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John Ferris
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Earth Sciences

ESS102H1

John Ferris

Winter

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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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