CHAPTER 7 QUANTUM THEORY OF THE ATOM Vanessa

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

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 2

THE WAVE NATURE OF LIGHT 3

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 l= 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 4

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

THE WAVE NATURE OF LIGHT EXAMPLE 7. 2 : WHAT IS THE FREQUENCY OF VIOLET LIGHT WITH A WAVELENGTH OF 408 nm? c = nl n= c l n = 3. 00 x 108 m/s 408 X 10 -9 m = 7. 35 x 1014 s-1 6

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 456 nm. What is the frequency? 7

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

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 m 2/s 2 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 9

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 E = hn = hc l h = 6. 626 ´ 10 -34 J × s 10

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. THE PHOTOELECTRIC EFFECT: E=h The ejection of electrons from the surface of a metal or from a material when light shines on it 11

QUANTUM EFFECTS & PHOTONS EXAMPLE 7. 3 : THE RED SPECTRAL LINE OF LITHIUUM OCCURS AT 671 nm (6. 71 x 10 -7 m). 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 12

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? 13

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. 14

THE BOHR THEORY OF THE HYDROGEN ATOM BOHR’S POSTULATE • The stability of the atom (H 2) • 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 n 2 RH = 2. 179 x 10 -18 J n = principle quantum number 15

THE BOHR THEORY OF THE HYDROGEN ATOM BOHR’S POSTULATE • The stability of the atom (H 2) • 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 16

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

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 = -RH 42 E f = - RH 22 DE = -RH - -RH 16 4 E = -4 RH + 16 RH = -RH + 4 RH = 3 RH = hn 64 16 16 18

THE BOHR THEORY OF THE HYDROGEN ATOM EXAMPLE 7. 4 : Cont… n = E = 3 RH = 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) 19

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). 20

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 589 nm? 21

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) For Light : l = For a Particle The de Broglie relation : l = h mc h mv = = h p 22

QUANTUM MECHANICS EXAMPLE 7. 5 : A) CALCULATE THE l (in m) OF THE WAVE ASSOCIATED WITH A 1. 00 kg MASS MOVING AT 1. 00 km/hr. v = 1. 00 km x 1000 m x 1 hr x 1 min = 0. 278 m/s hr 1 km 60 min 60 sec l= h = 6. 626 x 10 -34 kg. m 2/s 2. s = 2. 38 x 10 -33 m mv 1. 00 kg * 0. 278 m/s 23

QUANTUM MECHANICS EXAMPLE 7. 5 : cont… B) WHAT IS THE l (in pm) ASSOCIATED WITH AN ELECTRON WHOSE MASS IS 9. 11 x 10 -31 kg TRAVELING AT A SPEED OF 4. 19 X 106 m/s ? l= h = mv 6. 626 x 10 -34 kg. m 2/s 2. s 9. 11 x 10 -31 kg * 4. 19 x 106 m/s = 1. 74 x 10 -10 m = 174 pm 24

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. 25

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 4 p. SCHRODINGER’S EQUATION: Y 2 gives the probability of finding the particle within a region of space 26

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. 27

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. 28

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 29

QUANTUM NUMBERS According to QUANTUM MECHANICS: Each electron in an atom is described by 4 different quantum numbers: (n, l, m 1 and ms). The first 3 specify the wave function that gives the probability of finding the electron at various points in space. The 4 th (ms) refers to a magnetic property of electrons called spin ATOMIC ORBITAL: A wave function for an electron in an atom 30

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) 31

QUANTUM NUMBERS PRINCIPLE QUANTUM NUMBERS (n): This quantum number is the 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 32

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 33

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, +1 l = 2 giving ml = – 2, – 1, 0, 1, 2, and so on…. . . 34

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

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

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

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. a) 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. 38

QUANTUM MECHANICS EXAMPLE 7. 6 : cont… c) n = 2, l = 1, ml = 0, ms = +1/2 Permissible d) n = 2, l = 0, ms = +1 NOT permissible: The ms quantum number can only be +1/2 or -1/2. 39

QUANTUM MECHANICS EXERCISE 7. 7 : Explain why each of the following sets of quantum numbers is not permissible for an orbital: a) n = 0, l = 1, ml = 0, ms = +1/2 b) n = 2, l = 3, ml = 0, ms = -1/2 c) n = 3, l = 2, ml = +3, ms = +1/2 d) n = 3, l = 2, ml = +2, ms = 0 40

Electron Radial Distribution s Orbital Shapes: Holds 2 electrons 41

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

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

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 44

Effective Nuclear Charge 45

Example 1: 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 46

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? 47

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 48

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 49

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