Chem 402 Lecture 21: L21 3:8:17

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Washington University in St. Louis
University College - Chemistry
University College - Chemistry Chem 402
Barnes Alexander

8 March 2017 L21: Gibbs Free Energy and Maxwell Relations I. Fundamental Equations + Thermodynamic Data A. Criteria for Spontaneous Change (review) 1. Direction of spontaneous change οƒ  toward equilibrium οƒ  maximize accessible microstates οƒ  max S = π‘˜π‘™π‘›Ξ© uni 𝑒𝑛𝑖 2. Master Equation a. From the 1 law we have: π‘‘π‘ˆ = Γ°π‘ž + ð𝑀 = π‘‘π‘ž βˆ’ 𝑝𝑑𝑉 b. From the 2 law we have: 𝑑𝑆 β‰₯ Γ°π‘ž 𝑇 β€’ 𝑇𝑑𝑆 = π‘‘π‘ž π‘Ÿπ‘’π‘£ for equilibrium transformations β€’ 𝑇𝑑𝑆 > Γ°π‘ž for spontaneous processes where βˆ†S uni 0 π‘–π‘Ÿπ‘Ÿπ‘’π‘£ c. π‘‘π‘ˆ βˆ’ 𝑇𝑑𝑆 + 𝑝𝑑𝑉 = 0 at equilibrium, π‘‘π‘ˆ βˆ’ 𝑇𝑑𝑆 + 𝑝𝑑𝑉 < 0 for spontaneous processes 3. Criteria for Spontaneous Change in a System a. (𝑑𝑆) > 0,(𝑆 = π‘˜π‘™π‘›Ξ©) π‘ˆ,𝑉 β€’ Maximize entropy of the system at equilibrium b. (π‘‘π‘ˆ )𝑆,𝑉< 0,(π‘‘π‘ˆ = π‘‘π‘ž + 𝑑𝑀) β€’ Minimize internal energy of the system at equilibrium to maximize the entropy of the surroundings (heat and work out of system) c. (𝑑𝐻 )𝑆,𝑃< 0,(𝐻 = π‘ˆ + 𝑝𝑉) β€’ Minimize enthalpy of the system at equilibrium d. (𝑑𝐴 ) < 0,(𝐴 = π‘ˆ βˆ’ 𝑇𝑆) 𝑉,𝑇 β€’ Minimize Helmholtz Free Energy of the system at equilibrium e. (𝑑𝐺 )𝑇,𝑃< 0,(𝐺 = 𝐻 βˆ’ 𝑇𝑆) β€’ Minimize Gibbs Free Energy of the system at equilibrium f. Note: all to maximize the Suni β€’ T and S are absolute β€’ G and H are arbitrary because of U (since U is arbitrary, depends on where you set it), so you usually look at changes to these state variables B. Fundamental Equations 1. Setup a. With the free energies we introduced all our state functions for closed systems β€’ Helmholtz free energy: A = U – TS β€’stGibbs free energy: G = H – TS b. 1 law tells us that π‘‘π‘ˆ = Γ°π‘ž + ð𝑀 c. 2 Law tells us that TdS = dq rev β€’ Note, by using the above, the following fundamental equations all apply for systems at equilibrium, replace with a β€œ>” to show spontaneous change 2. U(S, V) οƒ  π‘‘π‘ˆ = 𝑇𝑑𝑆 βˆ’ 𝑝𝑑𝑉 a. π‘‘π‘ˆ = Γ°π‘ž + ð𝑀 = π‘‘π‘ž βˆ’ 𝑝𝑑𝑉 = 𝑇𝑑𝑆 βˆ’ 𝑝𝑑𝑉 3. H(S, p) οƒ  𝑑𝐻 = 𝑇𝑑𝑆 + 𝑉𝑑𝑝 a. H = U + pV b. 𝑑𝐻 = π‘‘π‘ˆ + 𝑝𝑑𝑉 + 𝑉𝑑𝑝 = 𝑇𝑑𝑆 βˆ’ 𝑝𝑑𝑉 + 𝑝𝑑𝑉 + 𝑉𝑑𝑝 = 𝑇𝑑𝑆 + 𝑉𝑑𝑝 4. A(T, V) οƒ  𝑑𝐴 = βˆ’π‘†π‘‘π‘‡ βˆ’ 𝑝𝑑𝑉 a. A = U – TS b. 𝑑𝐴 = π‘‘π‘ˆ βˆ’ 𝑇𝑑𝑆 βˆ’ 𝑆𝑑𝑇 = 𝑇𝑑𝑆 βˆ’ 𝑝𝑑𝑉 βˆ’ 𝑇𝑑𝑆 βˆ’ 𝑆𝑑𝑇 = βˆ’π‘†π‘‘π‘‡ βˆ’ 𝑝𝑑𝑉 5. G(T, p) οƒ  𝑑𝐺 = βˆ’π‘†π‘‘π‘‡ + 𝑉𝑑𝑝 a. G = H – TS b. 𝑑𝐺 = 𝑑𝐻 βˆ’ 𝑇𝑑𝑆 βˆ’ 𝑆𝑑𝑇 = 𝑇𝑑𝑆 + 𝑉𝑑𝑝 βˆ’ 𝑇𝑑𝑆 βˆ’ 𝑆𝑑𝑇 = 𝑉𝑑𝑝 βˆ’ 𝑆𝑑𝑇 c. Application in protein structural biology: β€’ Protein structures are usually determined around 90K and not room temp (298K) or body temp (317K) β€’ β€œTS” is not weighted as much at lower temperatures, and thus the structure imaged at 90K may not be the best for figuring out structures at biologically relevant temperatures β€’ β€œHigh sensitivity NMR at room temperature for biomolecular structure and dynamics” β€’ At room temperature or body temperature there i
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