ch 5 - Hard ­Sphere van der Waals Equa3on Eq. 5.4...

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Unformatted text preview: Hard ­Sphere van der Waals Equa3on Eq. 5.4 Excluded Volume: σ v0=(π/6)σ3 Van der Waals b coefficient: 8v0=(4π/3)σ3 Full van der Waals Equa3on of State Eq. 5.5 Eq. 4.15 Van der Waals Cohesive Energy Propor=onal to concentra=on Excess (interac=on) Energy: Total van der Waals Molar Energy: Comparison with Non ­Ideal Gas Data Taylor expansion of original van der Waals hard ­sphere equa=on: Exact hard ­sphere Taylor expansion: Compressibility factor: Packing frac=on: Carnahan ­Starling Equa=on of State: N. F. Carnahan and K. E. Starling, J. Chem. Phys. 51, 2, 635 (1969). How Accurate is the Carnahan ­Starling Equa3on? J. Kolafa, S. Labik, and A. Malijevsky, Phys. Chem. Chem. Phys. 6, 9, 2335 (2004). Carnahan ­Starling Van der Waals (CSvdW) Equa3on Eq. 5.13 Packing frac=on: −ε = − kBτ € D. Ben ­Amotz and D. R. Herschbach, J. Phys. Chem. 94, 3, 1038 (1990). How Accurate is the CSvdW Equa3on? Fit to experimental αT and κP at T=300K and P=0.1 MPa D. Ben ­Amotz and D. R. Herschbach, J Phys Chem 94, 1038 (1990) Molecular Sub-Group Increments for σ and τ D. Ben ­Amotz and K. G. Willis, J Phys Chem 97, 7736 (1993) CSvdW Chemical Poten3al Integrate  ­PdV (and differen=ate with respect to N) to get chemical poten=al Ideal gas Packing frac=on: Repulsive (excluded volume) AYrac=ve (cohesion) Liquid Cyclohexane: Extension of the CSvdW Equa3on to Solu3ons (with a dilute solute dissolved in a liquid solvent) Eq. 5.17 σ1 d= σ0 σ1 σ0 € τ ≈ τ1τ 0 € L. E. S. de Souza and D. Ben ­Amotz, J. Chem. Phys. 101, 11, 9858 (1994). A. D. GiY and D. Ben ­Amotz, J. Chem. Phys. 118, 14, 6427 (2003). Supercri3cal Fluids Carbondioxide (CO2) Large Scale Supercri3cal CO2 Extrac3on Used to extract essen3al oils from plants and caffeine from coffee… Homework Problem 7 (Chapter 5): Extraction of n-octane from coal using supercritical CO2 CO2 Coal Predict the pressure of CO2 at which the extraction will be thermodynamically favorable, ΔG < 0 Extract n-octane Pressure of n ­Octane from Fig. 4.4 [c] M Since [c] only changes by about 3% from 0 ­20 MPa we can use [c] ~ 5.8 M to es=mate ΔGs of n ­octane in n ­octane AND assume it is the same as the solva=on of n ­octane in coal Solva3on Gibbs Energy of n ­Octane in CO2 n ­octane gas fluid CO2 Solva3on Free Energy Packing Frac3on ΔGS = G ( fluid ) − G ( gas) = N A µ× = RT ( βµ× ) Graphs for Homework Problem 7 in homework set #11 T=350K Pressure can be used to tune the solva3ng power of CO2 SolvaZon Free Energy Pressure P* Determine P*, above which extrac=on will become spontaneous Solva3on Free Energy ΔG = ΔGR + ΔGA Cavity Formation Cohesive Energy (Reference Fluid) (Perturbation) Generalized van der Waals Equation ΔG = ΔG IG + ΔG X = ΔG IG + ΔG R + ΔG A = −RT ln K Use experimental K(T,P) to determine ΔG R and ΔG A etc. Raman Spectroscopy vs. T and P Chemical Reaction: A B Laser Raman B A Raman Shift, cm-1 Pressure Induced Hemiketal Forma3on + Acetone Methanol Hemiketal •  Reversible Equilibrium Reaction A. D. GiY and D. Ben ­Amotz, J. Chem. Phys. 118 (14), 6427 (2003). Development of Sta3s3cal Thermodynamics Ludwig Boltzmann 1844 ­1906 James Clerk Maxwell 1831 ­1879 Lars Onsager (1903  ­ 1976) Onsager, L. Chem. Rev. 1933, 13, 73. John “Jack” Kirkwood (1907  ­ 1959) Kirkwood, J. G. J. Chem. Phys. 1935, 3, 300. Kirkwood Reversible Work Theorem Total Interac3on Energy Solvent ­Solvent Solute ­Solvent 0 λ 1 Eq. 5.47 Force Distance Ben Widom (Cornell U.) Eq. 5.43 Eq. 5.54 Widom Inser3on Theorem Total Interac3on Energy Solvent ­Solvent Solute ­Solvent Pure Solvent Random Solute inser3on Acempts J. Chem. Phys. 100, 1456 (1994). − βΨ 0 µ = −k B T ln e × 1 = −k B T ln Pins Insertion probability € InserZon probability Hard ­Sphere Chemical PotenZals Cavity Size DistribuZon Chris Jarzynski (U. of Maryland) Jarzynski Irreversible Work Theorem Irreversible Gas Expansion Irreversible Solute Inser3on t T0 λ(t) P  ­ λ(t) λ(t) V Irreversible Work coupling parameter Eq. 5.55 Widom (fast coupling): Kirkwood (slow coupling): Displacement of a Constraint (control parameter), λ(t) Piston Volume displacement Molecular Stretching displacement λ(t) C. Bustamante, J. Liphardt, and F. Ritort, Phys Today 58, 43 (2005) "The nonequilibrium thermodynamics of small systems." The First Experimental Test of Jarzynski's Equa lity J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco, and C. Bustamante, Science 296, 1832 (2002). Thermally Irreversible Process What if T ≠ T0 ? System Bath T0 T χ(t) T0 T Thermally Irreversible Process 1st Law 2nd Law Dissipa3ve Work all experimentally measureable D. Ben-Amotz, J. M. Honig, Phys.l Rev. Lett. 96, 020602 (2006) “Average entropy dissipation in irreversible mesoscopic processes” >0 >0 >0 Generalized Carnot Engine If the system remains at constant T Wrev = 0 Wdiss = Wirr If Wdiss = 0 T0 T χ(t) t P V If Wdiss ≠ 0 χ(t) Entropy increase of the system during the expansion stroke Efficiency of an ideal Carnot engine ...
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This note was uploaded on 01/17/2011 for the course CHM 373 taught by Professor Ben-amotz during the Fall '10 term at Purdue University-West Lafayette.

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