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Chapter 7: The Quantum Mechanical Model of the Atom I. The Nature of Light: Its Wave-like Nature A. Light is a form of electromagnetic radiation 1. Composed of perpendicular oscillating waves, one for the ________ and one for the _________ field 1 B. All electromagnetic radiation waves move through space at the same, constant speed: 2 C. Wave Characteristics: 1. Wavelength (λ): distance from peak to peak 2. Amplitude: magnitude or intensity of wave 3 3. Frequency (ν) is the number of waves that pass a point in a given period of time 4. The total energy is proportional to the amplitude of the waves and the frequency a. Larger amplitude = b. Higher frequency = 4 5. Wavelength and Frequency are related by the speed of light: α. ν in s-1, λ in meters! b. Higher frequency = higher energy c. Higher wavelength = lower energy 5 D. Electromagnetic Spectrum 6 E. Wave Interference 1. The interaction between waves is called ____________. 2. When waves interact so that they add to make a larger wave it is called ___________interference. 7 4. When waves interact so they cancel each other it is called __________ interference. 8 F. Wave Diffraction 1. When traveling waves encounter an obstacle or opening in a barrier that is about the same size as the wavelength, they bend around it – this is called __________. 9 2. 10 3. The diffraction of light through two slits separated by a distance comparable to the wavelength results in an _________________ of the diffracted waves. 11 G. Ex: The relationship between ν and λ: The longest wavelength of red light is λ = 750 nm. What is its frequency? H. Ex: For the 3G cell phone network, ν = 3 GHz. What is the wavelength of this electromagnetic radiation? 12 II. The Particle-like Nature of Light A. Light (and all electromagnetic radiation) exists as discrete packets called photons, each with an energy: 1. Ex: For the 3G cell phone network, ν = 3 GHz. What is the energy of this electromagnetic radiation? 13 2. Light is quantized : only available in discrete, whole numbers of photons. A. B. Ex: A 100 watt light bulb radiates 100 J per second. If we assume that λ = 525 nm only, how many photons are emitted each second? 14 III. Atomic Spectra and the Bohr Atom A. Sunlight is white light : contains a continous spectrum of all visible colors (all λ), some infrared and some ultraviolet too. 15 B. How do rainbows form? 16 C. Emission of light by atoms: 1. Flame tests for aqueous metal ions. Na K Li 17 2. Gas discharge tubes. Hg He H2 18 Ba 3. Analyzing the light given off by H2 gas in a discharge tube. 19 4. Analyzing other gases. H2 He O2 Ne 20 5. Atoms can also absorb light for an absorption spectrum: 21 5. Emission and Absorption spectra for hydrogen, H2(g). 22 D. The Bohr Model of the Atom: 1. Bohr s major idea was that the energy states of the atom were _________, and that the amount of energy in the atom was related to the electron s position in the atom. 2. The electrons travel in orbits that are at a fixed distance from the nucleus. 3. Electrons emit radiation when they jump from an orbit with higher energy down to an orbit with lower energy 23 E. A picture of the Bohr Model for Hydrogen Atoms: 24 F. For the hydrogen atom, the Rydberg equation relates λ to the n values: % 1 1( 1 = 1.097 ×107 m−1'' 2 − 2 ** λ & n2 n1 ) 1. € 2. A negative sign on λ can mean that the light is being emitted. Ex: What is the wavelength of light emitted as an electron jumps from n = 4 to n = 1? 25 3. From λ and the Rydberg equation, the energy of an electron in a hydrogen atom can be calculated: hc E= λ € % 1 1( 7 −1 1 = 1.097 ×10 m '' 2 − 2 ** λ & n2 n1 ) € 26 4. Ex: Draw a ladder diagram of energy levels for the n = 1, 3 and ∞ energy levels in kJ/mol. Define n = ∞ as the zero energy state. 27 F. The Wave-like Nature of Matter 1. For an electron, (m = 9.1x10–31 kg), the wavelike nature of matter is significant… 28 2. The deBroglie wavelength of matter is: 3. Ex: What is the wavelength of an electron traveling at 2.2x106 m/s, the velocity of an electron in the hydrogen atom? 29 4. Ex: All matter has wave-like properties. What is my wavelength as I walk across the room at 1 m/s? 30 G. When you try to observe the wave nature of the electron, you cannot observe its particle nature – and vice-versa. 1. wave nature = 2. particle nature = H. The wave and particle nature of the electron are complementary properties. 31 I. Heisenberg Uncertainty Principle: the product of the uncertainties in both the position and speed of a particle was inversely proportional to its mass. 1. 2. 3. 4. x = position, v = velocity, m = mass in kg h = Planck s constant 32 5. Ex: An electron traveling at 2.2x106 m/s has an uncertainty of 2x105 m/s in its velocity. What is the minimum uncertainty in its position? (Δx)(mΔv) ≥ h 4π € 33 6. For electrons, the uncertainty in its position and/or velocity prevent exact knowledge of its future position. 7. Exact knowledge of the electrons position collapses the diffraction pattern. 34 IV. Quantum Mechanics tells us about the location of the electrons in the atom. A. Schrödinger 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. B. The solutions to Schrödinger s Equation give 4 quantum numbers that define orbitals that the electrons are in. We look at the first 3 quantum numbers this chapter. The last is in Ch. 8. 35 1. The principal quantum number, n, describes the main energy level (or shell) that an electron occupies. a. The principal energy levels are very similar to the Bohr Energy levels. 36 2. The angular momentum quantum number, l, describes the shape of the orbital. Each value of l has specific a specific shape. 3. The magnetic quantum number, ml, designates the orientation of the orbital within the sublevel. 4. The first principal energy level: 37 5. The second principal energy level: n = 2, l = 0 to 1 for l = 0, ml = 0 for l = 1, ml = –1, 0, 1 38 6. The third principal energy level: n = 3, l = 0 to 2 for l = 0, ml = 0 for l = 1, ml = –1, 0, 1 for l = 2, ml = –2, –1, 0, 1, 2 39 7. The fourth principal energy level: n = 4, l = 0 to 3 for l = 0, ml = 0 for l = 1, ml = –1, 0, 1 for l = 2, ml = –2, –1, 0, 1, 2 for l = 3, ml = –3, –2, –1, 0, 1, 2, 3 40 8. The relationship between 1s, 2s and 3s. 9. The relationship between 2s and 2p. 10. p and d orbitals have lobes with different phases. 41 D. How to get from ψ to orbitals 1. ψ is a function that is a solutionto the Schrödinger equation. It is called a wavefunction. 2. ψ2 is the probability density: 42 3. The radial distribution function represents the total probability at a certain distance from the nucleus. It multiplies ψ2 times the volume at a given distance r from the nucleus. 43 4. The 2s and 3s orbitals have nodes. 44 E. Describing an Orbital 1. Each set of n, l, and ml describes one orbital 2. Orbitals with the same value of n are in the same principal energy level. 3. Orbitals with the same values of n and l are said to be in the same sublevel. 45 4. Ex: What are the quantum numbers and names (for example, 2s, 2p) of the orbitals in the n = 4 principal level? How many orbitals exist? 46 E. Using Microsoft Excel to Plot ψ, ψ2 and ΔV*ψ2 1. ψ = wavefunction 1s: psi vs. radius 0.0016 0.0014 0.0012 psi 0.001 0.0008 0.0006 0.0004 0.0002 0 0 50 100 150 200 250 300 350 400 450 radius (pm) 47 2. ψ2 = probability density of finding the electron at a certain distance from the nucleus: 1s: psi^2 vs. radius 0.0016 0.0014 0.0012 psi^2 0.001 0.0008 0.0006 0.0004 0.0002 0 0 50 100 150 200 250 radius (pm) 48 300 350 400 450 ΔV*ψ2 = total radial probability at a given radius = radial distribution function 3. 1s: Total Radial Probability vs. Radius Total Radial Probability 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0 50 100 150 Radius (pm) 49 1s: Total Radial Probability vs. Radius Total Radial Probability 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0 50 100 150 200 250 300 Radius (pm) 50 200 250 300