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  1. ELECTRONIC STRUCTURE OF ATOMS/ RELATIONSHIP BETWEEN ATOMS AND QUANTUM THEORY (CHAPTER 7) ENERGY What have we learned ? Atoms + energy emit light 1) Atoms are the…
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  • 1. ELECTRONIC STRUCTURE OF ATOMS/ RELATIONSHIP BETWEEN ATOMS AND QUANTUM THEORY (CHAPTER 7) ENERGY What have we learned ? Atoms + energy emit light 1) Atoms are the building blocks of matter. A GOOD EXAMPLE IS: 2) Atoms are composed of a dense nucleus ) m mp f (protons and neutrons) with electrons far Different elements (atoms) emit removed from the nucleus. light of different colors (red, green, 3) Stoichiometry. yellow, orange, blue). 4) Elements are arranged in a periodic table. 5) Behavior of gases (laws and theory). 6) Basics of thermochemistry. What we don't know ? M* M + light excited emission atom 1) Why atoms combine to form molecules ? 2) Why the periodic table has its particular arrangement of the elements ? M + light M* 3) What is the detailed structure of an atom absorption excited (electrons) ? atom 4) Why atoms combine to form compounds with particular formulas ? Light interacts with atoms in specific ways – 5) Why elements (main group, Group A) form gain information on the electronic structure of stable monatomic ions with a particular atoms by studying this interaction. charge (e.g., Na+, Ca2+, O2-, Cl-) ? MOLECULAR FORMULAS • Atoms combine to form molecules with specific formulas. Light is electromagnetic radiation and is a • How atoms combine depends on the wave and it has wavelike characteristics. distribution or arrangement of electrons around the nucleus of the atom (i.e., electronic structure). l i ) Wave: Vibrating disturbance by which energy • Early experimental evidence aided in is transmitted. developing an understanding of atomic structure: Water waves, sound waves, etc. – J. J. Thomson’s discovery of the electron. Waves are characterized by: frequency – Rutherford’s Nuclear Model of the Atom. wavelength amplitude 1
  • 2. Figure 7.2 Frequency/Amplitude(intensity) of Various Waves Figure 7.2 WAVE CHARACTERISTICS Speed of wave propagation =  X  1  distance Wavelength (  lambda): The distance distance      speed speed (length) between adjacent peaks on the wave.  time  time SI unit: meter (m). Electromagnetic radiation (radiant energy, light). q y Frequency (  nu): The number of waves Speed of light (c) = 3.00 X 108 m/s (in a vacuum) p g ( ) (wavelengths) that pass a point per second. (186,000 miles/s or 671,000,000 mph) The SI unit of frequency is the hertz (Hz). This radiation does not need a medium for SI units: 1/s or s-1 or hertz (Hz). propagation - can travel through a vacuum. c =    = constant Amplitude (intensity): The vertical distance from the mid-line to the peak (intensity). If  increases, then  decreases (inverse relationship). 2
  • 3. Example: Frequency and Wavelength What is the frequency () of radiation that has a wavelength () of 1.5 X 10-2 m ? Start this problem with c =   Rearrange the equation and solve for : R h i d l f c 3.00 X 10 8 m/s     2 X 1010 s -1  1.5 X 10 - 2 m or 2 X 1010 Hz Figure 7.4 Electromagnetic Radiation Max Planck (1900) Maxwell (1873): Developed a theory for light. Studied radiation emitted by He proposed that light waves consists of two matter (objects). All objects components; a vibrating electric field and a emit electromagnetic radiation. vibrating magnetic field. Hence, the term “electromagnetic radiation”. This radiation involves the transmission of energy from one place to another. Postulated that atoms and molecules can emit or absorb energy (radiation) only in discrete quantities. Figure 7.3 Energy in matter is quantized, not continuous !! 3
  • 4. QUANTIZED ENERGY PHOTOELECTRIC What is a quantum of energy ? EFFECT – The smallest increment of energy that can be These observations of the emitted or absorbed in the form of photoelectric effect could not be electromagnetic energy (light) by matter. explained by the wave theory of light. Energy (E) of a single quantum is given by: E = h Albert Ei t i (1905) Considered that li ht i Alb t Einstein (1905): C id d th t light is E = energy (J) composed of a stream of particles that are  = frequency (1/s) called photons (or quanta-packets of h = Planck's constant (6.63 X 10-34 J · s) energy). – Units: E = (J · s)(1/s) = J (energy) Extended Planck’s idea’s. PHOTOELECTRIC EFFECT: Light can cause Photoelectric Effect electrons to be ejected from a metal surface. light elec t r o n met al surf ace There is a threshold frequency () for electron ejection and if light has a  less than the threshold, no electrons are ejected. At a  greater than the threshold, the number of electrons ejected is proportional to the light intensity. Figure 7.5 4
  • 5. Each photon of light has an energy, E: E = h ·  (Planck’s ideas) Energy of a photon is directly proportional to the frequency. Energy of a p gy f photon is important and dictates p how light interacts with matter. Electromagnetic radiation (light) has both wave-like and particle-like properties. The dual nature of light!!! Figure 7.4 Problem: Quantized Energy A typical laser pointer has red light - Energy of Electromagnetic Radiation wavelength 630 nanometers (6.3 X 10-7 meters). • Given by E = h = hc/ What is the energy of one photon of light with  = 630 nm? • Short wavelengths ()  high energy; e.g., X rays - very dama in to life e X-rays damaging hc (6.63 X 10 - 34 J s)(3.0 X 10 8 m/s) • Long wavelengths ()  low energy; E  hν   e.g., radio waves - safe λ (6.3 X 10 - 7 m)  3.16 X 10 -19 J Electromagnetic Spectrum ATOMIC SPECTRA Radio waves Experimental observations of the interaction of Some Facts radiation with matter. Microwave Inverse relationship Atom + h Atom* absorption Infrared between  and  Atom* Atom + h emission Visible  (lambda) ranges from 1013 to 10-3 nm Atoms absorb and emit light with discrete energy Ultraviolet (’s and ’s). They can have only particular  (nu) ranges from amounts of energy – energy of atoms is quantized. 104 Hertz (Hz) to X-rays 1020 Hz Energy levels of atoms - it's the electrons that Gamma rays are important - energy levels of the electrons.   5
  • 6. Atomic Spectrum of Hydrogen Atoms Simplest element, it has the simplest spectrum. See a series of lines in the atomic spectrum of hydrogen atoms. Only discrete ’s and ’ of li ht are absorbed O l di t ’ d ’s f light b b d or emitted by hydrogen atoms. This resulting spectrum is unique for hydrogen - different from all other elements. Experiment to View the Atomic Spectrum of H Atomic Spectra of Hydrogen Numerous lines ranging from the ultraviolet ( < 400 nm) to the infrared region ( > 800 nm). Figure 7.6 J. J. Balmer (1885): Studied the lines in the visible portion of the the spectrum (400 nm to 700 nm). energy energy emission H2 H H* H + light Continuous Spectrum Found a simple equation ( F d i l ti (empirical) for calculating i i l) f l l ti the emission wavelengths in the visible region: 1  1 1  Figure 7.6  RH  2  2  RH = Rydberg constant λ 2 n  = 109,678 cm-1 Each type of atom (element) has its own unique spectrum (lines). n = 3, 4, 5, and 6 6
  • 7. Atomic Spectra of Hydrogen General equation for calculating all emission Will the lines in the hydrogen spectrum: photon be absorbed ? 1  1 1  RH = Rydberg constant  RH  2  2  = 109,678 cm-1 , λ n   i nf  Restriction: ni and nf are whole numbers 1st Bohr orbit If of the proper ranging from 1 to infinity. 2nd Bohr orbit energy, the photon is absorbed and the The above equation is called the Rydberg 3rd Bohr orbit electron jumps to a equation. higher energy orbit. Different Series of Spectral Lines for H atoms Absorption 1st Bohr orbit Lyman series: nf = 1; ni = 2, 3, 4,…(ultraviolet) 2nd Bohr orbit Electron can return to Balmer series: nf = 2; ni = 3, 4, 5,…(visible) 3rd Bohr orbit the ground state by Paschen series: nf = 3; ni = 4, 5, 6, …(infrared) emitting a photon of Brackett series:nf = 4; ni = 5, 6, 7, …(infrared) the same energy. EXPLANATION OF H-LINE SPECTRUM Niels Bohr (1913), Danish physicist Developed a theoretical model for the Emission hydrogen atom. Bohr viewed the electron as moving around the nucleus in only fixed (discrete) radii or orbits (planets in a solar system). 1st Bohr orbit This placed restrictions on the sizes of the orbits 2nd Bohr orbit and the energy that the electron can have in a 3rd Bohr orbit given orbit (departure from classical physics). 7
  • 8. Potential Energy Figure 7.9 Bohr described the potential energy of the Bohr’s Model of H-atom electron in the hydrogen atom by a simple (?) n = 1: First Bohr orbit, ground state, lowest equation: potential energy. 2 4 2π me E If the hydrogen atom in the ground state n2 h2 absorbs energy, then the electron goes to a m = mass of the electron (9 109 X 10-31 kg) (9.109 higher energy (larger) orbit (n = 2, 3, 4, ...); g gy ( g ) ( ) e = charge of the electron (1.602 X 10-19 C, called excited states. (1.519 X 10-14 kg1/2 m3/2 s-1) For transitions in the Bohr Model of the H-atom h = Planck’s constant (6.63 X 10-34 J · s) n = quantum number (1, 2, 3, ….) n = 1 to n = 2 absorbs energy 2 2π me 4 same energy R'H   2.18 X 10-18 J n = 2 to n = 1 emits energy h2 Bohr’s Hydrogen Atom Bohr’s Equation of Energy  R'  E n  - 2H  n  1, 2, 3, ...  (integers)  n  R’H: constant (2.18 X 10-18 J) n: quantum number (has values from 1 to  ) Negative sign: Potential energy of the atom < potential energy of H+ and a free electron. b  R’H Lowest energy, n = 1 (E < 0) Highest energy, n =  (E = 0)  R'  E n  - 2H  n  1, 2, 3, ...  (integers)  n  8
  • 9. Example: What is the change in energy of the hydrogen atom when the electron goes from the n = 3 to the n = 1 level ? Use the equation for the change in energy for H- atom:  1 1  ΔE  R' H  2  2 R  n n   i f   1 1 ΔE  2.18 X 10 -18 J  2  2   1.94 X 10-18 J 3 1  A photon of light energy is emitted, since ni > nf , E is negative. Figure 7.9 Energy of Transitions in Bohr H-Atom What is the frequency () and wavelength () of this photon ? Use Einstein‘s equation and solve for : Energy of the  R' H  ΔE  hν Initial State: E i   2  n    i  ΔE 1.94 X 10 -18 J ν   2.93 X 1015 Hz h 6 63 X 10 - 34 J  s 6.63  R'  Calculate wavelength () : Energy of Final E f   2H  State:  n  c  λν  f  c 3.00 X 10 8 m/s λ   1.02 X 10 - 7 m  102 nm ν 2.93 X 1015 /s Energy of Transitions in Bohr H-Atom  - R'   - R'  ΔE  E f  E i   2 H    2 H   n   n   f   i   1 1  ΔE  R' H  2  2   hν R n n   i f  If ni > nf , then a photon is emitted, E is negative. If ni < nf , then a photon is absorbed, E is positive. 9
  • 10. Relationship between Bohr’s equation and the de Broglie’s Hypothesis (1924) Rydberg equation: • If electromagnetic radiation (light) can have particle-like properties (photon), then electrons Bohr’s equation: may also possess wave-like properties. • Electrons do indeed have both particle and  1 1  ΔE  R' H  2  2  wave-like properties !!! Electrons are similar to n n  g light.  i f  • From Einstein's theory of relativity: Rydberg Equation: E  mc 2 1  1 1  m  mass  RH  2  2  c  speed of light λ n   i nf  This equation relates energy and mass. Rydberg constant calculated from Bohr’s theory: de Broglie’s Hypothesis ΔE  hv  hc 1  hc   Using photon equation for light: λ λ hc c   E  h  R' H in J   c  RH in cm 1  hc   Substitute mc2 for E and solve for : 1 2.18 X 10-18 J  1m        6.63 X 10 J  s 3.00 X 10 m/s  100 cm  - 34 8 hc mc 2   109,600 cm -1  and Agrees well with the known Rydberg constant h value (109,678 cm-1).  mc Bohr’s Theory de Broglie’s Hypothesis • Only works for the hydrogen atom; not a general theory, it is limited and incomplete. • Notice that the product, mc, is defined as momentum (i.e., mass X velocity). • Used some elements of "classical physics", which do not apply to atomic and subatomic • This equation then relates wavelength, particles !!!! mass, and velocity. • Questions for Bohr: • For an electron, its speed must be less than – Why is energy of the electron quantized? the speed of light, c. We can simply – Why are electrons restricted to only certain substitute in for the speed (velocity) of the orbits? electron, v. – Why are atoms stable (exist)? 10
  • 11. The wavelength of an electron (a particle) is given by: de Broglie Wavelength For a Baseball h  h  mv mv v  velocity (m/s) For the baseball: m v  momentum (particle property) (p p p y) Units : mass is 0.14 kg and velocity is 40 m/s kg  m 2 6.63 x 10 -34 J  s s  m (distance, wavelength)   1.2 x 10 -34 m kg  m (0.14 kg)(40 m/s) s Electrons, protons, and neutrons experience Wave properties of particles only apply to atomic diffraction - wave property !!!! and subatomic particles – not to baseballs !!!! Diffraction Pattern for Al with X-rays and Electrons QUANTUM MECHANICS (WAVE MECHANCIS) Erwin Schrödinger (1926) Nobel Prize (1933) Applies wave properties of matter to explain pp p p p atomic properties. Bohr’s theory: Electrons move around the nucleus in fixed (discrete) orbits (e.g., solar system). electrons X-rays Wave mechanics: Treats electrons as waves – Figure 7.14 difficult to perceive. Example: This theory leads to a series of mathematical Calculate the de Broglie wavelength for an functions called wave functions. electron travelling at 1 % the speed of light (3.00 X 106 m/s). The electron wave function is called an orbital h (atomic orbital). Use de Broglie’s relationship:   m v For an electron, The t Th term “orbital” i used to distinguish f “ bit l” is d t di ti i h from Bohr’s ideas of fixed orbits. m = 9.11 X 10-31 kg; v = 3.00 X 106 m/s 6.63 X 10 - 34 J  s λ  2.43 X 10 -10 m or 243 pm 9.11 X 10- 31 kg3.00 X 106 m/s  Follow the units through!!!! 11
  • 12. Interference of Waves Electron Waves Two waves with the same frequency add together if they are “in phase” (their peaks exactly match). Constructive interference. 4 wavelengths 3.6 wavelengths destructive interference Interference of Waves These electron wave functions are quantized. They must have integral numbers of Two waves with the same wavelengths, otherwise destructive interference frequency but exactly will result in annihilation of the electron wave. “out of phase” (peak of one wave matches a valley of the other wave) cancel f h h ) l out (destroyed). Energy changes in an atom - simply changes in wave functions (patterns) which are quantized. Destructive interference. Electrons exist only in those regions in which their waves reinforce each other. In wave mechanics, the electrons do not orbit the Allowed (Stable) Orbital: nucleus in fixed, circular orbits (Bohr’s ideas). 4 wavelengths – Rather, the electron wave function takes on a constructive interference particular shape, a probability of finding the electron in a particular volume of space. An atomic orbital is characterized by: Forbidden (Unstable) Orbital energy 4.5 wavelengths – size destructive interference shape orientation in space 12
  • 13. The “p” Orbitals: Are not spherical, consist of TYPES (SHAPES) OF ATOMIC ORBITALS two lobes of electron density about the nucleus. The “s” Orbitals: Have a spherical shape: The “s” Orbitals: Have a spherical shape: There are three individual “p” orbitals, differ only in their orientation in space: Figure 7.18 Figure 7.20 The three individual “p” orbitals, represented The “s” Orbitals: Have a spherical shape: in an atom: 13
  • 14. The “d” Orbitals: Have a more complex shape (four QUANTUM NUMBERS lobes of electron density about the nucleus). Five individual “d” orbitals, differ only in their • PRINCIPAL QUANTUM NUMBER (n) Determines orientation in space: the energy and size of an orbital. – Allowed values  1, 2, 3,…  (positive integers) • As n increases the energy and size of the increases, Figure 7.21 orbital increases. • All orbitals that have the same value for "n" are in the same "shell". n = 1 shell (first shell) n = 2 shell (second shell) n = 3 shell (third shell) etc. The “f” Orbitals: Have a very complex shape (eight Angular Momentum (Shape) Quantum Number (l): lobes of electron density about the nucleus). This quantum number determines the shape of There are seven individual “f” orbitals that vary the orbital. only in their orientation in space. Divides a shell into smaller groups called subshells. Only certain "l" values are possible-limited by "n": Can have integral values from zero to n – 1 (0, 1, 2, 3, (0 1 2 3 ....n - 1) Shells Subshells n=1 l=0 n=2 l = 0, 1 n=3 l = 0, 1, 2 Angular Momentum (Shape) Quantum Number (l) Typically, a letter code is used to specify the Quantum Numbers “l” values: l 0 1 2 3 4 5…. Bohr’s theory used a single quantum number (n) to describe an orbit. letter s p d f g h…. Quantum mechanics theory (wave mechanics) Letter designations  Nature of spectral has three quantum numbers (n, l, ml) to describe feature: an atomic orbital. s  Sharp-very narrow p  Principle-very strong d  Diffuse-spread out f  Fundamental 14
  • 15. Angular Momentum (Shape) Quantum Number (l) MAGNETIC QUANTUM NUMBER (ml): To designate a subshell, we need to specify • For a "d" subshell: both "n" (number) and "l" (letter): l = 2 then ml = -2, -1, 0, 1, 2 (five ml values) Shell Subshell designation n=1 l=0 1s There are five orbitals i a "d" subshell. • Th bit l in b h ll n=2 l=0 2s n=2 l=1 2p • For the "f" subshell: l = 3 then ml = -3, -2, -1, 0, 1, 2, 3 n=3 l=2 3d (seven ml values) • There are seven orbitals in a "f" subshell. Angular Momentum (Shape) Quantum Number (l) • For a given "n" value, there are small differences in energy for various "l" values. • Order of increasing energy of subshells in the same shell: s < p < d < f < g…. increasing energy 4s < 4p < 4d < 4f MAGNETIC QUANTUM NUMBER (ml) REPRESENTATION OF ORBITALS • Describes the orientation in space relative to (shapes and sizes) other orbitals of that same subshell (l). – Divides a subshell into individual orbitals. Electron wave form (function) considers a • Possible ml values :(-l, -l + 1, ....+l) probability of finding the electron around the • For the "s" subshell: nucleus. l = 0 then ml = 0 • There is only a single orbital in an “s” subshell. Define the size and shape of an orbital by a • For the "p" subshell: 90% probability diagram (90% of the electron density lies within these boundaries, 90 % l = 1 then ml = -1, 0, 1 (three ml values) chance that the electron will be found within this volume). • There are three orbitals in a "p" subshell. 15
  • 16. The “s” Orbitals: n = 1 to infinity; l = 0; ml = 0 There are three individual “p” orbitals in a “p” Have a spherical shape: subshell, differ only in their orientation in space: The “d” Orbitals: n  3; l = 2; ml = -2, -1, 0, 1, 2 Five individual orbitals in a “d” subshell. They have a more complex shape, differ only in their orientation in space. Each have the same energy, size, and 1s < 2s < 3s shape. Increasing size and energy The “p” Orbitals: Five individual “d” orbitals, differ only in their orientation in space: n  2; l = 1; ml = -1, 0, 1 Three individual orbitals in a “p” subshell. They all have the same shape, size, and energy - differ only in their orientation in space. Consist of two lobes of electron density about the nucleus. Figure 7.21 16
  • 17. THE “f” ORBITALS: MAGNETIC PROPERTIES OF THE ELECTRON n  4 ; l = 3; ml = -3,-2, -1, 0, 1, 2, 3 Seven individual orbitals in the “f” subshell. Very complex shape (8 lobes of electron density) and differ only in their orientation in space. Each have the same energy, size and shape. Orbitals with the same energy, i.e., the three p orbs, five d orbs, etc. are sa
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