Real Solutions
Lecture 7
Review: Chemical Potential
in Ideal Solutions
æ ¶G ö
çè
÷ø = V
¶P T
• In terms of partial molar quantities
• For an ideal gas:
æ ¶ µi ö
çè
÷ø = vi
¶P T
RT
æ ¶µi ö
=
çè
÷
¶P ø T ,ideal
P
µiP - µio = RT ln
P
P˚
• Integrating from P˚ to P:
• Where P˚ is the pressure of pure substance in its
‘standard state’ and µ˚ is the chemical potential of i
in that state. In that case, P/P˚ = Xi and:
µi,ideal = µio + RT ln Xi
Review: Henry’s Law
• In dilute solutions, most substances exhibit Henry’s Law
behavior as their mole fractions approach 0 (Xi ⟶ 0).
• Henry’s Law is:
Pi = hiXi
o where hi is Henry’s Law ‘constant’. It can be (usually is) a function of T and P
and the nature of the solution, but is independent of the concentration of i.
• In a way analogous to our derivation of chemical
potential for ideal solutions, we can express Henry’s Law
as:
µi =µi˚ + RT ln hiXi
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
o Letting µ* = µ˚ + ln h
µi =µi* + RT ln Xi
o 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.
We want to find a way to
deal with real solutions
within the framework we
have already constructed
for ideal ones.
Fugacities
• We found for an ideal solution:
µiP - µio = RT ln
P
P˚
• We define fugacity to have the same relationship to chemical
potential as the partial pressure of an ideal gas:
ƒ
µi = µio + RT ln oi
ƒi
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:
ƒ
lim i = 1
P®0 P
i
• 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.
H2O & CO2 Fugacity
Coefficients
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).
o Now we have stuffed that non-ideal behavior into the activity coefficient!
• 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
G = åG i,excess = RT å Xi ln li
i
i
Electrolyte solutions are solutions in which the solute
dissociates to form ions, which facilitate electric
conduction. Seawater and all natural waters are
electrolyte solutions. These solutions are of enormous
importance in many geologic processes.
Solvent is the substance present in greatest abundance.
Solute refers to the remaining substances present in solution.
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 in solution, water also
insulates ions by surrounding
them with a solvation shell.
• First solvation shell usually
has 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
Aside: Back to Rum ‘n
Coke
• Some covalently bonded
substances like sugars and
ethanol will readily dissolve
in water because, like
water, they contain polar
hydrogen-oxygen bonds.
These substances are called
hydrophilic.
• Other covalently bonded
substances, such as
hydrocarbons and lipids
(fats, oils) do not dissolve in
water because their bonds
are not polar. These are
known as hydrophobic
substances.
Some definitions and
conventions
• Concentrations
o
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.
• pH
o
o
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.