Modeling Tools for Environmental Engineers and Scientists

ackward reactions can expressed in terms of their respective reaction rateconstants and the molar concentrations of the species by:r forward = k 1 [A] and r backward = k 2 [B] (4.45)Therefore, at equilibrium, equating the two rates, the concentrations of thetwo species at equilibrium are related by:[ B] eq = k 1 = K eq (4.46)[ A]eqk2where K eq is known as the equilibrium constant for the reaction. Equilibriumconstants play an important role in environmental modeling in that they arekey inputs in determining equilibrium distributions of participating reactantsand products in many environmental systems. It has to be noted that K eq isstrongly dependent on the system temperature, for example, in the case ofself-ionization of water, K eq = 0.45 × 10 –14 at 15ºC and K eq = 1.47 × 10 –14at 30ºC.The directions of reversible reactions depend on the energy of the systemand can be established through a thermodynamic analysis. It can be shownthat K eq is related to the Gibbs’ standard free energy of the products minus theGibbs’ standard free energy of the reactants, ∆G o , by the following equation:∆G o = –RT ln K eq (4.47)where R is the Ideal Gas Constant and T is the absolute temperature. Thisequation enables K eq values to be calculated from ∆G o . Tabulated values ofK eq can also be found in Chemical Property Handbooks. The application ofthe above concepts is illustrated in Worked Example 4.7.Worked Example 4.7(1) Calculate the equilibrium constant for the dissociation of carbonic acid.(2) Derive the equations that can be used to plot the concentrations of thevarious carbonate species in a closed aqueous system as a function of pHfor a given total carbon mass, C T .Solution(1) The equilibrium constant can be calculated from ∆G o using Equation(4.47). By definition, ∆G o can be found from the following:∆G o = (v i ∆G o f ) products – (v j ∆G o f ) reactantsij© 2002 by CRC Press LLC