5 - Degeneracy, Boltzman, Rotational Spectroscopy

5 - Degeneracy, Boltzman, Rotational Spectroscopy - 1 2...

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Unformatted text preview: 1 2 Today: Degeneracies of energy levels Boltzmann Distributions Degeneracies & populations of energy levels Molecular rotational spectroscopy quantum numbers rotational states reduced mass & moment of Inertia rotational constants rotational absorption (or emission) spectra equilibrium bond lengths selection rules ¡ general and quantum Molecular Vibration Spectroscopy (intro) 3 Today: Degeneracies of energy levels Boltzmann Distributions use these to calculate expected populations of energy levels These three concepts are general ¡ apply to all classes of spectroscopy and more 4 For ¡allowed¢ absorption or emission processes, what determines absorption or emission intensity? Absorbance or emission O , cm-1 , s-1 , or eV¡s Answer: Many things, but we will only discuss degeneracies 5 Degenerate (n.) somebody regarded as immoral or corrupt (adj.) in a condition that is worse than an original or previous state (adj.)QUANTUM PHYSICS describes a system in which different quantum states have equal energy 6 Degeneracies & Populations of Quantum Levels E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 7 particle in a box systems at temperature T The lowest energy state is the ground state An electron in this state can be spin up or spin down ¡ these two possibilities imply that the degeneracy of the ground state is 2. We say that g 1 = 2, where 1 means we are referring to the n= 1 quantum state. 7 Degeneracies & Populations of Quantum Levels E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 7 particle in a box systems at temperature T If we have a statistical number (> than the 7 shown) P in a Box systems. At temperature T, what fraction of these systems have an electron in the n=2 state? 8 h = 6.626e-34 J·s (kg m 2 s-1 ) = Planck¡s constant k B = 1.381e-23 J·K-1 T = temperature (in K) For an energy level E k , the population of that level (for the case above, the probability that that level is occupied) is given by: T k E k k B k e g n ¡ v Degeneracies & Populations of Quantum Levels This exponential function leads to what is called a Boltzmann distribution ¢ very important in many fields ¢ Chem Eng; Chem; Physics; Astronomy, (even Biology) etc. E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 1 st molecule 2 nd molecule 3 rd molecule g k = 2 for each level shown here 9 Boltzmann Distributions Ludwig Boltzmann (shown here smiling) Molecular velocity, energy level, etc. 10 ' E 1,2 = E 2 ¡ E 1 h = 6.626e-34 J·s (kg m 2 s-1 ) = Planck¢s constant k B = 1.381e-23 J·K-1 T = temperature (in K) A more relevant number is the population ratio...
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This note was uploaded on 11/10/2011 for the course CH 1b taught by Professor Natelewis during the Winter '09 term at Caltech.

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5 - Degeneracy, Boltzman, Rotational Spectroscopy - 1 2...

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