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# Lect11 - Lecture 11 Equilibrium Between Particles Agenda...

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Physics 213: Lecture 11, Pg 1 Lecture 11 Lecture 11 Equilibrium Between Particles Equilibrium Between Particles Agenda for today Agenda for today Free Energy and Chemical Potential Simple defects in solids Ideal gases, revisited Reference for this Lecture: Elements Ch 11 Reference for Lecture 12: Elements Ch 12

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Physics 213: Lecture 11, Pg 2 Why do we define Free Energy, F = U - TS? Why do we define Free Energy, F = U - TS? Answer: Equilibrium corresponds to maximum S tot = S reservoir + S small system . The free energy incorporates the effect of the reservoir by its temperature T. In minimizing F (equivalent to maximizing S tot ) we don’t have to deal explicitly with S reservoir . Consider exchange of gas between two containers: The derivative of free energy with respect to particle number is so important in determining an equilibrium condition that we define a special name and symbol for it: For these 2 subsystems exchanging particles , we see that the condition for “chemical equilibrium” is: V 1 V 2 2 1 μ = μ chemical potential of subsystem “i” = i i i dN dF = μ Maximum Total Entropy Minimum Free Energy Equal chemical potentials 0 dN dF dN dF dN dF dN dF dN dF 2 2 1 1 1 2 1 1 1 = - = + = Equilibrium condition: (dF/dN 1 = 0) dN dF dN dF 2 2 1 1 = 1 N equilibrium value F = F 1 +F 2 Small system at temperature T
Physics 213: Lecture 11, Pg 3 The path ahead. .. The path ahead. .. We have thus far studied systems in which volumes can be exchanged this led to mechanical equilibrium and equal pressures in the two volumes. We then studied systems in which energy can be exchanged this led to thermal equilibrium and equal temperatures of the systems. Now we consider systems in which particles can be exchanged, e.g., Particles can move from place to place Particles can combine (chemistry, vacancy-interstitial recombination, electron-hole recombination, nuclear reactions…) into new types. This will lead to “chemical equilibrium”, in which the “chemical potential” is equalized, the free energy of the system is minimized, and the total entropy is maximized (which after all is the most probably state of affairs. ..) And lots and lots of applications. .. We start with a concrete example. ..

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Physics 213: Lecture 11, Pg 4 Simple defects in solids (1) Simple defects in solids (1) Atoms can pop out of position and sit in interstitial sites. These defects (and others) play a crucial role in the thermal weakening of materials, etc. In thermal equilibrium, how many are there? (determines how hot is safe for some structure to get) We calculate the number N I of interstitials to minimize F Let’s say that an atom can either go to the surface, making a ‘normal’ site, or sit at an interstitial site, with extra energy I . For big crystals, the surface energy and entropy changes are
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Lect11 - Lecture 11 Equilibrium Between Particles Agenda...

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