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- Chapter6_Musfeldt_Larger View.pdf

Jonathan R.

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Chapter 6: Electronic Structure of Atoms Why is Quantum Mechanics Necessary? Solution to the Schrodinger Equation Gives Orbitals Atomic Quantum Numbers and Spectroscopy Chapter 6 homework problems are available on Mastering Chemistry Today: The wave nature of light Quantized energy and photons? the development of the theory was motivated by unresolved experimental issues Line spectra and the Bohr model To understand the electronic structure of atoms, one must understand the nature of electromagnetic radiation. ? The distance between corresponding points on adjacent waves is the wavelength (?). ? The number of waves passing a given point per unit of time is the frequency (?). ? So if velocity is constant, shorter wavelength waves have higher frequency and longer wavelength waves have lower frequency. ? Expressing that inverse relationship between wavelength and frequency: Consider a wave? short ?, high ? long ?, low ? ? ? c ? Notice: we assume that all electromagnetic radiation travels at the same velocity speed of light = c = 3.00 ? 10 8 m/s ? ? c ? rearrange ???c best units = Hz = sec -1 The Electromagnetic Spectrum Organizes Light in Order of Increasing Wavelength (or Decreasing Frequency) ? Note that the electromagnetic spectrum spans an enormous range ? Visible light is just a small portion of the spectrum (400 ? 700 nm) the visible spectrum 114 9 8 103.4 ) 10 1 (700 /100.3 ? ??? sx nm m nm smxc ? ? violet light: shorter ?, higher ? red light: longer ?,lower? ? = 400 nm ? = 700 nm 114 9 8 105.7 ) 10 1 (400 /100.3 ? ??? sx nm m nm smxc ? ? Let?s step back an examine why light should have wavelike (and particle-like) properties ? A particle travels in a trajectory with a precise position and precise velocity at each instant ? Any type of motion can be excited to an a state of arbitrary energy Neither are true in quantum mechanics! Classical Mechanics: Quantum effects only when: Very small and light Moving very fast Macroscopic motion easiest to treat using classical mechanics; Microscopic motion easiest with quantum mechanics Failures of Classical Mechanics ? Frequency dependence of black body badiation ? Heat capacity of solids at low temperature ? Photoelectric effect ? Diffraction of electrons ? Cathode ray effect ? Discrete character of spectra??? Experimental conclusion: Light is a wave and a particle! Wave-like behavior should already be clear. Let?s take a look at the ?particle-like properties? ? The wave nature of light does not explain how an object can glow when its temperature increases. ? Max Planck explained it by assuming that energy comes in packets called quanta. Other particle-like aspects ? Einstein used this assumption to explain the photoelectric effect. ? He concluded that energy is proportional to frequency: E = h? where h is Planck?s constant, 6.626 ? 10 ?34 J-s. The Nature of Energy ? Therefore, if one knows the wavelength of light, one can calculate the energy in one photon, or packet, of that light ? Note: light ?particles? are called ?photons? ? You can see that energy is ?quantized in units of h ? ? hc hE ?? monochromatic laser light Continuous energies vs. quantization of energy Classical mechanics allows continuous energies Quantum mechanics insists on only certain allowed energies. A Quick Example ... UV A and UV b are two well-known components of natural radiation that can cause irreversible damage to the skin. The wavelengths of UV A and UV B are 320-400 nm and 290-320 nm, respectively. Which has higher energy? Energy ? ? hc hE ?? Summarizing the main points so far ? Light is both a wave and a particle, exhibiting characteristics of one or the other depending on how it is probed ? ?Wave-particle duality? ? This dual nature also extends to matter (de Broglie?s relation) But let?s go back to the discrete character of line spectra Another mystery in the early 20th century involved the emission spectra observed from atoms and molecules. What are these ?line spectra?? ? A white light source gives a continuous spectrum ? Atoms and molecules are different: their spectrum is a series of lines or discrete wavelengths. ? This is called a ?line spectrum?. What about the Atomic Spectra? ? Atomic ?line spectra? were a major ?stumbling block? for classical theories ? Do the discrete spectra ?tell us something? about the nature of energy quantization in an atom? ?Yes! ? Bohr?s conclusion: only certain energies are ?allowed? for an electron in the hydrogen atom; electron energy levels are ?quantized? Niels Bohr adopted Planck?s assumption and explained the phenomena in this way: 1. Electrons in an atom can only occupy certain orbits. And these orbits correspond to certain energies. 2. Electrons in permitted orbits have specific, ?allowed? energies; their energies are ?stable? and given by a principle quantum number. 3. Energy is only absorbed or emitted in such a way as to move an electron from one ?allowed? energy state to another 4. The energy of each state is E = h?, and the transition energy is ?E = E f -E i The energy absorbed or emitted from the process of electron promotion or demotion can be calculated: ? ? ? ? ? ? ? ? ???? ? ? ? ? ? ? ? ? ???? ? 22 18 22 11 10178.2 11 ifif H nnnn RE Here, R H is the Rydberg constant, 2.178 ? 10 ?18 J, and n i and n f refer to the initial and final states. For a hydrogen line spectrum: ? ? hc hE ?? And since E is related to wavelength and frequency, the latter can be easily calculated. Energies are Quantized n = 1, 2, 3, 4, ... Z = atomic number ?Quantization?comes from the boundary counditions of the problem. ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ??? ? 2 2 2 2 18 6.1310178.2 n Z eV n Z JE n ? Note that energy goes as 1/n 2 ? Energy level separation gets smaller at larger quantum numbers Energy differences between levels are easy! Absorption or Emission ? ? ? ? ? ? ? ? ???? ? ? ? ? ? ? ? ? ???? ? 22 218 22 2 11 )(10178.2 11 )( ifif H nn Z nn ZRE ? ? ? hc c hh nn ZE if ??? ? ? ? ? ? ? ? ? ????? ? 22 218 11 )(10178.2 These energy levels were verified experimentally! Experimental evidence for energy quantization ? Balmer, Paschen, Lyman Series for Hydrogenic Atoms ? These sharp spectral lines correspond to various n-level transitions. Let?s do an example: What is the emission wavelength from sodium atoms from their n = 2 state to their ground state? ? ? ? ? ? ? ? ? ???? ? ? ? ? ? ? ? ? ???? ? 22 218 22 2 11 )(10178.2 11 )( ifif H nn Z nn ZRE Step 1: identify n i and n f for an emission process Step 2: identify Z for Na Step 3: calculate the energy difference Step 4: convert to wavelength Summarizing ? Classical mechanics can not describe many experimental observations ? Quantum mechanics describes the quantized motion of particles and waves ? Light is a wave and a particle ? Bohr model explains atomic line spectra Please start work on the Chapter 6 electronic homework problems! Jan Musfeldt Microsoft PowerPoint - Chapter6_Musfeldt.ppt

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