Chapter 7. The Quantum Mechanical Model of the Atom

Transcription

1 Chapter 7 The Quantum Mechanical Model of the Atom

2 Quantum Mechanics The Behavior of the Very Small Electrons are incredibly small. Electron behavior determines much of the behavior of atoms. Directly observing electrons in the atom is impossible, the electron is so small that observing it changes its behavior.

3 The Beginnings of Quantum Mechanics The field of quantum mechanics began with the studies of physicists in the early the 20th century. Max Planck (1918) Albert Einstein (1921) Neils Bohr (1922) Arthur Compton (1927) Louis de Broglie (1929) Werner Heisenberg (1932) P. A. M. Dirac (1933) Erwin Schrödinger (1933)

4 The Path from Classical Theory to Quantum Theory Classical Physical Theory Matter particulate massive Quantum Theory Energy continuous wavelike Could energy be discontinuous and quantized? Could matter have wavelike properties? Observation Theory Conclusion Black-body radiation Photoelectric effect Atomic line spectra Planck Einstein Bohr Energy is quantized Light has particulate behavior Energy of atoms is quantized

5 The Electromagnetic Spectrum RedOrangeYellowGreenBlueViolet Shorter wavelengths of energy have higher energy than longer wavelengths: Radiowaves have the lowest energy. Gamma rays have the highest energy.

6 The Relationship Between Wavelength and Frequency ν = c λ ν = frequency c = speed of light λ = wavelength

7 The Electromagnetic Spectrum RedOrangeYellowGreenBlueViolet Shorter wavelengths of energy have higher energy than longer wavelengths: Energy Radiowaves have the lowest energy. Increases Gamma rays have the highest energy.

8 Blackbody Radiation Intensity increases from right to left on the curve as wavelength decreases. As the wavelength continues to decrease, however,intensity reaches a maximum and then drops off to zero. Max Planck

9 Blackbody Radiation The dependence of the intensity of blackbody radiation on wavelength at two different temperatures. Intensity increases from right to left on the curve as wavelength decreases. As the wavelength continues to decrease, intensity reaches a maximum and then drops off to zero.

10 ν = c λ = x Jᐧs

11 The Photoelectric Effect Observation: Many metals emit electrons when a light shines on their surface. Classic wave theory: Light energy is transferred to the electron, which is then ejected from the metal. Prediction: If the wavelength of light is made shorter, or the light waves intensity made brighter, more electrons should be ejected.

12 The Photoelectric Effect A beam of white light is dispersed into its wavelength components by a quartz prism and falls on a metal sample (potassium, in this case). Light of the highest frequencies (violet and ultraviolet) produces the most energetic photoelectrons (longest arrows). Light of lower frequencies (for example, orange) results in less energetic photoelectrons (shorter arrows).

13 The Photoelectric Effect Light with a frequency lower than 4.23 x s-1 (710 nm) produces no photoelectric effect at all on potassium, regardless of how bright (intense) the light is.

14 White Light Produces a Continuous Spectrum

15 Atomic Spectra When atoms or molecules absorb energy, that energy is often released as light energy. When that emitted light is passed through a prism, a pattern of particular wavelengths of light is seen that is unique to that type of atom or molecule the pattern is called an emission spectrum. non-continuous can be used to identify the material

16 The Electromagnetic Spectrum RedOrangeYellowGreenBlueViolet Shorter wavelengths of energy have higher energy than longer wavelengths: Radiowaves have the lowest energy. Gamma rays have the highest energy.

17 The Relationship Between Wavelength and Frequency ν = c λ ν = frequency c = speed of light λ = wavelength

18 The Electromagnetic Spectrum RedOrangeYellowGreenBlueViolet Shorter wavelengths of energy have higher energy than longer wavelengths: Energy Radiowaves have the lowest energy. Increases Gamma rays have the highest energy.

19 Exciting Gas Atoms to Emit Light with Electrical Energy

20 Identifying Elements with Flame Tests Na K Li Ba

21 Emission Spectra

22 Emission vs. Absorption Spectra Spectra of Mercury

23 Examples of Spectra

24 The Bohr Model of the Atom Neils Bohr ( ) The energy of the atom is quantized. The amount of energy in the atom is related to the electron s position in the atom. Quantized means that the atom could only have very specific amounts of energy. Bohr correlated these allowed energy levels with allowed radii of electron orbits.

25 The Bohr Model-Potential Energy of Electrons Energy Radius

26 The Bohr Model-Allowed Radii of Electron Orbits Z = atomic number ao = a constant derived from the mass and charge of the electron, Planck s constant, and Coulomb s constant n = state of the atom n can only be an integer!!

27 The Bohr Model-Potential Energy of Electrons Energy Radius

28 The Bohr Model-The Hydrogen Atom En = ev/n n=1 n=2 n=3 n=4 n=5 radius n = n 2 rb 1rb 4rb 9rb 16rb 25rb nm nm nm nm 1.32 nm

29 The Bohr Model-The Hydrogen Atom

30 Bohr s Model The electrons travel in orbits that are at a fixed distance from the nucleus (stationary states). The energy of the electron is proportional to the distance the orbit was from the nucleus. Electrons emit radiation when they jump from an orbit with higher energy down to an orbit with lower energy. Emitted radiation is a photon of light. The distance between the orbits determines the energy of the photon of light produced.

31 Bohr s Model

32 2-Slit Interference

33 Diffraction

34 Wave Behavior of Electrons Louis de Broglie ( ) Particles could have wave-like character. The wavelength of a particle was inversely proportional to its momentum (m x v). Because it is so small, the wave character of electrons is significant.

35 Wave Behavior of Electrons allowed forbidden

36 1. Calculate the wavelength of an electron traveling at 2.65 x 10 6 m/s (Masselectron = 9.11 x kg)

37 2. Determine the wavelength of a neutron traveling at 1.00 x 10 2 m/s (Massneutron = x g)

38 3. Determine the wavelength of a 80 kg human traveling at 1.00 m/s 8.0 x 10 2 = 8.3 x m

39 Complementary Properties When you try to observe the wave nature of the electron, you cannot observe its particle nature and vice-versa. wave nature = interference pattern particle nature = position The wave and particle nature of the electron are complementary properties. (As you know more about one you know less about the other.)

40 Uncertainty Principle Werner Karl Heisenberg ( ) The product of the uncertainties in both the position and speed of a particle was inversely proportional to its mass. x = position, Δx = uncertainty in position v = velocity, Δv = uncertainty in velocity m = mass (m)(δx)(δv) h/4π This means that the more accurately you know the position of a small particle, such as an electron, the less you know about its speed and vice-versa.

41 For an electron, m = 9 x kg, (Δx)(Δv) 6 x 10-5 Re = 3 x m; for an uncertainty in position of 1 radius, the uncertainty in velocity is approximately 20,000,000,000 m/sec. For an neutron, m = 2 x kg, (Δx)(Δv) 3 x 10-8 Rn = 1 x m; for an uncertainty in position of 1 radius, the uncertainty in velocity is approximately 300,000 m/sec. For an 80 kg man, m = 80 kg, (Δx)(Δv) 2 x Rhuman = 1 m; for an uncertainty in position of 1 radius, the uncertainty in velocity is approximately m/sec.

42 Determinacy vs. Indeterminacy According to classical physics, particles move in a path determined by the particle s velocity, position, and forces acting on it. Because we cannot know both the position and velocity of an electron, we cannot predict the path it will follow. The best we can do is to describe the probability an electron will be found in a particular region using statistical functions. Electron energy and position are complementary because KE = ½mv 2. Statistical Mechanics For an electron with a given energy, the best we can do is describe a region in the atom of high probability of finding it.

43 Trajectory vs. Probability

44 Schrödinger s Equation Erwin Schrödinger ( ) Schödinger s Equation allows us to calculate the probability of finding an electron with a particular amount of energy at a particular location in the atom. Solutions to Schödinger s Equation produce many wave functions, Ψ. A plot of distance vs. Ψ 2 represents an orbital, a probability distribution map of a region where the electron is likely to be found.

45 Solutions to the Wave Function, Ψ Calculations show that the size, shape, and orientation in space of an orbital are determined by three integer terms in the wave function. These integers are called quantum numbers. principal quantum number, n angular momentum quantum number, l magnetic quantum number, ml

46 The Quantum Numbers n, l, ml

47 Principal Quantum Number, n Corresponds to Bohr s energy level. The principal quantum number n can be any integer 1. The larger the value of n, the more energy the orbital has. Energies are defined as being negative. (An electron would have E = 0 when it just escapes the atom.) The larger the value of n, the larger the orbital. As n gets larger, the amount of energy between orbitals gets smaller.

48 Principal Energy Levels in Hydrogen

49 Angular Momentum Quantum Number, l The angular momentum quantum number determines the shape of the orbital. l can have integer values from 0 to (n 1). Each value of l is called by a particular letter that designates the shape of the orbital. s orbitals (l =0) are spherical p orbitals (l =1) are like two balloons tied at the knots d orbitals (l =2) are mainly like four balloons tied at the knot f orbitals (l = 3) are mainly like eight balloons tied at the knot

50 Magnetic Quantum Number, ml The magnetic quantum number is an integer that specifies the orientation of the orbital. Values are integers from l to +l including zero. Gives the number of orbitals of a particular shape. Ex: when l = 2, the values of ml are 2, 1, 0, +1, +2; which means there are five orbitals with l = 2

51 The Hierarchy of Quantum Numbers for Atomic Orbitals Name, Symbol (Property) Allowed Values Quantum Numbers Principal, n (size, energy) Positive integer (1, 2, 3,...) Angular momentum, l (shape) 0 to n Magnetic, m l (orientation) -l,,0,,+l

52 Describing Orbitals

53 Describing an Orbital Each set of n, l, and ml describes one orbital. (the position of one electron) Orbitals with the same value of n are in the same principal energy level (or principal shell). Orbitals with the same values of n and l are said to be in the same sublevel (or subshell).

54 4. What are the quantum numbers and names of the orbitals in the n = 1 principal level? How many orbitals exist? n = 1 l : 0 n =1, l = 0 (s) ml : 0 1 orbital 1s total orbitals: 1 = 1 2 : n 2

55 5. What are the quantum numbers and names of the orbitals in the n = 2 principal level? How many orbitals exist? n = 2 l : 0,1 n = 2, l = 0 (s) ml : 0 1 orbital 2s n = 2, l = 1 (p) ml : 1,0,+1 3 orbitals 2p total of 4 orbitals: = 2 2 : n 2

56 6. What are the quantum numbers and names of the orbitals in the n = 3 principal level? How many orbitals exist? n = 3 l : 0,1,2 n = 3, l = 0 (s) ml : 0 1 orbital 3s n = 3, l = 1 (p) ml : 1,0,+1 3 orbitals 3p n = 3, l = 2 (d) ml : 2, 1,0,+1,+2 5 orbitals 3d total of 9 orbitals: = 3 2 : n 2

57 7. What are the quantum numbers and names of the orbitals in the n = 4 principal level? How many orbitals exist? n = 4 l : 0, 1, 2, 3 n = 4, l = 0 (s) n = 4, l = 1 (p) n = 4, l = 2 (d) n = 4, l = 3 (f) ml : 0 1 orbital 4s ml : 1,0,+1 3 orbitals 4p ml : 2, 1,0,+1,+2 5 orbitals 4d ml : 3, 2, 1,0,+1,+2,+3 7 orbitals 4f total of 16 orbitals: = 4 2 : n 2

58 Electron Arrangement of Elements

59 Quantum Mechanical Explanation of Atomic Spectra Each wavelength in the spectrum of an atom corresponds to an electron transition between orbitals. When an electron is excited, it transitions from an orbital in a lower energy level to an orbital in a higher energy level. When an electron relaxes, it transitions from an orbital in a higher energy level to an orbital in a lower energy level. When an electron relaxes, a photon of light is released whose energy equals the energy difference between the orbitals. Each line in the emission spectrum corresponds to the difference in energy between two energy states.

60 Energy Transitions in Hydrogen

61 Bohr Model of H Atoms

62 Quantum Leaps

63 Predicting the Spectrum of Hydrogen The wavelengths of lines in the emission spectrum of hydrogen can be predicted by calculating the difference in energy between any two states. For an electron in energy state n, there are (n 1) energy states it can transition to, therefore (n 1) lines it can generate. Both the Bohr and Quantum Mechanical Models can predict these lines very accurately for a 1-electron system.

64 Energy Transitions in Hydrogen The energy of a photon released is equal to the difference in energy between two energy levels. ΔEelectron = Efinal state Einitial state Eemitted photon = ΔEelectron It can be calculated by subtracting the energy of the initial state from the energy of the final state. R H

65 8. Calculate the wavelength of light emitted when the hydrogen electron transitions from n = 3 to n = 2 n i, n f ΔE atom E photon λ ΔE atom = E photon ( - ) x (1/4-1/9 = 0.139) Ephoton = ( x J) = x x J J 3.03 x J 6.56

66 9. Calculate the wavelength of light emitted when the hydrogen electron transitions from n = 2 to n = 1 n i, n f ΔE atom E photon λ ΔE atom = E photon Ephoton = ( 1.64 x J) = 1.64 x J (1-¼ = 0.750) The unit is correct, the wavelength is in the UV, which is appropriate because it has more energy than 3 2 (in the visible).

67 9. Calculate the wavelength of light emitted when the hydrogen electron transitions from n = 6 to n = 5. n i, n f ΔE atom E photon λ ΔE atom = E photon Ephoton = ( x J) = x J (1/25-1/36 = 0.122) The unit is correct, the wavelength is in the infrared, which is appropriate because it has less energy than 3 2 (in the visible).

68 Wavelengths Associated with Hydrogen Electron Transitions 7.46 x10-6 m x10-7 m x10-7 m 2 1

69 The Shapes of Atomic Orbitals

70 The Shapes of Atomic Orbitals The l quantum number primarily determines the shape of the orbital. l can have integer values from 0 to (n 1) Each value of l is called by a particular letter that designates the shape of the orbital. s orbitals are spherical p orbitals are like two balloons tied at the knots d orbitals are mainly like four balloons tied at the knot f orbitals are mainly like eight balloons tied at the knot

71 Each principal energy level has one s orbital Lowest energy orbital in a principal energy state Spherical Number of nodes = (n 1) l = 0, the s orbital

72 1s, 2s and 3s Orbitals 1s 2s 3s

73 l = 1, p orbitals Each principal energy state above n = 1 has three p orbitals ml = 1, 0, +1 Each of the three orbitals points along a different axis px, py, pz 2 nd lowest energy orbitals in a principal energy state Two-lobed One node at the nucleus, total of n nodes

74 p orbitals

75 l = 2, d orbitals Each principal energy state above n = 2 has five d orbitals ml = 2, 1, 0, +1, +2 Four of the five orbitals are aligned in a different plane the fifth is aligned with the z axis, d z 2, d xy, d yz, d xz, d z2 y2 3 rd lowest energy orbitals in a principal energy level Mainly four-lobed Planar nodes higher principal levels also have spherical nodes

76 d orbitals

77 l = 3, f orbitals Each principal energy state above n = 3 has seven d orbitals ml = 3, 2, 1, 0, +1, +2, +3 4 th lowest energy orbitals in a principal energy state Mainly eight-lobed Planar nodes higher principal levels also have spherical nodes

78 f orbitals

79 Why are Atoms Spherical?