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Solution thermodynamics theory—Part I Chapter 11

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topics Fundamental equations for mixtures Chemical potential Properties of individual species in solution (partial properties) Mixtures of real gases Mixtures of real liquids

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A few equations For a closed system Total differential form, what are (nV) and (nS) Which are the main variables for G?? What are the main variables for G in an open system of k components?

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G in a mixture (open system)

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G in a mixture of k components at T and P How is this equation reduced if n =1?

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2 phases (each at T and P) in a closed system Apply this equation to each phase Sum the equations of both phases, take into account that In a closed system:

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We end up with How are dn i and dn i related at constant n?

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For 2 phases, k components at equilibrium For all i = 1, 2,…k Thermal equilibrium Mechanical equilibrium Chemical equilibrium

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In order to solve the VLE problem Need models for i in each phase Examples of models of i in the vapor phase Examples of models of i in the liquid phase

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Now we are going to learn: Partial molar properties Because the chemical potential is a partial molar property At the end of this section think about this – What is the chemical potential in physical terms – What are the units of the chemical potential – How do we use the chemical potential to solve a VLE (vapor-liquid equilibrium) problem

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Partial molar property Solution property Partial property Pure-species property

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example Open beaker: ethanol + water, equimolar Total volume nV T and P Add a drop of pure water, n w Mix, allow for heat exchange, until temp T Change in volume ?

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Total vs. partial properties See derivation page 384

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Derivation of Gibbs-Duhem equation

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Gibbs-Duhem at constant T&P Useful for thermodynamic consistency tests

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Binary solutions See derivation page 386

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Obtain dM/dx1 from (a)

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Example 11.3 We need 2,000 cm 3 of antifreeze solution: 30 mol% methanol in water. What volumes of methanol and water (at 25 o C) need to be mixed to obtain 2,000 cm 3 of antifreeze solution at 25 o C Data:

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solution Calculate total molar volume of the 30% mixture We know the total volume, calculate the number of moles required, n Calculate n 1 and n 2 Calculate the total volume of each pure species needed to make that mixture

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Note curves for partial molar volumes

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From Gibbs-Duhem: Divide by dx1, what do you conclude respect to the slopes?

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Example 11.4 Given H=400x x 2 +x 1 x 2 (40x 1 +20x 2 ) determine partial molar enthalpies as functions of x 1, numerical values for pure-species enthalpies, and numerical values for partial enthalpies at infinite dilution Also show that the expressions for the partial molar enthalpies satisfy Gibbs-Duhem equation, and they result in the same expression given for total H.

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