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Lect12 - Lecture 12 Applications of Free-Energy Minimum...

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Physics 213: Lecture 12, Pg 1 Lecture 12 Lecture 12 Applications of Free-Energy Minimum Applications of Free-Energy Minimum Agenda for today Agenda for today Law of Atmospheres, revisited Semiconductors Re fe re nc e  fo r this  Le c ture : Ele m e nts   C h 12 Re fe re nc e  fo r Le c ture  13: Ele m e nts   C h 13 
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Physics 213: Lecture 12, Pg 2 Last time: Free Energy, F = U - TS Last time: 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
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Physics 213: Lecture 12, Pg 3 Chemical potential of particles with Chemical potential of particles with potential energy potential energy (e.g., (e.g., atoms in a gravitational field or atoms bound into atoms in a gravitational field or atoms bound into molecules) molecules) The potential energy gets added to the translational energy: + = PE kT 2 3 N U ln T n kT PE n μ = + n = N/V particle density Atom in gravity: ln mgh Molecule with binding energy : ln T T n kT n n kT n μ μ = + ÷ = - ∆ ÷ So, the chemical potential ( μ = dF/dN) gains an additional contribution: d(N·PE)/dN = PE. (For now we are ignoring the rotation and vibration of molecules.) PE = potential energy per particle The potential energy makes an additional contribution (N·PE) to the free energy: F = U - TS. The resulting chemical potential in this case is therefore: Examples Remember: Chemical potential is the change in free energy when we add one particle (at constant V,T).
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Physics 213: Lecture 12, Pg 4 The Law of Atmospheres: Revisited The Law of Atmospheres: Revisited Problem: Find how the gas density n and pressure p of the gas vary with height h from the earth’s surface? Compare region 2 at height h with region 1 at height 0. For every state in region 1 there’s a corresponding state in region 2 with mgh more energy (bigger u(T)) Assume that the temperature is in equilibrium. That means it’s equal everywhere: T 1 =T 2 .
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