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ESS102H1
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John Ferris
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Earth Sciences

ESS102H1

John Ferris

Winter

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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
ΔG = ΔH −TΔS (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 ΔG = 0. If
the reaction is spontaneous then ΔG < 0, whereas ΔG > 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
∂G
μ i (2)
∂ni P,T
1 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.
aA+bB ⇔ cC + dD (3)
When the reaction advances to the point of equilibrium, G does not change and ΔG = 0 r
for the reaction. From the definition of chemical potential, the point of minimum G is
described by
ΔG =rcμ + dC − aμ −Dμ = 0 A B (4)
The chemical potential of an individual species μ iis related to the standard
0
potential of the species in its pure form μ i
0
μ i μ +iRT ln{i} (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-1r
liquid {i} = 1 by definition. In real solutions, activity and actual molal, mol kg ,
concentrations of chemical species i in solution are related by an activity coefficient γ .
i
{i} = γ i i (6)
An important consideration that derives from the definition of species activity is
that activity coefficients approach unity in very dilute solutions
γ ⇒1 and {i}⇒ m
i i
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
0
μ i μ +iRT ln f m (7)
The fugacity of gas species m is related to the partial pressure of the gas p and a fugacity
χ
coefficient m
fm= χ pm m (8)
At low pressures, the fugacity and partial pressure become equivalent
χ m1 and f ⇒ p m m
2 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
0 0
ΔG =r0 = c(μ + RTCln{C}) + d(μ + RT ln{DD)
0 0
− a(μ +ART ln{A}) −b(μ + RT lB{B})
0 0 0 0
cμ C dμ − aD −bμ =A−RT(clnBC}+ d ln{D}− aln{A}−bln{B}) (9)
c d
ΔG = −RT ln {C} {D}
{A} {B} b
From the discussion above, the left side of the Eqn. 9 is equivalent to the standard
0 o
free energy of the reaction, ΔG at 25 C 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
c d
{C} {D}
K = a b (10)
{A} {B}
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
ΔG 0
K = exp− (11)
RT
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., ΔG ≠ 0 ) is formulated from Eqn. 9 as
r
0 0 0 0
ΔG =rcμ + dC − aμ −Dμ + RTAcln{C}B d ln{D}− aln{A}−bln{B})
(12)
{C} {D} d
ΔG = ΔG + RT ln
r {A} {B} b
0
As noted above, the sum of standard chemical potentials in Eqn. 12 is ΔG ; however, the
equilibrium constant K is replaced by the non-equilibrium reaction quotient Q
3 {C} {D} d
Q = a b (13)
{A} {B}
The simplified form of Eqn. 12 is
ΔG = ΔG + RT lnQ (14)
r
Under any conditions, comparison of the actual composition of a solution in terms
of Q with the equilibrium constant K provides a test for chemical equilibrium (i.e., ΔGr=
0 when Q = K)
ΔG = RT ln Q
r K (15)
The reason molal concentrations are used formally in chemical thermodynamics is
that concentrations expressed in mol kg of solvent are independent of temperature and
pressure. In aqueous systems, concentrations are more commonly reported in terms
molarity (M), which is defined as mole L of solution. The difference between the two
concentration scales is small in dilute solutions, especially with respect to uncertainties
associated with determining equilibrium constants or estimation of activity coefficients.
The standard free energy of a reaction can be determined from Eqn. 11 if the
equilibrium constant is known. Alternatively, the quantity can be calculated from the
relationship
0 0 0
ΔG r ∑ G f −products G f −reactants (16)
0
where ΔG f is the standard free energy of formation. Standard free energies of formation
are tabulated in many chemical thermodynamic references.
Example 2.1
Determine the equilibrium constant (referredoto in the case of minerals as the solubility
product) for the dissolution of calcite at 25 C.
2+ 2-
CaCO =3Ca + CO 3
The mass action equilibrium constant, respecting the unit activity of solid calcite, is
2+ 2−
K ={Ca }{CO } 3
The standard free energies of formation for the individual species in the reaction are
4 Species ΔG (kJ/mol)
f
2+
Ca -553.6
CO 3- -527.0
CaCO (3alcite) -1128.4
From Eqns. 11 and 16
0 0 0
ΔG =r ∑ ΔG f −products ΔG f −reactants553.6) + (−527.0)−(−1128.4) = 47.8kJ/mol
− ΔG r − 47.8 −8.37
logK = = = −8.37 and K =10
2.303RT 5.708
2.2 Effect of Temperature
The change in K with respect to temperature is given by the van’t Hoff equation
d lnK ΔH 0
= 2 (17)
dT RT
which yields after integration
ΔH 0 1 1
lnK =2 − + lnK 1 (18)
R T1 T2
The equilibrium constant at T2can be predicted accurately as long as the temperature is
within + 10 to 15 C of 25 C. At higher values of T ,2the fundamental assumption
0
inferred from Eqn 17 and Eqn 18 that the enthalpy of the reaction ΔH is constant breaks
down. In this case, the enthalpy changes with respect to temperature at constant pressure
according to
∂ΔH 0
= ΔC P (19)
∂T P
When the heat capacity ΔC 0 is in

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