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Chapter 7: QUANTUM THEORY OF THE ATOM

Vanessa N. Prasad-Permaul Valencia Community College CHM 1045. Chapter 7: QUANTUM THEORY OF THE ATOM. THE WAVE NATURE OF LIGHT. Frequency, : The number of wave peaks that pass a given point per unit time (1/s) Wavelength, : The distance from one wave peak to the next (nm or m)

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Chapter 7: QUANTUM THEORY OF THE ATOM

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  1. Vanessa N. Prasad-Permaul Valencia Community College CHM 1045 Chapter 7: QUANTUM THEORY OF THE ATOM

  2. THE WAVE NATURE OF LIGHT Frequency, : The number of wave peaks that pass a given point per unit time (1/s) Wavelength, : The distance from one wave peak to the next (nm or m) Amplitude: Height of wave Wavelength x Frequency = Speed (m) x (s-1) = c (m/s) The speed of light waves in a vacuum in a constant c = 3.00 x 108 m/s

  3. THE WAVE NATURE OF LIGHT

  4. THE WAVE NATURE OF LIGHT • EXAMPLE 7.1 : WHAT IS THE WAVELENGTH OF • THE YELLOW SODIUM EMISSION, WHICH HAS A FREQUENCY OF 5.09 X 1014 S-1? • c = nl • =c • n • l=3.00 x 108 m/s • 5.09 x 1014 s-1 • = 5.89 x 10 -7 m • = 589 x 10-9 m • = 589 nm

  5. THE WAVE NATURE OF LIGHT EXERCISE 7.1 : The frequency of the strong red line in the spectrum of potassium is 3.91 x 1014 s-1. What is the wavelength of this light in nanometers? 5

  6. THE WAVE NATURE OF LIGHT • EXAMPLE 7.2 : WHAT IS THE FREQUENCY OF VIOLET LIGHT WITH A WAVELENGTH OF 408nm? • c = nl • n =c • l • n=3.00 x 108 m/s • 408 X 10-9 m • = 7.35 x 1014 s-1 6

  7. THE WAVE NATURE OF LIGHT EXERCISE 7.2 : The element cesium was discovered in 1860 by Robert Bunsen and Gustav Kirchoff, who found to bright blue lines in the spectrum of a substance isolated from a mineral water. One of the spectral lines of cesium has a wavelength of 456nm. What is the frequency? 7 7

  8. THE WAVE NATURE OF LIGHT THE ELECTROMAGNETIC SPECTRUM • Several types of electromagnetic radiation make up the electromagnetic spectrum

  9. QUANTUM EFFECTS & PHOTONS Atoms of a solid oscillate of vibrate with a definite frequency E = h  E = hc /  h = Planck’s constant, 6.626 x 10-34 J s E = energy 1 J = 1 kg m2/s2 When a photon hits the metal, it’s energy (hn) is taken up by the electron. The photon no longer exists as a particle and it is said to be absorbed

  10. QUANTUM EFFECTS & PHOTONS • Max Planck (1858–1947): proposed the energy is only emitted in discrete packets called quanta (now called photons). The amount of energy depends on the frequency: E = energy = frequency  = wavelength c = speed of light h = planck’s constant hc 34 - E h h 6 . 626 10 J s = n = = ´ × l

  11. QUANTUM EFFECTS & PHOTONS Albert Einstein (1879–1955): Used the idea of quanta to explain the photoelectric effect. • He proposed that light behaves as a stream of particles called photons • A photon’s energy must exceed a minimum threshold for electrons to be ejected. • Energy of a photon depends only on the frequency. E = h  THE PHOTOELECTRIC EFFECT: The ejection of electrons from the surface of a metal or from a material when light shines on it

  12. QUANTUM EFFECTS & PHOTONS EXAMPLE 7.3 : THE RED SPECTRAL LINE OF LITHIUUM OCCURS AT 671nm (6.71 x 10-7m). CALCULATE THE ENERGY OF ONE PHOTON OF THIS LIGHT. n = c = 3.00 x 108 m/s = 4.47 x 1014 s-1 l 6.71 x 10-7 m E = hn = 6.63 x 10-34 J.s * 4.47 x 1014 s-1 = 2.96 x 10-19 J

  13. QUANTUM EFFECTS & PHOTONS • EXERCISE 7.3 : The following are representative wavelengths in the infrared, ultraviolet and x-ray regions of the electromagnetic spectrum, respectively: • 1.0 x 10-6 m, 1.0 x 10-8 m and 1.0 x 10-10 m. • What is the energy of a photon of each radiation? • Which has the greatest amount of energy per photon? • Which has the least?

  14. THE BOHR THEORY OF THE HYDROGEN ATOM • Atomic spectra: Result from excited atoms emitting light. • Line spectra: Result from electron transitions between specific energy levels. • Blackbody radiation is the visible glow that solid objects emit when heated.

  15. THE BOHR THEORY OF THE HYDROGEN ATOM • BOHR’S POSTULATE • The stability of the atom (H2) • The line spectrum of the atom • ENERGY-LEVEL POSTULATE: An electron can only have specific energy level values in an atom called ENERGY LEVELS • E = RH where n = 1, 2, 3 • n2 • RH = 2.179 x 10-18 J • n = principle quantum number

  16. THE BOHR THEORY OF THE HYDROGEN ATOM • BOHR’S POSTULATE • The stability of the atom (H2) • The line spectrum of the atom • TRANSITIONS BETWEEN ENERGY LEVELS: An electron in an atom can change energy only by going from one energy level to another energy level. By doing so, the electron undergoes a transition. • An electron goes from a higher energy level (Ei) to a lower energy level (Ef) emitting light: • -DE = -(Ef - Ei) • DE = Ei - Ef

  17. THE BOHR THEORY OF THE HYDROGEN ATOM ENERGY LEVEL DIAGRAM OF THE HYDROGEN ATOM

  18. THE BOHR THEORY OF THE HYDROGEN ATOM • EXAMPLE 7.4 : WHAT IS THE WAVELENGTH OF THE LIGHT EMITTED WHEN THE ELECTRON IN A HYDROGEN ATOM UNDERGOES A TRANSITION FROM ENERGY LEVEL n = 4 TO LEVEL n = 2. • Ei = -RHEf = - RH • 42 22 • DE = -RH - -RH • 16 4 • E = -4RH + 16RH = -RH + 4RH = 3RH = hn • 64 16 16

  19. THE BOHR THEORY OF THE HYDROGEN ATOM EXAMPLE 7.4 : Cont… • = E = 3RH = 3 * 2.179 x 10-18 J = 6.17 x 1014 s-1 • h 16* h 16 * 6.626 x 10-34 J.s • l = c = 3.00 x 108 m/s = 4.86 x 10-7 m • n 6.17 x 10 14 s-1 • = 486 nm • (the color is blue-green)

  20. THE BOHR THEORY OF THE HYDROGEN ATOM EXERCISE 7.4 : Calculate the wavelength of light emitted from the hydrogen atom when the electron undergoes a transition from level 3 (n = 3) to level 1 (n = 1).

  21. THE BOHR THEORY OF THE HYDROGEN ATOM EXERCISE 7.5 : What is the difference in energy levels of the sodium atom if emitted light has a wavelength of 589nm?

  22. h h : For Light l = = mc p h h For a Particle : l = = mv p QUANTUM MECHANICS • Louis de Broglie (1892–1987): Suggested waves can behave as particles and particles can behave as waves. This is called wave–particle duality. m = mass in kg p = momentum (mc) or (mv) The de Broglie relation

  23. QUANTUM MECHANICS • EXAMPLE 7.5 : • CALCULATE THE l (in m) OF THE WAVE ASSOCIATED WITH A 1.00 kg MASS MOVING AT 1.00km/hr. • v = 1.00 km x 1000m x 1hr x 1min = 0.278m/s • hr 1km 60min 60 sec • l = h = 6.626 x 10-34 kg.m2/s2.s = 2.38 x 10-33m • mv 1.00kg * 0.278m/s

  24. QUANTUM MECHANICS • EXAMPLE 7.5 : cont… • B) WHAT IS THE l (in pm) ASSOCIATED WITH AN ELECTRON WHOSE MASS IS 9.11 x 10-31kg TRAVELING AT A SPEED OF 4.19 X 106 m/s ? • = h = 6.626 x 10-34 kg.m2/s2.s • mv9.11 x 10-31kg * 4.19 x 106 m/s • = 1.74 x 10-10 m • = 174pm

  25. QUANTUM MECHANICS EXERCISE 7.6 : Calculate the l (in pm) associated with an electron traveling at a speed of 2.19 x 106 m/s.

  26. QUANTUM MECHANICS QUANTUM MECHANICS ( WAVE MECHANICS): The branch of physics that mathematically describes the wave properties of submicroscopic particles UNCERTAINTY PRINCIPLE: A relation that states that the product of the uncertainty in position and the uncertainty in momentum (mass times speed) of a particle can be no smaller than Planck’s constant divided by 4p. SCHRODINGER’S EQUATION: Y2gives the probability of finding the particle within a region of space

  27. Quantum Mechanics • Niels Bohr (1885–1962): Described atom as electrons circling around a nucleus and concluded that electrons have specific energy levels. • Erwin Schrödinger (1887–1961): Proposed quantum mechanical model of atom, which focuses on wavelike properties of electrons.

  28. Quantum Mechanics • Werner Heisenberg (1901–1976): Showed that it is impossible to know (or measure) precisely both the position and velocity (or the momentum) at the same time. • The simple act of “seeing” an electron would change its energy and therefore its position.

  29. Quantum Mechanics • Erwin Schrödinger (1887–1961): Developed a compromise which calculates both the energy of an electron and the probability of finding an electron at any point in the molecule. • This is accomplished by solving the Schrödinger equation, resulting in the wave function

  30. QUANTUM NUMBERS According to QUANTUM MECHANICS: Each electron in an atom is described by 4 different quantum numbers: (n, l, m1 and ms). The first 3 specify the wave function that gives the probability of finding the electron at various points in space. The 4th (ms) refers to a magnetic property of electrons called spin ATOMIC ORBITAL: A wave function for an electron in an atom

  31. Quantum Numbers • Wave functions describe the behavior of electrons. • Each wave function contains four variables called quantum numbers: • Principal Quantum Number (n) • Angular-Momentum Quantum Number (l) • Magnetic Quantum Number (ml) • Spin Quantum Number (ms)

  32. QUANTUM NUMBERS • PRINCIPLE QUANTUM NUMBERS (n): • This quantum number is the one on which the energy of the electron in an atom principally depends; it can have any positive value (1, 2, 3 etc..) • The smaller n, the lower the energy. • The size of an orbital depends on n; the larger the • value of n, the larger the orbital. • Orbitals of the same quantum number (n) belong • to the same shell which have the following letters: • Letter: K L M N • n: 1 2 3 4

  33. Quantum Numbers • ANGULAR MOMENTUM QUANTUM NUMBER (l): Defines the three-dimensional shape of the orbital. • For an orbital of principal quantum number n, the value of l can have an integer value from 0 to n – 1. • This gives the subshell notation: Letter: s p d f g l: 0 1 2 3 4

  34. Quantum Numbers • Magnetic Quantum Number (ml): Defines the spatial orientation of the orbital. • For orbital of angular-momentum quantum number, l, the value of ml has integer values from –l to +l. • This gives a spatial orientation of:l = 0 giving ml = 0 l = 1 giving ml = –1, 0, +1l = 2 giving ml = –2, –1, 0, 1, 2, and so on…...

  35. Quantum Numbers • Magnetic Quantum Number (ml): –l to +l S orbital 0 P orbital -1 0 1 D orbital -2 -1 0 1 2 F orbital -3 -2 -1 0 1 2 3

  36. Quantum Numbers Table of Permissible Values of Quantum Numbers for Atomic Orbitals

  37. Quantum Numbers • Spin Quantum Number:ms • The Pauli Exclusion Principle states that no two electrons can have the same four quantum numbers.

  38. QUANTUM MECHANICS • EXAMPLE 7.6 : State whether each of the following sets of quantum numbers is permissible for an electron in an atom. If a set is not permissible, explain. • n = 1, l = 1, ml = 0, ms = +1/2 • NOT permissible: The l quantum number is equal to n. • IT must be less than n. • b) n = 3, l = 1, ml = -2, ms = -1/2 • NOT permissible: The magnitude of the ml quantum • number (that is the ml value, ignoring it’s sign) must be • greater than l.

  39. QUANTUM MECHANICS • EXAMPLE 7.6 : cont… • n = 2, l = 1, ml = 0, ms = +1/2 • Permissible • n = 2, l = 0, ml = 0, ms = +1 • NOT permissible: The ms quantum number can only be • +1/2 or -1/2.

  40. QUANTUM MECHANICS EXERCISE 7.7 : Explain why each of the following sets of quantum numbers is not permissible for an orbital: • n = 0, l = 1, ml = 0, ms = +1/2 • n = 2, l = 3, ml = 0, ms = -1/2 • n = 3, l = 2, ml = +3, ms = +1/2 • n = 3, l = 2, ml = +2, ms = 0

  41. Electron Radial Distribution • s Orbital Shapes: Holds 2 electrons

  42. Electron Radial Distribution • p Orbital Shapes: Holds 6 electrons, degenerate

  43. Electron Radial Distribution • d and f Orbital Shapes: d holds 10 electrons and f holds 14 electrons, degenerate

  44. Effective Nuclear Charge • Electron shielding leads to energy differences among orbitals within a shell. • Net nuclear charge felt by an electron is called the effective nuclear charge (Zeff). • Zeff is lower than actual nuclear charge. • Zeff increases toward nucleus ns > np > nd > nf

  45. Effective Nuclear Charge

  46. Example1: Light and Electromagnetic Spectrum • The red light in a laser pointer comes from a diode laser that has a wavelength of about 630 nm. What is the frequency of the light? c = 3 x 108 m/s

  47. Example 2: Atomic Spectra • For red light with a wavelength of about 630 nm, what is the energy of a single photon and one mole of photons?

  48. Example 3: Wave–Particle Duality • How fast must an electron be moving if it has a de Broglie wavelength of 550 nm? me = 9.109 x 10–31 kg

  49. Example 4: Quantum Numbers • Why can’t an electron have the following quantum numbers? (a) n = 2, l = 2, ml = 1 (b) n = 3, l = 0, ml = 3 (c) n = 5, l = –2, ml= 1

  50. Example 5: Quantum Numbers • Give orbital notations for electrons with the following quantum numbers: • n = 2, l = 1 (b) n = 4, l = 3 (c) n = 3, l = 2

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