from the nucleus where the probability of finding the electron goes to zero. The number of radial nodes is given by the relationship # nodes = (n - ) - 1

  40. Size and Nodes For s-Orbitals 2s 3s # radial nodes = 1 # radial nodes = 2

  41. Spin Quantum Number (ms) The spin quantum number indicates whether the electron is spinning clockwise or counterclockwise, and can take on two values, +1/2 and - 1/2. Direct evidence for this quantum number was first found experimentally by Stern and Gerlach.

  42. Multi-electron Atoms There are two differences between hydrogen and other atoms, which have more than one electron. 1) Each electron in a multi-electron atom has its own set of four quantum numbers. These electrons must obey the Pauli exclusion principle. 2) The energy for an electron in a multi-electron atom depends on both n and . We label the various energy levels for the electrons by their value for n and the letter used to represent the value for . Example: n = 2,  = 0 2s orbital n = 3,  = 0 3s orbital n = 3,  = 2 3d orbital This dependence of energy on both n and  is a consequence of electron-electron repulsion.

  43. The Pauli Exclusion Principle Each electron in a multi-electron atom has its own set of four quantum numbers. An early observation concerning these quantum numbers was made by Wolfgang Pauli, and is called the Pauli Exclusion Principle. No two electrons in an atom or ion can have the same set of four quantum numbers. Note that as many as three of the quantum numbers can be the same, but at least one quantum number must be different. As we will see, the arrangement of elements in the periodic table and their periodic properties are in a real sense a consequence of the exclusion principle.

  44. Energy Ordering For Orbitals The usual ordering of energy for orbitals is as indicated below. Notice that the ordering is not strictly in terms of the quantum number n 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f ...

  45. Mnemonic Device For Energy Ordering We may use the following mnemonic device for the order in energy of the orbitals. The order in which the labels are crossed out below is the order of energies. 1s 2s 2p 3s 3p 3d 4s 4p 4d 4f 5s 5p 5d 5f … 6s 6p 6d 6f … So 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f ...

  46. Rules For Adding Electrons to Atoms There are three rules that must be followed when adding electrons to a multielectron atom to find the lowest energy state (ground state) of the atom. 1) Pauli principle - No two electrons can have the same set of four quantum numbers. 2) Aufbau principle - Electrons add to the lowest energy available orbital until that orbital is filled. 3) Hund’s rule - Electrons add in such a way as to make as many of the electrons as possible “spin up” (ms = 1/2). Electron configuration - A list of each electron containing orbital, in order of energy. A superscript to the right of the orbital is used to indicate how many electrons are present in the orbital.

  47. Element n  m ms electron configuration H (1 e-) 1 0 0 1/2 1s1 He (2 e-) 1 0 0 1/2 1s2 1 0 0 - 1/2 Li (3 e-) 1 0 0 1/2 1s22s1 1 0 0 - 1/2 2 0 0 1/2

  48. Element n  m ms electron configuration O (8 e-) 1 0 0 1/2 1s22s22p4 1 0 0 - 1/2 2 0 0 1/2 2 0 0 - 1/2 2 1 1 1/2 2 1 0 1/2 2 1 -1 1/2 2 1 1 - 1/2 Maximum number of electrons in an orbital s-orbital 2 = 2(2 + 1) p-orbitals 6 d-orbitals 10 f-orbitals 14

  49. Orbital Filling Diagram An orbital filling diagram is a picture where the orbitals for an atom are indicated by lines, and electrons by arrows pointing up (“spin up”) or down (“spin down”). Such a diagram is useful in identifying the total number of unpaired electron spins in an atom. Example: (Note – Can also use boxes for the orbitals.) O(8 e-) 1s22s22p4 __ __ __ __ __ 1s 2s 2p 2 unpaired electron spins Notice that if we are only interested in identifying unpaired electron spins we need only examine those orbitals that are partially filled. All electron spins will be paired up for filled orbitals. Also notice how we use Hund’s rule in filling the orbitals.

  50. Magnetic Properties of Atoms An atom placed in a magnetic field will be attracted or repelled by the field depending on the number of unpaired electron spins in the atom. There are two cases diamagnetic - Weakly repelled by an external magnetic field. Occurs when there are no unpaired electron spins. paramagnetic - Strongly attracted by an external magnetic field. Occurs when there are one or more unpaired electron spins. The number of unpaired spins can be determined by measuring the strength of the attraction felt by the atoms.

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