Thermodynamics
• “the branch of science that deals with
energy levels and the transfer of energy
between systems and between different
states of matter”
What is Energy???
• “It is important to realize that in physics today, we have no
knowledge of what energy is. We do not have a picture that
energy comes in little blobs of a definite amount. It is not that
way.” –Richard Feynman
• HOWEVER  Feynman goes on to elaborate that energy has
meaning as a way to define, and quantify, changes which bring
about reactions – between systems, energy levels, or states of
matter.
• “How seriously must we take the physical existence of this
energy? No more and no less than any other bookkeeping
practices.” –Richard Feynman
Thermodynamics
• Thermodynamics answers the following question:
• For any reaction - defined by a set of reactants
and products set in exactly defined conditions
(temperature, pressure, concentration, etc.)  will
that reaction go forward spontaneously or not??
• For ANY geochemical reaction, if thermo says
NO, rest assured the reaction will not proceed.
• Thermo says NOTHING about the speed a
reaction occurs!!
Equilibrium
• Anything at equilibrium is theoretically
undergoing forward and reverse reactions:
• A+B↔C
– A + B  C AND C  A +B
• Equilibrium has 2 criteria:
– Reaction does not appreciably change in time
– Perturbation of that equilibrium will result in a
return to the equilibrium
Equations can be ‘added’ together, equilibrium constants also get
‘added’ together!
Convenient way to rewrite reactions (to look at more appropriate
reactions or to use things you’ve more directly measured….)
log Keq
• CaCO3(calcite) = Ca2+ + CO32-
-8.48
• CO2(g) + H2O = H2CO30
-1.47
• H2CO30 = H+ + HCO3-
-6.35
• H+ + CO32- = HCO3-
+10.33
CaCO3(calcite) + CO2(g) + H2O = Ca2+ + 2 HCO3-
-5.97
Assessing equilibrium
[ products]
n
K
i
[reactants]
n
i
[ products]
Q
[reactants]
n
i
n
i
Q  reaction quotient, aka Ion Activity Product
(IAP) is calculated from knowing activity of all
components of a reaction
K  aka Keq, we get from thermodynamic data – it
is one number defined AT EQUILIBRIUM
Equilibrium for any reaction is when Q = K
Where do K’s come from?
• Measure directly – experimental
determination of conditions at equilibrium
• Use thermodynamic data – K is directly
related to free energy of reaction – DGR
Hydroxylapatite
• Ca5(PO4)3(OH) = OH- + 3 PO43- + 5 Ca2+
• Log K = -59.0351 at 25ºC
Aqueous Complexes
•
•
Combinations of ions to form dimers,
trimers, etc., are complexes
Why do we care??
1. Complexation of an ion also occurring in a
mineral increases solubility
2. Some elements occur as complexes more
commonly than as free ions
3. Adsorption of elements greatly determined
by the complex it resides in
4. Toxicity/ bioavailability of elements depends
on the complexation
How do we know about all those
species??
• Based on complexation  how any ion
interacts with another ion to form a
molecule, or complex (many of these are
still in solution)
• Yet we do not measure how much
CaNO3+, CaF+, or CaPO4- there is in a
particular water sample
• We measure Ca2+  But is that Ca2+ really
how the Ca exists in a water??
Defining Complexes
• Use equilibrium expressions:
• cC + lHL  CL + lH+
c
 n
[CL] [ H ]
i 
c
l
[C ] [ HL ]
• Where B is just like Keq!
Equilibrium
• Equilibrium Constant, K (or Keq) describes
conditions AT equilibrium (where DGR=0)
K
n
[
products
]

i
n
[
reactants
]

i
DGR -DG0R = RTlnK
DG0R = -RTlnK
DG0R= SDG0R products – SDG0R reactants
DGR= SDGR products – SDGR reactants
DG0
• Energy at STANDARD STATE
• = 25°C, 1 bar Pressure, 1 molal
concentration for each
Speciation
• Any element exists in a solution, solid, or
gas as 1 to n ions, molecules, or solids
• Example: Ca2+ can exist in solution as:
Ca++
Ca(H3SiO4)2
Ca(O-phth)
CaB(OH)4+
CaCH3COO+
CaCO30
CaCl+
CaF+
CaH2SiO4
CaH3SiO4+
CaHCO3+
CaNO3+
CaOH+
CaPO4CaSO4
CaHPO40
• Plus more species  gases and
minerals!!
Mass Action & Mass Balance
c
 n
[CL] [ H ]
i 
c
l
[C ] [ HL ]
mCa   mCa L
2
2 n
x
• mCa2+=mCa2++MCaCl+ + mCaCl20 + CaCL3- +
CaHCO3+ + CaCO30 + CaF+ + CaSO40 +
CaHSO4+ + CaOH+ +…
• Final equation to solve the problem sees the
mass action for each complex substituted into
the mass balance equation
Mineral dissolution/precipitation
• To determine whether or not a water is saturated with
hydroxyapatite, we could write a dissolution reaction
such as:
Ca5(PO4)3(OH) = OH- + 3 PO43- + 5 Ca2+
• We could then determine the equilibrium constant:
K
1
a 5Ca 2 a 3 PO43 aOH

a1hydroxyapatite
• If K = -59.04, can determine how much Ca2+ and PO43might dissolve at any pH and T
Activity
• Sometimes called ‘effective concentration’,
which is misleading and reflects a poor
understanding of the property…
• Think of more of the effect the rest of a
solution has on how easily two ions come
together..
Activity
• For solids or liquid solutions:
ai=Xigi
• For gases:
ai=Pigi = fi
Xi=mole fraction of component i
Pi = partial pressure of component i
mi = molal concentration of component i
• For aqueous solutions:
ai=migi
Activity Coefficients
• Where do they come from??
• The standard state for dissolved ions is
actually an infinitely dilute solution…
• Activity of phases - gases, minerals, and bulk
liquids (H2O) are usually pretty close to 1 in
waters
• Dissolved molecules/ ions have activity
coefficients that change with concentration
(ions are curved lines relating concentration
and activity coefficients, molecules usually
more linear relation)
Application to ions in solution
• Ions in solutions are obviously nonideal
mixtures!
ai = gimi
• The activity coefficient, gi, is found via
some empirical foundations
• Dependent on the other ions in water…
Dissolved species gi
• First must define the ionic strength (I) of the
solution the ion is in:
I   mi z i
2
i
Where mi is the molar concentration of species i
and zi is the charge of species I
Activity Coefficients
• Debye-Huckel approximation (valid for I:
log g 
2
Az I
o
1
2
1
2
1 a BI
• Where A and B are constants (depending
on T), z is charge, I is ionic strength, and å
is a measure of the effective diameter of
the ion
Different ways to calculate gi
•
•
•
•
Limiting law
Debye-Huckel
Davies
TJ, SIT
models
• Pitzer, HKW
models
Neutral species
• Setchnow equation:
• Logan=ksI
For activity coefficient (see table 4-2 for
selected coefficients)
Mass Action & Mass Balance
c
 n
[CL] [ H ]
i 
c
l
[C ] [ HL ]
mCa   mCa L
2
2 n
x
• mCa2+=mCa2++MCaCl+ + mCaCl20 + CaCL3- +
CaHCO3+ + CaCO30 + CaF+ + CaSO40 +
CaHSO4+ + CaOH+ +…
• Final equation to solve the problem sees the
mass action for each complex substituted into
the mass balance equation
• Equations for each ion – iterative solution…
Speciation Models
• PHREEQC, or WebPHREEQ, is a USGS
program that solves, simultaneously
(iteratively really), all of the mass action
and mass balance equations for a water’s
chemical composition