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Real Solutions Lecture 7

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Three Kinds of Behavior Looking at the graph, we see 3 regions: 1. Ideal: µ i =µ i ˚ + RT ln X i 2. Henry’s Law: µ i =µ i ˚ + RT ln h i X i µ i =µ i ˚ + RT ln h i X i + RT ln h i Letting µ* = µ˚ + ln h µ i =µ i * + RT ln X i µ* is chemical potential in ‘standard state’ of Henry’s Law behavior at X i = Real Solutions o Need a way to deal with them.

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Fugacities We define fugacity to have the same relationship to chemical potential as the partial pressure of an ideal gas: o Where ƒ˚ is the ‘standard state’ fugacity. We are free to chose the standard state, but the standard state for µ˚ and ƒ˚ must be the same. We can think of this as the ‘escaping tendency’ of the gas. The second part of the definition is: Fugacity and partial pressure are the same for an ideal gas. We can imagine that at infinitesimal pressure any gas should behave ideally.

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Fugacity Coefficient We can express the relationship between pressure and fugacity as: ƒ = ΦP where Φ is the fugacity coefficient which will be a function of T and P. o For example, see fugacity coefficients for H 2 O and CO 2 in Table 3.1.

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Activities Fugacities are useful for gases such as H 2 O and CO 2, but we can extent the concept to calculate chemical potentials in real liquid and solid solutions. Recalling: We define the activity as: Hence o Same equation as for an ideal solution, except that a i replaces X i. We have retained our ideal solution formulation and stuffed all non-ideal behavior into the activity. Activity can be thought of as the effective concentration.

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Activity Coefficients We’ll express the relationship between activity and mole fraction as: a i = λ i X i The activity coefficient is a function of temperature, pressure, and composition (including X i ). For an ideal solution, a i = X i and λ i = 1.

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Rational and Practical Activity Coefficients The rational activity coefficient, λ, relates activity to mole fraction. Although mole fraction is the natural thermodynamic concentration unit, other units, such as moles (of a solute) per kilogram or liter or solution are more commonly used (because they are easily measured). In those units, we use the practical activity coefficient, γ.

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Excess Functions Comparing real and ideal solutions, we can express the difference as: G excess = G real – G ideal Similarly for other thermodynamic functions, so that: G excess = H excess – Ts excess Also And

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Water and Electolyte Solutions

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Water Water is a familiar but very unusual compound. o Highest heat capacity (except ammonia) o Highest heat of evaporation o Highest surface tension o Maximum density at 4˚C o Negative Clayperon Slope o Best solvent Its unusual properties relate to the polar nature of the molecule.

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Solvation The polar nature of the molecule allows it to electrostatically shield ions from each other (its high dielectric constant), hence dissolve ionic compounds (like salt). Once is solution, it also insulates ions by surrounding them with a solvation shell. First solvation shell usually 4 to 6 oriented water molecules (depending on ion charge) tightly bound to ion and marching in lock step with the ion. o Outer shell consists of additional loosely bound molecules.

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Solvation Effects Enhances solubility Electrostriction: water molecules in solvation shell more tightly packed, reducing volume of the solution. Causes partial collapse of the H-bonded structure of water. Non-ideal behavior

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Some definitions and conventions Concentrations o Molarity: M, moles of solute per liter o Molality: m, moles of solute per kg o Note that in dilute solutions these are effectively the same. pH o Water, of course dissociates to form H + and OH –. o At 25˚C and 1 bar, 1 in 10 7 molecules will do so such that a H + × a OH – = pH = -log a H + Standard state convention a˚ = m = 1 (mole/kg) o Most solutions are very non-ideal at 1 m, so this is a hypothetical standard state constructed by extrapolating Henry’s Law behavior to m = 1. Reference state (where measurements actually made) is infinite dilution.

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Example: Standard Molar Volume of NaCl in H 2 O Volume of the solution given by Basically, we are assigning all the non- ideal behavior on NaCl. o Not true, of course, but that’s the convention.

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How do deal with individual ions We can’t simply add Na + to a solution (positive ions would repel each other). We can add NaCl. How do we partition thermodynamic parameters between Na + and Cl – ? For a salt AB, the molarity is: m A = ν A m AB and m B = ν B m AB For a thermodynamic parameter Ψ (could be µ) Ψ AB = ν A Ψ A + ν B Ψ B So for example for MgCl 2 :

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Practical Approach to Electrolyte Activity Coefficients Debye-Hückel and Davies

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Debye-Hückel Extended Law Assumptions o Complete dissociation o Ions are point charges o Solvent is structureless o Thermal energy exceeds electrostatic interaction energy Debye-Hückel Extended Law Where A and B are constants, z is ionic charge, å is effective ionic radius and I is ionic strength:

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Debye-Hückel Limiting & Davies Laws Limiting Law (for low ionic strength) Davies Law: o Where b is a constant (≈0.3). Assumption of complete dissociation one of main limiting factors of these approaches: ions more likely to associate and form ion pairs at higher concentrations.

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