Quantum Theory and Electromagnetic Radiation

1. Chapter 6 Quantum Atom

2. Electronic Structure • Chemical properties of elements depend on the arrangement and the energies of the electrons. • Topics in this chapter: • Energy • Electromagnetic radiation • Quantum theory • The Hydrogen atom • Many-electron atoms

3. History Thomson: • discovered the electrical nature of matter and the nature of electricity itself (1900). Measured the e/m ratio. Millikan: determined the charge on the electron Rutherford Gold Foil • atom consists of a nucleus surrounded by electrons (1911).

4. The Wave Nature of Light • Visible light: type of electromagnetic radiation • Electromagnetic radiation: carries energy through space. Called also: Radiant Energy • Speed of all types of electromagnetic radiation: 3.00 x 108 m/s in vacuum. • Similar to up/down movement of water waves. • Radiation: the emission and transmission of energy through space as a wave. • Electromagnetic wave: has a magnetic field component and an electrical field component

6. Properties of Waves • Wave: progressive, repeating disturbance that spreads through a medium from a point of origin to more distant points. • Wavelength (λ): the distance between identical points on successive waves. Units? • Frequency ( v): the number of waves passing a given point per second (Hertz-Hz) 1 hertz (Hz) = 1 cycle/second or sec-1 • Amplitude: the distance from the middle to the peak of the trough. • Speed:(c) c = λ ν

8. Waves - Simulation • Applet: • http://www.surendranath.org/Applets.html

9. Electromagnetic Radiation Electromagnetic waves: originate from the movement of electric charges. This movement produces oscillations in electric and magnetic fields that are propagated over distances. • http://micro.magnet.fsu.edu/primer/java/polarizedlight/emwave/

11. Electromagnetic Spectrum Electromagnetic Spectrum: The complete range of wavelengths and frequencies. Continuous Spectra: ROY G BIV

12. Waves and Electromagnetic Spectrum: Examples • Which light has the higher frequency: the bright red brake light of an automobile or the faint green light of a distant traffic signal? • The yellow light given off by a sodium vapor lamp used for public lighting has a wavelength of 589 nm. What is the frequency of this radiation? (5.09 x 1014 s-1) • (a) A laser used in eye surgery to fuse detached retinas produces radiation with wavelegthn of 640.0 nm. Calculate the frequency of this radiation. (b) An FM radio station broadcasts electromagnetic radiation at frequency of 103.4 MHz. Calculate the wavelength of this radiation. (4.688 x1014 s-1; 2.901 m)

14. Continuous and Line Spectra • Emission Spectra: • Continuous: ROY G BIV. • Line Spectra: emitted by excited atoms and passed through a prism • Line Spectra: • Different atoms emit different combination of line spectra • Monochromatic: single wavelength • Used to identify the elements

15. Planck’s Quantum Theory • When solids are heated they emit radiation at all temperatures • Wavelength distribution of the radiation depends on temperature (depends on the energy of solid particles about fixed point) • Prevailing laws of physics (wave theory) could explain with separate theory for ,low temperatures and separate theory for high temperatures. No unifying theory was available. • Max Planck found unified theory that explained this problem

16. From Classical Physics to Quantum Theory • Planck (1900): atoms and molecules emit (absorb) energy only in certain definite (whole-number multiples) amounts called QUANTA. • Quantum mechanics is born • Theory of duality of wave-particle started

17. Planck's Quantum Theory (1900) • Atoms and molecules can emit or absorb energy only in discreet quantities • Quantum (fixed amount): the smallest quantity of energy that can be emitted (or absorbed) in the form of electromagnetic radiation. • E = hν E = • Planck’s constant: h = 6.626 x 10-34 J-s • Energy is emitted in whole number multiples of h. • Energy changes can occur only in discrete quantities. Variation of energy is discontinuous. • Quantum Chemistry: is the application of quantum mechanics to problems in chemistry

18. The difference between continuous and quantized energy levels can be illustrated by comparing a flight of stairs with a ramp.

19. The Photoelectric Effect and Photons 7.1 Planck's Quantum Theory (1900) • http://www.ifae.es/xec/phot2.html • http://lectureonline.cl.msu.edu/~mmp/kap28/PhotoEffect/photo.htm • http://cwx.prenhall.com/petrucci/medialib/media_portfolio/text_images/040_PhotoelectEff.MOV Fig. 7.5

20. The Photoelectric Effect and Photons • Light shining on a clean metal surface causes emission of electrons. • For each metal there is a threshold (minimum) of frequency below which no electrons will be emitted.

21. Photoelectric Effect: Einstein • Cannot be explained by the wave theory • Einstein: beam of light (radiant energy) is a stream of tiny energy packets called photons. • Each photon possesses energy given by E = hν • and for photoelectric effect hν = KE + BE, • BE is the binding energy of the electrons in the substance (metal)

22. Quantum Atom: Examples • Calculate the energy, in joules, of one photon of yellow light whose wavelength is 589 nm. (2.03 x 105 J/mol) • (a) A Laser emits light with frequency of 4.69 X 1014 s-1. What is the energy of one photon of the radiation from this laser? (b) If the laser emits a burst or pulse of energy containing 5.0 x 1017 photons of this radiation, what is the total energy of the pulse? (c) If the laser emits 1.3 x10-2 J of energy during a pulse, how many photons are emitted during this pulse? (3.11 x 10-19 J/photon; 0.16 J; 4.2 x 1016 photons)

23. Emission Spectra: Four Lines of the Hydrogen Atom (Balmer) Fig. 7.6

24. Line Spectra of Hydrogen Atom Excited hydrogen atoms return to their lowest energy state, the ground state, and emit photons of certain energies, and thus certain colors.

25. Bohr’s Postulates for H-Atom • The angular momentum (h/2 ) values that depend on the value of n = 1, 2, 3, …is quantized, it could have only certain specified values determined by the value of n = 1,2 ,3, ….. • An electron in a permitted orbit has a specific energy (E1, E2, E3, …) and is in an “allowed” energy state. • Electron can move from one stable orbit to another only by absorbing or releasing an amount of energy exactly equal to the difference between the energies of the two orbits and it can be expressed as E = hv

26. Bohr’s Model of the Atom • Bohr’s Calculation: • Started with his three postulates • Used classical equations for motion • Used classical equations for interacting electrical charges • Derived an equation for the electron in hydrogen atom z, nuclear charge R; Rydberg’s Constant = 2.18 x 10-18 J

27. The Energy States of Hydrogen Atom: Interpretation of Bohr’s Equation z for H atom =1 n, principal quantum number; n = 1,2, 3……,∞ n= 1, is closest to the nucleus n = 1 is also called the ground state, the most stable state. In all other energy levels: EXCITED STATE. The lowest energy (n=1) is the MOST negative energy As n increases, the energy is more positive (less negative: it is never a positive quantity) At infinity (total ionization, the electron is free), E = 0

28. Bohr’s Equation ΔE = Ef – Ei = Ephoton = hν • nf and ni are the principal numbers of initial and final states of the atom • Knowing the transition of the electron, we can calculate the energy for the emitted photon, and either the frequency or wavelength or both of the emitted photon.

29. Bohr’s Model of the Atom • Demonstrates transfer of electrons and energy of the different level (wave and particle model) • http://www.walter-fendt.de/ph14e/bohrh.htm

31. Emission Process in an Excited atom of Hydrogen Absorption of photons emission

32. Interpretation of Line Spectra • Atoms which have gained or absorbed extra energy from some excitation energy source (flame, electric discharge etc.) release that energy in the form of light • The light atoms give off contain very specific wavelengths called a line spectrum • light given off = emission spectrum • Each element has its own line spectrum which can be used to identify it. • Ground state: the lowest energy state of the atom. • Excited state of an electron: it is in a higher energy state than the ground state.

33. Fig. 7.11

34. Bohr’s Model of the Atom

35. Limitations of the Bohr Model • Only explains hydrogen atom spectrum and other 1 electron systems (He+1) • Neglects interactions between electrons • Assumes circular or elliptical orbits for electrons - which is not true

36. Bohr’s Equation: examples 1. Using the resources in your packet and the textbook, predict which of the following transitions produces the longest wavelength spectral line: n = 2 to n = 1; n = 3 to n = 4; n = 2 to n = 3? (n = 4 to n = 2) 2. Calculate the energy and wavelength of the atom when electron is being moved from ni = 3 to nf = 1. Is energy emitted or released? (ΔE = -1.94 x 10-18 J; λ = 1.03 x 10-7 m)

37. Bohr’s Equations: Examples 3. What is the energy and wavelength of a photon emitted during a transition from ni =5 state to nf = 2 state in the hydrogen atom? In what part of the spectrum will this photon be? [-4.58 x10-19J,434 nm] 4. How much energy will be emitted when 1 mole of the above photons are emitted? [275.7 kJ/mol] 5. Calculate the energy released when an electron moves from n= 3 to n=5 in a He+1 ion

38. The Dual Nature of the Electron: Matter Waves Louis de Broglie (1923) • If light (wave) behaves as particles (photons), then particles should behave as waves. • Postulated matter waves • Wavelength related to momentum • Matter waves in atoms are standing waves • The length of the wave must fit the orbital of the electron exactly.

39. Standing Waves in Strings • Generated by plucking a string. • Do not move along the string. • Standing waves are complete waves • All standing waves have nodes that do not move (amplitude =0) • The greater the frequency, the greater is the number of nodes. • Only certain number of nodes are allowed in any given length of string

41. Standing Waves - Simulations • Electron microscopy is based on electrons behaving as waves. Circular Standing Waves (optional) http://www.colorado.edu/physics/2000/quantumzone/debroglie.html

42. DeBroglie • Calculate the wavelength of a particle with a mass of 9.1 x 10-31 kg that is moving with the speed of 1.5 x 108 m/s. • Calculate the wavelength of a tennis ball (mass = 30.0 g) that is moving with the speed of 15.0 m/s.

43. Wave Mechanical Model of the Atom • Energy can be treated as particles (quanta): Planck • Light could be treated as particles (photons): Einstein • Electrons (and all particles) can be treated as waves: de Broglie • Wave mechanics: the treatment of atomic structure through the wavelike properties of the electron: Schrödinger and wave equation • Wave function: an acceptable solution to Schrödinger’s wave equation is called. It can only be solved for simple systems, but approximated for others. Describes the behavior of the wave in space. • A wave function, Ψ (psi) represents an energy state of the atom. Electrons are like 3-dimensional waves. • Wave mathematics is used to calculate probability densities of finding the electron in a particular region in the atom.

44. Schrödinger Equation: Differential Equation HΨ = E Ψ Potential Energy of system Total quantized energy of the atomic system Concepts of Schrodinger Equation http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/schr.html

45. Solutions & Meaning of Schrödinger Equation • Regions in space of high probability for finding the electron - these are called orbitals • ORBITAL: a particular, unique wave equation for a given electron defined by three integer terms (quantum numbers). • The square of the function, Ψ2, is the 90% probability of finding an electron in a particular volume of space in an atom. • three-dimensional • Each electron also has a fourth quantum number to represent the direction of spin (derived separately)

46. Wave Function: Orbital, describes the motion of the electron Ψ Probability, Ψ Distance from nucleus

47. Electron Density: Radial probability of finding an electron at a specific location in a volume ΔV (given by |Ψ2| ΔV Ψ2 Sum of all Probabilities, Ψ2 Distance from nucleus

48. The Heisenberg Uncertainty Principle • A wave extends in space and its location is not precisely defined. • For very small objects (subatomic particles) the dual nature of matter places limitations on the precision of knowing both the place and the momentum of the object at the same time. • Bohr’s Model fails because of this principle. Heisenberg Uncertainty Principle

49. Heisenberg Uncertainty Principle 1. The electron has a mass of 9.11 x 10-31 kg and moves at an average speed of about 5.0 x 106 m/s in a hydrogen atom. Assuming that we know the speed of the electron to an uncertainty of 1% (5.0 x 104 m/s), calculate the uncertainty in position. Compare to the radius of the hydrogen atom (2 x 10-10 m). Do we know where the electron is? (1 x 10-9 m) 2. The speed of an electron is measured to be 5.00 x 103 m/s to an accuracy of 0.003%. Find the uncertainty in determining the position of this electron. The mass of the electron is 9.11 x 10-31 kg. (0.385 mm)

50. Heisenberg Uncertainty: Examples 3. What is the uncertainty in the position of an electron mass 9.31 x 10-31 kg with an uncertainty in the speed of 0.100 m/s (5.67x10-4 m) 4. What is the uncertainty in the position of a baseball, mass .145 kg with an uncertainty in the speed of 0.100 m/s (3.64 x 10-33 m)