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2 Mass Action.doc

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Earth Sciences
John Ferris

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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