### Lecture 7

```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
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
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.
```