# ESS102H1 Lecture Notes - Pitzer Equations, Davies Equation, Sodium Chloride

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14 Mar 2013
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2 Mass Action
2.0 Introduction
The chemical compositions of many aqueous systems are determined by
numerous physical, chemical, and biological processes. This complexity is constrained
fundamentally by the Laws of Thermodynamics, which govern the exchange of energy
between systems and their surroundings. Adoption of the principles of thermodynamics
to the study of chemical changes in aqueous systems is especially advantageous in that it
provides both a conceptual and quantitative framework within which to evaluate the
chemistry of pristine and contaminated waters.
The intent of this chapter is not to develop the equations of thermodynamics, but
to extract and emphasize relationships that may be applied quantitatively in studies of
biogeochemical processes.
2.1 Chemical Thermodynamics
Chemical thermodynamics is concerned with the thermodynamic description of
systems subject to chemical change. This is true of abiotic as well as biologically
mediated transformations. Considering that chemical reactions near the surface of Earth
usually take place under constant-temperature constant-pressure conditions, an important
result of combining the first and second Laws of Thermodynamics for a finite change at
constant pressure and temperature is
STHG =
(1)
This relationship establishes a very useful correlation between the three state
functions of a system and finite changes in the Gibbs free energy (G), enthalpy (H), and
entropy (S). For reversible chemical processes at equilibrium, that is a system plus its
surroundings in which S has attained a maximum value, G is minimized and G = 0. If
the reaction is spontaneous then G < 0, whereas G > 0 for an impossible process.
A critical issue in chemical thermodynamics that extends from the derived
relationships of state functions is the establishment of reference conditions against which
the tendency for reactions to precede can be measured. This begins with the definition of
chemical potential, which is a partial molar property related to G and the number of
moles n of any given species i in a system, which at constant pressure and temperature is
given by
TP
i
in
G
,
=
µ
(2)
1
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The chemical potential relationship permits definition of the equilibrium point of
reactions. In the case of a general reaction involving species A, B, etc. with
stoichiometric coefficients a,b, etc.
dDcCbBaA ++
(3)
When the reaction advances to the point of equilibrium, G does not change and Gr = 0
for the reaction. From the definition of chemical potential, the point of minimum G is
described by
µµµµ
(4)
The chemical potential of an individual species
i
µ
is related to the standard
potential of the species in its pure form
0
i
µ
}ln{
0iRT
ii +=
µµ
(5)
Here R is the universal gas constant, T is the absolute temperature in degrees Kelvin, and
{i} is the dimensionless activity of species i in solution. For pure phases be it a solid or
liquid {i} = 1 by definition. In real solutions, activity and actual molal, mol kg-1,
concentrations of chemical species i in solution are related by an activity coefficient
γ
i.
{i} =
γ
i mi(6)
An important consideration that derives from the definition of species activity is
that activity coefficients approach unity in very dilute solutions
ii
mi }{ and 1
γ
In the case of gases, the term fugacity is used instead of activity. The chemical
potential of a gas species m is taken conventionally to be
mii fRT ln
0+=
µµ
(7)
The fugacity of gas species m is related to the partial pressure of the gas p and a fugacity
coefficient
mmm pf
χ
=
(8)
At low pressures, the fugacity and partial pressure become equivalent
mmm pf and 1
χ
2
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The definition of chemical potential in terms of standard conditions permits a
more explicit evaluation of the equilibrium state. Considering Eqn. 4 and Eqn. 5, the
equilibrium condition for the general reaction above becomes
ba
dc
BA
DCr
BA
DC
RTG
BRTbARTa
DRTdCRTcG
}{}{
}{}{
ln
})ln{}ln{}ln{}ln{(
})ln{(})ln{(
})ln{(})ln{(0
0
0000
00
00
=
+=+
++
+++==
µµµµ
µµ
µµ
(9)
From the discussion above, the left side of the Eqn. 9 is equivalent to the standard
free energy of the reaction, G 0 at 25 oC and 1 atm of pressure. Similarly, the multiple
on the right side of Eqn. 9 is a product scaled by RT that reduces to the familiar mass
action equation of chemical reactions, which provides a thermodynamic definition for the
equilibrium constant K
ba
dc
BA
DC
K}{}{
}{}{
=
(10)
The essential result that arrives out of Eqn. 9 and Eqn. 10 is that the mass action
equilibrium constant is a function of an exponential argument of the standard free energy
of the reaction
= RT
G
K
0
exp
(11)
The equilibrium constant of chemical reactions is crucial to the evaluation of how
chemical reactions progress. Specifically, the change in free energy for the advancement
of a reaction that is not at equilibrium (i.e.,
0r
G
) is formulated from Eqn. 9 as
ba
dc
r
BA
DC
RTGG
}{}{
}{}{
ln
})ln{}ln{}ln{}ln{(
0
0000
+=
+++=
µµµµ
(12)
As noted above, the sum of standard chemical potentials in Eqn. 12 is G0 ; however, the
equilibrium constant K is replaced by the non-equilibrium reaction quotient Q
3
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