Chemistry 141 Wednesday, November 1, 2017 Lecture 24 Chemistry 11 - Lecture 11 9/30/2009 Chemistry 141 Wednesday, November 1, 2017 Lecture 24 Matter Waves and the H Atom

The Balmer Series n n (Hz) l (nm) 3 4.5711014 656.3 Johann Balmer worked out that the frequencies of the light emitted by excited H-atoms satisfy the equation below n n (Hz) l (nm) 3 4.5711014 656.3 4 6.1721014 486.1 5 6.9121014 434.0 6 7.3141014 410.2 For n = 3, 4, 5, … where n2 > n1 RH = 1.096776×107 m-1 (Rydberg’s constant)

The Bohr atom Bohr postulated that electrons traveled in circular orbits Only orbits with certain radii were allowed n can only be an integer!! The energy of an electron in a given orbit is: E depends on n Energy is also quantized Note that E is negative!

Atomic energy levels Energy levels are like a staircase A change from lower to higher energy level corresponds to absorption of energy A change from higher to lower energy level corresponds to emission of energy E=0 energy of level n=4 energy to move between levels energy n=3 “excited states” n=2 n=1 “ground state”

Bohr atom energies Brackett Paschen Energy Balmer Lyman Energy n =  -13.6 eV n = 5 n = 6 Lyman -3.40 eV -1.51 eV -0.85 eV -0.54 eV -0.38 eV 0 eV Balmer Paschen Brackett Energy Bohr atom energies Energy 1 eV = 1 electron volt = 1.60210-19 J

Example What is the wavelength of the photon absorbed when going from the n=2 energy level to the n=4 energy level?

Bohr atom: does it work? What next? Made accurate predictions for H and He+ line spectra Theoretical basis of Bohr’s model is questionable There is no physical explanation for why the electron “orbits” are stable Bohr simply threw in idea of “allowed orbits” to classical physics to get agreement with experiments Only works for one-electron atoms Predictions for multielectron atoms were incorrect What next?

Standing wave animation Standing waves Node The wave must fit inside the confined region There must be an even number of half wavelengths: • n =1, 2, 3, … Standing wave animation x More standing waves Length

de Broglie’s great leap de Broglie (1923): Could electrons behave like standing waves? l=h/mv m = mass of particle λ = wavelength v = velocity of particle For particles (matter)

de Broglie wavelength of an electron What is the wavelength of an electron with a velocity of 3×106 m/s (me = 9.1039×10-31 kg)? Compare this with the radius of the hydrogen atom (r = 5×10-11 m).

Diffraction of waves When a wave passes through an aperture about the same size as its wavelength or smaller, it exhibits diffraction wave appears to ‘bend’ around edges of aperture particles that don’t hit wall pass through aperture in a straight line http://www.falstad.com/ripple/

interference of water waves Interference of Waves Constructive vs. Destructive interference of water waves + = in phase out of phase Image from: http://www.phys.unsw.edu.au/~epe/1111/lectures/1111.L7.small.html

Diffraction and Interference of Light Young’s experiment demonstrates the wavelike nature of light A light beam passes through two slits, creating two light sources The waves from the two sources interfere with one another This creates an interference or diffraction pattern on the screen Interference is evidence of wave behavior Figure taken from: http://micro.magnet.fsu.edu/primer/lightandcolor/interference.html

Electron diffraction X-rays Electrons Davisson and Germer first observed diffraction of electrons fired at a nickel crystal (an ordered array of atoms) George Thomson (J.J. Thomson’s son) fired electrons at a thin piece of metallic foil Davisson and Thomson shared the Nobel prize in physics in 1937 Amazingly, J.J. Thomson received the Nobel prize for demonstrating that the electron is a particle, while his son received the Nobel prize for demonstrating that the electron acts as a wave! “Lucky accident” because they were initially firing the electrons at an amorphous sample of nickel, but a bottle of liquid air exploded, breaking open the sample chamber and causing the nickel surface to be oxidized by the air rushing onto the heated surface. To remove the oxide, they heated the nickel at high temperatures for a long time, this is called “annealing” and not only removes the oxide layer, but also causes large crystals to form, as opposed to having many small crystals. When they looked at this sample, they obtained something resembling a diffraction pattern, then they optimized this to get a clean diffraction pattern.

Matter Waves Scientists at IBM accidentally observed standing waves from the interference of electrons on a copper surface with defect sites They then decided to intentionally create electron standing waves by arranging iron atoms in a ring, thereby confining the space the electrons can roam in, and forming standing waves Images from: http://www.almaden.ibm.com/vis/stm/atomo.html

Wavelengths of larger ‘particles’ What is the wavelength of a baseball (mass = 0.145 kg) moving at a velocity 18 m/s?

What does it all mean? Light and matter both exhibit characteristics of both waves and particles Large pieces of matter exhibit mostly particle-like behavior Small pieces of matter (electrons) exhibit both particle and wave behavior Vanishingly small pieces of matter (photons) exhibit mainly wave-like behavior Matter and energy are not distinct – energy is a form of matter (or is it the other way around?!)

A very deep question: Waves do not have a location. If a small particle (i.e. an electron) behaves like a wave, can we specify its location?

Heisenberg’s Uncertainty Principle The Uncertainty Principle (Heisenberg, 1927) states: There is an inherent uncertainty in our ability to specify an electron’s exact location there is a fundamental limit to our ability to describe the properties of particles that are very small Instead of certainty, we must rely on probabilities to tell us the likely location of these small particles

When does the uncertainty principle matter? What is the uncertainty in the position of an electron (mass = 9.11x10-31 kg) moving with a speed of 5x106 m/s? Assume the uncertainty in the speed is 1%.