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Real Solutions Lecture 7 Three Kinds of Behavior • Looking at the graph, we see 3 regions: • 1. Ideal: µi =µi˚ + RT ln Xi • 2. Henry’s Law: µi =µi˚ + RT ln hiXi µi =µi˚ + RT ln hiXi + RT ln hi • Letting µ* = µ˚ + ln h µi =µi* + RT ln Xi • µ* is chemical potential in ‘standard state’ of Henry’s Law behavior at Xi = 1. • 3. Real Solutions o Need a way to deal with them. Fugacities • We define fugacity to have the same relationship to chemical potential as the partial pressure of an ideal gas: µi = µio + RT ln ƒi ƒio 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: ƒi =1 P®0 P i lim • Fugacity and partial pressure are the same for an ideal gas. • We can imagine that at infinitesimal pressure any gas should behave ideally. 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 H2O and CO2 in Table 3.1. Activities • Fugacities are useful for gases such as H2O and CO2, but we can extent the concept to calculate chemical potentials in real liquid and solid solutions. ƒ • Recalling: µi = µio + RT ln i ƒio • We define the activity as: • Hence ai = ƒi ƒ io µi = µio + RT lnai o Same equation as for an ideal solution, except that ai replaces Xi. • 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. Activity Coefficients • We’ll express the relationship between activity and mole fraction as: ai = λiXi • The activity coefficient is a function of temperature, pressure, and composition (including Xi). • For an ideal solution, ai = Xi and λi = 1. 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. Excess Functions • Comparing real and ideal solutions, we can express the difference as: Gexcess = Greal – Gideal • Similarly for other thermodynamic functions, so that: Gexcess = Hexcess – Tsexcess • Also G i,excess = RT ln li • And åG i i,excess = RT å Xi ln li i Water and Electolyte Solutions 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. 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. 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 Some definitions and conventions • Concentrations o o o • pH o o Molarity: M, moles of solute per liter Molality: m, moles of solute per kg Note that in dilute solutions these are effectively the same. Water, of course dissociates to form H+ and OH–. At 25˚C and 1 bar, 1 in 107 molecules will do so such that aH+ × aOH– = 10-14 pH = -log aH+ • 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. Example: Standard Molar Volume of NaCl in H2O • Volume of the solution given by aq V = nw V w + nNaCl V NaCl • Basically, we are assigning all the nonideal behavior on NaCl. o Not true, of course, but that’s the convention. 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: • mA = νAmAB and mB = νBmAB • For a thermodynamic parameter Ψ (could be µ) • ΨAB = νAΨA + νBΨB • So for example for MgCl2: µMgCl2 = µMg+ + 2µCl - Practical Approach to Electrolyte Activity Coefficients Debye-Hückel and Davies Debye-Hückel Extended Law • Assumptions o o o o Complete dissociation Ions are point charges Solvent is structureless Thermal energy exceeds electrostatic interaction energy • Debye-Hückel Extended Law -Az 2 I log g i = i 1+ Bå I • Where A and B are constants, z is ionic charge, å is effective ionic radius and I is ionic strength: I= 1 mi zi2 å 2 i Debye-Hückel Limiting & Davies Laws • Limiting Law (for low ionic strength) log g i = -Azi2 I • Davies Law: é I ù -Azi2 ê - bI ú ë 1+ I û 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.