Real SolutionsLecture 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 hµi =µi* + RT ln Xi

• µ* is chemical potential in ‘standard state’ of Henry’s Law behavior at Xi = 1.

• 3. Real Solutionso 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:

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.

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.

• Recalling:

• We define the activity as:

• Hence

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

• And

Water and Electolyte Solutions

Water• Water is a familiar but

very unusual compound.o Highest heat capacity (except

ammonia)o Highest heat of evaporationo Highest surface tensiono Maximum density at 4˚Co Negative Clayperon Slopeo 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

• Concentrationso Molarity: M, moles of solute per litero Molality: m, moles of solute per kgo 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 107 molecules will do so such that

aH+ × aOH– = 10-14

pH = -log aH+

• Standard state conventiona˚ = 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

• Basically, we are assigning all the non-ideal 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:

Practical Approach to Electrolyte Activity

CoefficientsDebye-Hückel and Davies

Debye-Hückel Extended Law

• Assumptionso Complete dissociationo Ions are point chargeso Solvent is structurelesso 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:

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.