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Unformatted text preview: 1 2 For allowed absorption or emission processes, what determines absorption or emission intensity? Absorbance or emission , cm1 , s1 , or eVs Answer: Many things, but we will only discuss degeneracies 3 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. 4 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 (more than the 7 shown) PiBox systems. At temperature T, what fraction of these systems have an electron in the n=2 state? 5 E 1 E 2 E 1 E 2 h = E 2 E 1 h = 6.626e34 Js (kg m 2 s1 ) = Plancks constant k B = 1.381e23 JK1 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 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. 6 Boltzmann Distributions Ludwig Boltzmann (shown here smiling) Molecular velocity, energy level, etc. 7 E 1 E 2 E 1 E 2 h = E 2 E 1 h = 6.626e34 Js (kg m 2 s1 ) = Plancks constant k B = 1.381e23 JK1 T = temperature (in K) A more relevant number is the population ratio between the ground and excited states, since this is what is typically measured: T k E B e g g n n = " ' " ' Degeneracies & Populations of Quantum Levels k B T at 298K 200 cm1 If we plotted this for all of the states n of energy E n , we would have the Boltzmann distribution 8 h = 6.626e34 Js (kg m 2 s1 ) = Plancks constant k B = 1.381e23 JK1 T = temperature (in K) T k E B e g g n n = " ' " ' Rotational energy levels separated by a few cm1 s ground and several higher (excited) levels will be populated at room T Vibrational energy levels 10 210 3 cm1 s mostly just the ground state is populated at room T Electronic energy levels 10 410 5 cm1 s pretty much only ground state is populated at room T k B T at 298K 200 cm1 Degeneracies & Populations of Quantum Levels 9 Here is a pretty good problem to work out: Take a particleina1Dbox system, at room temperature, in which the particle is an electron,...
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This note was uploaded on 06/05/2008 for the course CH 1a taught by Professor Lewis during the Winter '08 term at Caltech.
 Winter '08
 Lewis

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