Lecture18A - Maxwell Relations and Non-ideality (22.3-5)...

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Unformatted text preview: Maxwell Relations and Non-ideality (22.3-5) There are lots of mathematical tricks and games we can play V S U U dU TdS PdV dS dV S V ∂ ∂ ⎛ ⎞ ⎛ ⎞ = − = + ⎜ ⎟ ⎜ ⎟ ∂ ∂ ⎝ ⎠ ⎝ ⎠ V S U U T and P S V ∂ ∂ ⎛ ⎞ ⎛ ⎞ = = − ⎜ ⎟ ⎜ ⎟ ∂ ∂ ⎝ ⎠ ⎝ ⎠ but since S V U U S S V ⎡ ⎤ ⎡ ⎤ ∂ ∂ ∂ ∂ ⎛ ⎞ ⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ ⎢ ⎥ ∂ ∂ ∂ ⎠ ⎝ ⎠ ⎦ ⎣ ⎦ then this is a Maxwell relation V S S V V S S V T P S ∂ ∂ ∂ ∂ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎣ ⎦ ∂ ∂ ⎛ ⎞ ⎛ ⎞ = − ⎜ ⎟ ⎜ ⎟ ∂ ⎠ ⎝ ⎠ S V V S ∂ ∂ ⎝ ⎠ ⎝ ⎠ Similarly for P S dA PdV SdT V ∂ ∂ ⎛ ⎞ ⎛ ⎞ = − − → = ⎜ ⎟ ⎜ ⎟ ∂ V T T V ⎜ ⎟ ⎜ ⎟ ∂ ∂ ⎝ ⎠ ⎝ ⎠ Lecture 18 1 Now for equations of state (which involve relationships between Pressure, Temperature, and Volume), it is very useful to express entropy and other functions using these new equations. constant Temp S P P S dV ∂ ∂ ∂ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ → = ⎟ ⎜ ⎟ ⎜ ⎟ at constant Temp T V V dS dV V T T = → = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ∂ ∂ ∂ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 2 If V is very large our gas is ide V P d V ∂ ⎛ ⎞ 1 1 so If V is very large, our gas is ideal. V V S d V T Δ = ⎜ ⎟ ∂ ⎝ ⎠ ∫ V P ∂ ⎞ ( ) , ' (will write id for ideal) ideal ideal V V S T V S dV T ∂ ⎛ ⎞ − = ⎜ ⎟ ∂ ⎝ ⎠ ∫ T a P R ⎞ 2 for vdW gas: V RT a P R P V b V T V b ∂ ⎛ ⎞ = − → = ⎜ ⎟ − ∂ − ⎝ ⎠ ( ) ' , l n id V id V dV V b S T V S R R V b V b − − = = − − ∫ Lecture 18 2 id For an ideal gas: V P RT R T T V V ∂ ∂ ⎛ ⎞ ⎛ ⎞ = = ⎜ ⎟ ⎜ ⎟ ∂ ∂ ⎝ ⎠ ⎝ ⎠ 2 1 2 1 ' ln for an isothermal process V V V dV S R R V V Δ = = ∫ arlier (from “fun” with math) we showed using: 2 1 or ln V S nR V Δ = Earlier (from fun with math) we showed using: then solving for dS: T U U dU TdS PdV dT dV V ∂ ∂...
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This note was uploaded on 04/01/2012 for the course CHEM 444 taught by Professor Gruebele,m during the Spring '08 term at University of Illinois, Urbana Champaign.

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Lecture18A - Maxwell Relations and Non-ideality (22.3-5)...

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