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Quantum Mechanics Chapter 7 §4-5

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The de Broglie Relation 1924 1924 All matter has a wave-like nature… All matter has a wave-like nature… Wave-particle duality… Wave-particle duality… All matter and energy exhibit wave-like and particle- like properties.

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The de Broglie Relation The de Broglie Equation relates the wavelength of a particle to its momentum. The de Broglie Equation relates the wavelength of a particle to its momentum. Wavelength Planck’s constant 6.626x10 -34 Js Velocity, m/s Mass, kg

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The de Broglie Relation Compare the wavelengths of (a) an electron traveling at a speed of one-hundredth the speed of light with (b) that of a baseball of mass 0.145 kg having a speed of 26.8 m/s (60.0 mi/hr). Compare the wavelengths of (a) an electron traveling at a speed of one-hundredth the speed of light with (b) that of a baseball of mass 0.145 kg having a speed of 26.8 m/s (60.0 mi/hr). (a) the electron What is the mass of an electron? What is the electron speed if it is one-hundredth the speed of light?

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The de Broglie Relation Compare the wavelengths of (a) an electron traveling at a speed of one-hundredth the speed of light with (b) that of a baseball of mass 0.145 kg having a speed of 26.8 m/s (60.0 mi/hr). Compare the wavelengths of (a) an electron traveling at a speed of one-hundredth the speed of light with (b) that of a baseball of mass 0.145 kg having a speed of 26.8 m/s (60.0 mi/hr). (b) the baseball

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The de Broglie Relation Compare the wavelengths of (a) an electron with (b) that of a baseball. Compare the wavelengths of (a) an electron with (b) that of a baseball. What does that mean? (a)The electron (2.43x10 -10 m) (a)The baseball (1.71x10 -34 m)

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The Schroedinger Equation Schroedinger combined Planck’s photons, Einstein’s wave-particle duality, and de Broglie’s idea that all energy and matter follow the wave particle duality into one equation (the wave function) for the electron: Schroedinger combined Planck’s photons, Einstein’s wave-particle duality, and de Broglie’s idea that all energy and matter follow the wave particle duality into one equation (the wave function) for the electron: No, you don’t have to memorize it. This created the basis for Quantum mechanics.

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Quantum Mechanics Quantum Mechanics of an atom are divided into four quantum numbers: Quantum Mechanics of an atom are divided into four quantum numbers: n m m s m s Principle Quantum Number – the number that represents the energy level Angular Momentum Quantum Number – Azimuthal – the number that represents the subshell Magnetic Quantum Number – the number that represents the orbital within the subshell Spin Quantum Number – the number that represents the electron’s spin

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Quantum Mechanics Quantum Mechanics of an atom are divided into four quantum numbers: Quantum Mechanics of an atom are divided into four quantum numbers: n m m s m s Electron Spin: m s = – 1 / 2 OR + 1 / 2 Energy level: n = 1 - Subshell: Based on which energy level the electron is in; = 0 - (n-1) Orbital: Based on which subshell the electron is in; m = – - +

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Quantum Mechanics Energy level n SubshellOrbital m Electron Spin m s 10 (s)0– 1 / 2 OR + 1 / 2 20 (s)0– 1 / 2 OR + 1 / 2 21 (p)–1, 0, +1– 1 / 2 OR + 1 / 2 30 (s)0– 1 / 2 OR + 1 / 2 31 (p)–1, 0, +1– 1 / 2 OR + 1 / 2 32 (d)–2, –1, 0, +1, +2– 1 / 2 OR + 1 / 2 40 (s)0– 1 / 2 OR + 1 / 2 41 (p)–1, 0, +1– 1 / 2 OR + 1 / 2 42 (d)–2, –1, 0, +1, +2– 1 / 2 OR + 1 / 2 43 (f)–3, –2, –1, 0, +1, +2, +3– 1 / 2 OR + 1 / 2

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Let’s Practice Determine the quantum numbers for… Determine the quantum numbers for… Na Na First, write out the electron configuration. 1s 2 2s 2 2p 6 Next, write out the four quantum numbers for the last electron in the electron configuration: n = = m = m s = 3s 1 3 Since the energy level is 3 0 Since the subshell is s, which is indicated by the number 0 0 Since the orbital is in the s subshell, so the only possible value is 0. ± 1 / 2 Spin must follow Pauli’s exclusion principle

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Let’s Practice Determine the quantum numbers for… Determine the quantum numbers for… W First, write out the electron configuration. 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 10 4p 6 5s 2 4d 10 5p 6 6s 2 4f 14 Next, write out the four quantum numbers for the last electron in the electron configuration: n = = m = m s = 5d 4 5 Since the energy level is 5 2 Since the subshell is d, which is indicated by the number 2 +1 Since the orbital is in the 4 th orbital in the d subshell: –2. –1, 0 +1, +2. ± 1 / 2 Spin must follow Pauli’s exclusion principle

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Let’s Practice Determine the quantum numbers for… Determine the quantum numbers for… Br Br First, write out the electron configuration. 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 10 Next, write out the four quantum numbers for the last electron in the electron configuration: n = = m = m s = 4p 5 4 Since the energy level is 4 1 Since the subshell is p, which is indicated by the number 1 0 Since the orbital is in the 2 nd orbital in the p subshell: –1, 0 +1. ± 1 / 2 Spin must follow Pauli’s exclusion principle

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Quantum Mechanics Bohr’s theory established the concept of atomic energy levels but did not thoroughly explain the “wave-like” behavior of the electron.

Quantum Mechanics Bohr’s theory established the concept of atomic energy levels but did not thoroughly explain the “wave-like” behavior of the electron.

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