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3 Kinetics.doc

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Earth Sciences
John Ferris

3 Chemical Kinetics and System Dynamics 3.0 Introduction The use of equilibrium mass action models in understanding chemical process in aqueous systems permit a quantitative description of boundary conditions associated with the attainment of equilibrium. A limitation to this approach is that it gives no information concerning reaction pathways or the time needed to reach equilibrium. Instead, these important problems are addressed by chemical kinetics and system dynamics. Several key questions need to be answered in studies of reaction rates and system dynamics. First, is the reaction sufficiently fast and reversible to be regarded as chemical equilibrium controlled? Second, is the reaction homogeneous (occurring wholly within a gas or liquid phase) or heterogeneous (involving reactants in a gas and a liquid, or a liquid and a solid phase)? Slow reversible, irreversible, and heterogeneous reactions are those most likely to require interpretation using kinetic models. Third, is there a specific volume in which chemical equilibrium can be assumed to have been achieved for many possible reactions? This may be called the local equilibrium assumption. Fourth, is the system under consideration closed (only energy is exchanged with the surroundings) or open (energy and matter are exchanged with the surroundings)? 3.1 Elementary and Overall Reactions In kinetics a fundamental distinction is made between elementary and overall chemical reactions. Specifically, an elementary reaction is one that describes an exact reaction mechanism or pathway. Four examples include 2+ 2− (1) CaCO = 3a +CO 3 CO +OH = HCO − (2) 2 3 CO 3−+ H = HCO 3− (3) (4) + − H 2 = H +OH Conversely, an overall reaction does not indicate the reaction mechanisms involved or the pathway. In this case, an example would be the reaction arising from the combination of Eqns. 1 to 4 CaCO +C3 + H O2= Ca 2 2++ 2HCO 3− (5) 1 Rates of overall reactions can be predicted only if the rates of the component elementary reactions are known. Rates of elementary reactions are often proportional to the concentrations of reactants, but this may not always be the case for overall reactions. 3.2 Rate Laws An implicit condition associated with chemical equilibria of reversible reactions in closed systems is that concentrations of reactants and products do not change with respect to time. This circumstance is satisfied when the forward and reverse reaction rates are equivalent. In the case of a general reaction A ⇔ B (1) the equilibrium mass action relationship from chemical thermodynamics under dilute conditions where activity coefficients are taken to be unity and square brackets indicate molar concentrations is K = [B] (2) [A] with forward d[A] = −k fA] (3) dt and reverse d[B] = −k rB] (4) dt rates of reaction written in differential form (i.e., change in concentration with respect to reaction time) with forward k afd reverse k rare constants, respectively. The equivalency of the reaction rates satisfies the equilibrium mass action relationship k [A] = k [B] f r (5) such that K = [B] = kf (6) [A] kr 2 Example 3.1 The formation of the aqueous FeSO compl4x is described by the reaction 3+ 2− kf + Fe + SO 4 ↔ FeSO 4 kr -1 The equilibrium constant for the reaction is K = eq5 L mol , and the forward rate constant k f 6.37 x 10 L mol s at 25 C. Calculate the value of the reverse rate constant k r k 3 K = f = 205 = 6.37×10 eq k k r r kr= 31s -1 The concepts of molecularity and reaction order are important in the development of rate expressions for stoichiometric reactions. Considering the general reaction aA+bB = cC + dD (7) the expression for the forward reaction rate is d[A] d[B] a b R f = = −k fA] [B] (8) dt dt In this instance, the forward reaction has a molecularity of two and an order equivalent to the sum of the stoichiometric coefficients, (a + b). The reverse reaction is similar with a rate expression d[C] d[D] c d R r = = −k fC] [D] (9) dt dt and a molecularity of two, but different with a reaction order of (c + d). The total reaction order is said to be the sum of the stoichiometric coefficients (a + b + c + d). 3 Experimental evaluation of the general rate expressions illustrated by Eqn. 8 and Eqn. 9 are explored more easily under conditions far from equilibrium; however, when a reversible reaction is close to equilibrium, the overall rate of change in reactant (or product) concentrations depends on the rates of both the forward and reverse reactions. Considering again the reaction given by Eqn. 7, the anticipated rate of change in reactant concentrations is R =fk [C]r[D] − k [A] [B] f a b (10) where the directionality of the reaction (i.e., consumption of reactants) is expressed by the negative sign of the forward reaction (on the right side of the equation). A similar overall rate expression can be written for the products of the reaction by reversing the signs of the respective forward and reverse rates R = k [A] [B] − k [C] [D] c d (11) r f r Many reactions in aqueous systems involve simply one or two reactants other than liquid water or a mineral solid, both of which are characterized by unit activity as defined above. This is reasonable considering that the probability of multiple simultaneous encounters between varieties of dissolved reacting species in solution is quite small. In recognition of this, a generalized reaction involving reactant A and water that gives rise to product P takes the simple form of A+ H O ⇔ P (12) 2 The corresponding expression for the rate of the forward reaction is d[A] = −k [Af (13) dt This is a first order reaction that is easily integrated to give the concentration of reactant [A] ts a function of any time t and the initial reactant concentration [A] 0 [A] t[A] exp0−k t) f (14) The great utility of Eqn. 28 is that a semi-logarithmic plot of ln[A] as a functiot of time is linear with a y-intercept of ln[A] and sl0pe of – k. Applicatifn of regular linear regression of ln[A] on t thus provides an estimate for k, as well as fonfirmation of [A] 0 assuming that this concentration is known ahead of time. Moreover, the appropriateness of a first order rate expression for the reaction can be assessed by examining the correlation coefficient from the linear regression with a high degree of confidence evidenced by r values greater than 0.90. Defining the characteristics of second and third order reactions is somewhat more complicated in that multiple concentrations or additional stoichiometric coefficients 4 appear in the differential rate equations. For example, consider a second order version of Eqn. 26 A+ B + H O ⇔ P (15) 2 The rate expression inferred from Eqn. 27 for the consumption of reactant A is d[A] = −k [f][B] (16) dt The integrated form with different initial concentrations of reactants A and B [B] [A] [A] t t 0 exp([A] −0B] )(k 0) f (17) [B] 0 This formulation requires more information to evaluate experimentally with [A] being t dependent on two variables, namely the corresponding concentration of B at time t, as well as the advancement of the reaction with respect to time t. If measurements of both A and B are available over time, then linear regression of ln [A] / [B] on t will tield the second order rate constant k and the intercept ln [A] / [B] at t = 0. f 0 0 If the concentration of reactant B is held constant at all times such that [B] = [B] t 0 then d[A] ∗ = −k [f][B] = −k [A] f (18) dt The derived rate expression illustrated in Eqn. 18 now resembles a first order expression where kf is recognized as a pseudo-first order rate constant dependent on the concentration of B ∗ k f k [B]f (19) Some reactions are zero order in the concentration of reactants; that is, the reaction rate is independent of the reactant. For example, the rate expression would be d[A] = −k (20) dt and in integrated form [A] t[A] − k0 (21) 5 3.3 Dependence of Reaction Rates on Temperature Even when reactant concentrations are maintained, chemical reactions are observed to proceed at different rates at different temperatures. This means that the magnitude of rate constants must be temperature dependent. The relationship between the value of a rate constant and absolute temperature is given by the Arrhenius equation − E a k = Aexp (21) RT in which k is a rate constant, A is taken to be a constant independent of temperature, and E as the activation energy. Differentiation of Eqn. 21 gives d lnk Ea
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