Lecture6Ch161Feb11

Lecture6Ch161Feb11 - Summary of last lecture Thermal...

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Summary of last lecture Thermal equilibrium and absolute temperature Heat reservoir Sharpness of energy distribution, fluctuations of macroscopic quantities Dependence of density of states on external parameters (formulation of the problem and announcement of the result)
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Key Concepts and Lessons: a) Dependence of the density of states of external parameters and generalized forces - derivation of key relation b) Exact and Inexact Differentials, Function of State. Thermodynamic potentials c) All laws of thermodynamics d) Equations of state, example of ideal gas Reading: Reif, Ch3
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Dependence of Density of States on external parameters When external parameters change density of states changes as well. How? We can use the same analogy with flowing fluid. When we change an external parameter x some states which had energy less than E may have energy greater than E. How many such states exist? Consider x-derivative of energy levels: Y = E r ( x ) x and denote Ω Y E , x ( ) the number of states with energy E , E + δ E ( ) whose x-derivative of E r is fixed at Y Ω E , x ( ) = Ω Y E , x ( ) Y When x changes all states with a given value of Y move by Ydx in energy All states lying in the Ydx sliver below E will cross and their energy will become greater than E
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Cont… The number of states which crossed the boundary is then Ydx E σ Y E ( ) = Ω Y E , x ( ) δ E Ydx E ( ) = Ω Y E , x ( ) E Ydx Y = Ω E , x ( ) E Y dx where Y = 1 Ω E , x ( ) Ω Y E , x ( ) Y Y Y = E r X = X where X is the mean generalized force associated with parameter x
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Cont… The total net flow of states in the energy interval is then: ∂Ω ( E , x ) x dx = σ ( E ) − σ ( E + δ E ) = d ( E ) E E = E Ω Y ( ) = − − ∂Ω E Y − Ω Y E The second term in the last equality is << than the first term when N>>1 (prove:5pts) and we get: ln Ω x = ln Ω E Y = β X = X kT
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Finally we get our gem for quasi-static processes: ln Ω x = ln Ω E Y = β X = X kT d ln Ω = ln Ω E dE + ln Ω x α = 1 n dx d ln Ω = dE + X = 1 n dx = dE + dW ( )
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Relation of heat and entropy: A bridge between Micro and Macro kTdS = dE + dW
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Balancing the Books: First Law Now we recap the law of conservation of energy: Internal energy of a system can change as a result of heat exchange with another system(s) and work done by (on) the
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Lecture6Ch161Feb11 - Summary of last lecture Thermal...

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