110c-lecture9 - 1. Helmholtz Free Energy (A). A U TS A = U...

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1. Helmholtz Free Energy (A). A U – TS (combine 1 st and 2 nd Law at const T and V) Δ A = Δ U - T Δ S < 0 (spontaneous process at const T and V) Δ A = w rev =maximum work (isothermal, reversible, i.e. T Δ S=q rev ) 2. Gibbs Free Energy (G). TS H G (combine 1 st and 2 nd Law at const T and P) Δ G = Δ H – T Δ S < 0 (spontaneous process at const T and P) Δ G = w nonPV (isothermal, reversible, i.e. T Δ S=q rev ) 3. Fuel Cell Extracts Electrical Work ( Δ G=w nonPV ) from H 2 (g) + O 2 (g) H 2 O(l). Δ Hº = -285 kJ/mol and T Δ Sº = -48.7 kJ/mol at T=298 K, w nonPV = -nFE = Δ G Δ G = Δ H -T Δ S = -237 kJ/mol efficiency = w/q = Δ G/ Δ H = 83% vs 30% gas engine H 2 2 e - 2H + ½O 2 O 2- H 2 O H 2 + O 2 h ν Solar Fuel Cell
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What if Fuel Cell reaction is run in a piston and cylinder and does only PV work? H 2 ( g ) + 1 2 O 2 ( g ) ⎯ → H 2 O ( l ) () ( ) J w V V nRT V dV nRT dV V nRT PdV w PV PV 3 1 2 2 1 2 1 2 1 10 33 . 28 7 . 36 018 . 0 ln 15 . 298 314 . 8 5 . 1 ln × = = = = = = mol kJ G / 1 . 237 = Δ KJ H 8 . 285 = Δ ( ) KJ RT n PV gas 717 . 3 = Δ Δ ( ) KJ PV H U 1 . 282 = Δ Δ = Δ ( ) KJ S T TS 7 . 48 = Δ = Δ kJ q 4 . 310 1 . 282 33 . 28 = = Efficiency = w PV /q (q = -w PV + Δ U) Efficiency of PV work = w PV /q = 28.33/310 = 9% Efficiency of Fuel Cell (w nonPV /q) = Δ G/ Δ H = 237/285 = 83% W non-PV = 0
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2nd crossed partial derivatives are equal dT T A dV V A SdT PdV dA V T + = = , = V A P and = T A S from calculus: = = = T A V T V A V T A V A T ∂∂ 2 2 T V V S T V A T P V T A = = δ δδ 2 2 T V V S T P = eqn22.19 p888 A = U – TS = (q + w) – TS dA = δ q+ δ w –TdS SdT = TdS PdV TdS SdT dA = – PdV SdT Maxwell’s Relations Provide Useful Formulas V T T P V S = V nR T P V = dV T P dS V = = 2 1 V V dV V nR dS 1 2 ln V V nR S = Δ Ideal gas Maxwell Relation from A (PV work only) dT T A dV V A dA V T + =
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Maxwell Relations Derived from G For a reversible process with only PV work, SdT VdP dG SdT TdS VdP PdV PdV TdS dG SdT TdS VdP PdV w q dG = + + = + + + = δ (substituting δ q = TdS and δ w = -PdV) Express G as a function of P and T dT T G dP P G dG P T + = Comparing the last two eqns gives: T P G V = and P T G S = P T T G P P G T = () T T P P S P S T V = = * SdT TdS VdP PdV dU dG TS PV U G + + = + = P nR T V P = dP T V dS P = 1 2 2 1 2 1 ln P P nR P dP nR dP T V S P P P P P = = = Δ P T T V P S = 1 2 ln P P nR S = Δ
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This note was uploaded on 06/22/2010 for the course CHEM 21360 taught by Professor Ame during the Spring '09 term at East Los Angeles College.

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110c-lecture9 - 1. Helmholtz Free Energy (A). A U TS A = U...

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