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

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

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

0=−−+=∆ BADCr badcG

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

m

χ

mmm pf

χ

=

(8)

At low pressures, the fugacity and partial pressure become equivalent

mmm pf ⇒⇒ and 1

χ

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

ba

dc

BADC

BA

DCr

BA

DC

RTG

BbAaDdCcRTbadc

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.,

0≠∆ r

G

) is formulated from Eqn. 9 as

ba

dc

r

BADCr

BA

DC

RTGG

BbAaDdCcRTbadcG

}{}{

}{}{

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

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