3. CHEMICAL REACTION
KINETICS
One must wait until the evening to see how
splendid the day has been.
-Sophocles
Contents
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3.1 The Law of Mass Action
3.2 Rate Constants and Temperature
3.3 Reaction Order and Testing Reaction Rate Expressions
3.3.1 Zero-Order Reactions
3.3.2 First-Order Reactions
3.3.3 Second-Order Reactions
3.3.4 Other Reaction Orders
3.3.5 Michaelis-Menton Enzyme Kinetics
3.4 Consecutive Reactions
3.5 Reversible Reactions
3.6 Parallel Reactions, Cycles and Food Webs
3.7 Transition State Theory
3.8 Linear Free-Energy Relationship
Problems
3.1 LAW OF MASS ACTION
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1867, Guldberg and Waage
The rate of a reaction is proportional to the product of the
concentration of each substance participating in the reaction raised
to the power of its stoichiometric coefficients
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[ ] : chemical concentration (activity) in solution
If the reaction proceeds to chemical equilibrium, the rate of the
forward reaction becomes equal to the reverse reaction
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The equilibrium constant
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Elementary reactions : occur in a single step; the law of mass action
holds
Simple unimolecular reaction where 1 mole of chemical A decomposes
to form 1 mole of B irreversibly
Bimolecular elementary reactions
Trimolecular elementary reactions are less common and and more
complicated stoichiometric equations than trimolecular do not occur.
3.2 RATE CONSTANTS AND TEMPERATURE
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The rate constant carries its own units necessary to convert the mass
law expression into a reaction rate
For the first-order decay reaction, the units
on k are inverse time (T-1)
But for the 2nd-order reaction, k: L3M-1T-1(L mol-1 s-1 or L mg-1 d-1)
In Eyring's transition state theory, a reaction must overcome an
activation energy before it can proceed. Figure 3.1 shows, that the
reactant mixture has a certain energy content (internal energy) derived
from its chemical potential at a given temperature and pressure. If the
reaction occurs, the system proceeds through a peak in energy, a
metastable transition state that may involve an activated complex
(ABC#)
Figure 3.1 Diagram for transition reaction A + BC → AB + C. The free
activation energy ΔG# is necessary to form the activated complex ABC#,
which is in equilibrium with the reactants. The products AB+C are
formed from the dissociation of ABC#
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Reaction rates increase with increasing temperature
Svante Arrhenius: the relationship between the reaction rate constant and
temperature
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A is a constant that is characteristic of the reaction, Eact is the activation
energy (J mol-1 or cal mol-1), T is the absolute temperature in K, and R is the
universal gas constant (8.314 J mol-1 K-1 or 1.987 cal J mol-1 K-1). A plot of ln
k versus 1/T reveals the Eact from the slope of the straight line
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If both temperatures are known in chemical reaction, the equations for
k equated, and the constant A drops out as follows
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Generally chemical reactions occur in the temperature range from 0 to
35 ℃  Eact/RT1T2 ≈ constant, and the equation simplifies to
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θ is a constant temperature coefficient > 1.0 and usually within the
range 1.0-1.10, and k20 is the rate constant at the reference temperature
20 ℃
Figure 3.2 Arrhenius plot of reaction
rate constant at any temperature.
Activation energies for the reaction
can be obtained from the slope of the
line.
Figure 3.3 Effect of
temperature on reaction rate.
Example 3.1 Effect of Temperature on Reaction Rate Constants
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The Q10 rule in biology states that for a 10℃ increase in temperature,
the rate of the reaction will approximately double. Solve for the
activation energy and q value necessary for a doubling of the reaction
rate constant from 20 → 30 ℃. .
Solution: From the eq’s proposed above,
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Solving, we find
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Solving for θ, we find
Table 3.1 Effect of Temperature on Reaction Rate Constants
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Enzymes are catalysts that speed the rate of reaction but are not
consumed in the reaction.
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S – substrate, E – enzyme, SE – substrate-enzyme complex, P – product
The role of the enzyme is to lower the activation energy of the reaction
in Figure 3.1, resulting in a greater probability that reactants will
interact successfully to form products.
Homogeneous catalysts are dissolved in the aqueous phase together with
the reactants.
Heterogeneous catalysts are usually solid surfaces, and surface
coordination reactions are one of the steps in the overall reaction. The
surfaces bind a soluble reactant and create an activated complex.
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Table 3.2
Catalysts in
Selected
Aquatic
Chemical
Reactions
3.3 REACTION ORDER AND TESTING
REACTION RATE EXPRESSIONS
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In the arbitrary reaction between species A, B, and C, the overall
reaction order is defined as the sum of the exponents in the rate
expression (a + b + c).
For a reaction rate that can be written as an elementary reaction:
The overall reaction would be said to be of order a + b + c, but the
reaction rate could also be said to be a order in reactant A, b order in
reactant B, and c order in reactant C .
Most elementary reactions are either zero, first, or second order.
When reactions occur in a series of steps, fractional order reactions
are observed.
Methods for estimating rate constant for these several kinds of
reactions are described below.
3.3.1 Zero-Order Reactions
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If we consider irreversible degradation,
reaction rate does not depend on the
concentration of reactant in solution. k is the
rate constant of the zero-order reaction .
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For a zero-order reaction, integration of the rate expression results
in a straight line, and the rate constant k0 can be determined as the
slope of the line.
From the results of the batch experiment, we can determine two
important facts about the reaction. .
 The proposed rate expression is correct if the line is straight (the
measurements fall on a straight line to within some acceptable
statistical limit). .
 The rate constant can be obtained from the slope of the line.
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3.3.2 First-Order Reactions
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FOR: the reaction rate is proportional to the concentration of the
reactant to the first power
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Solving the above equation for A by separation of varlables and
integrating
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Equation for B can also be integrated, but it is one ordinary differential
equation with two unknowns (A and B). So, substituting for known A
and solving,
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The solution for exponential growth reaction:
Figure 3.4 Summary of simple reaction kinetics from batch reactor
Examples of FOR:
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Radioisotope decay.
Biochemical oxygen demand in a stream.
Sedimentation of noncoagulating solids.
Death and respiration rates for bacteria and algae.
Reaeration and gas transfer.
Log growth phase of algae and bacteria (production reaction).
Probably the only one that is "exactly" first order is radioisotope decay.
But the other reactions may be sufficiently close to first-order reactions
that we may assume the reaction mechanism as an approximation.
3.3.3 Second-Order Reactions
One-reactant
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For the second-order
reaction with one reactant:
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Nonlinear ordinary
differential equation:
Two-reactants
Autocatalytic
 1/A versus time will yield a straight line with a slope of k2.
 Second-order reaction with two reactants:
 A plot of ln(A/B) versus time should yield a straight line with the slope
of -k2(B0 – A0)
Figure 3.4 Summary of simple reaction kinetics from batch reactor
3.3.4 Other Reaction Orders
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N-order reaction
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If the reaction is not elementary but multi-step one, It may be
fractional order (0 < n < 1) or some other noninteger order. Fractional
order kinetics occur in precipitation and dissolution reactions.
For example, in the dissolution of oxides and aluminosilicate minerals
during chemical weathering, the reaction is surface-controlled by the
slow detachment of the central metal ion (activated complex) into
solution .
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⊪OH = hydrous oxide or aluminosilicate mineral; z = charge on the
central metal ion and the number of protons bound to the central metal
atom; M = central metal ion of valence z+; ⊪ = renewed surface
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0<m<1, m=nxz
3.3.5 Michaelis-Menton Enzyme Kinetics
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Enzyme kinetics often result in rather complicated rate expressions.
The classic case of Michaelis-Menton enzyme kinetics follows a twostep reaction mechanism as follows.
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E is the enzyme, S is the substrate, ES is the enzyme-substrate complex,
and P is the product of the reaction.
Note that the enzyme is a catalyst that speeds the rate of the reaction
(lowers the activation energy) but is not consumed in the reaction.
The rate of formation of ES complex
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Rate of formation of products is
first order in the ES complex.
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Steady state : d[ES]/dt = 0, k3 << k2
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Total enzyme in the system (E + ES)  E = ET – ES
The total enzyme ET is seen to increase the rate of the reaction
(catalyze it), but it is not consumed in the reaction. The formation rate
of product increases with increasing ET concentration. .
If the product P is cellular synthesis (cell biomass), then k3[ET]
represents the maximum growth rate of the product, and we obtain
the final expression for Michaelis-Menton kinetics:
μmax the maximum growth rate of the product (cells)
The reaction rate expression in the above equation is intermediate
between the 1st and 2nd order cases. . At low substrate concentrations
(S<< KM), it is second order overall. At high substrate concentrations
(S>>KM) it is first order overall and represents a log-growth phase.
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The growth rate is a maximum when S>>KM (the substrate
concentration is very large), and it is first order with respect to
substrate concentration for small substrate concentrations. Figure 3.5
is a plot of the growth rate as a function of substrate concentration.
Figure 3.5 Michaelis-Menton enzyme kinetics showing maximum growth
rate μmax and half-saturation constant (Michaelis constant) KM.
Figure 3.6 Lineweaver-Burk plot to linearize data using Michaelis-Menton
enzyme kinetics to obtain the parameters μmax and KM. It is a doublereciprocal plot of growth rate and substrate concentration.
3.4 CONSECUTIVE REACTIONS
Nitrification and carbonaceous biochemical oxygen demand (CBOD) in a
stream are examples, where D is the dissolved oxygen deficit that is created
when CBOD exerts itself. Ammonia-nitrogen is oxidized to nitrite-nitrogen,
which is, in turn, oxidized to nitrate-nitrogen. Because etch species is expressed
in terms of nitrogen, the stoichiometric coefficients are unity. Bacteria catalyze
the reactions in the above equations. For consecutive nitrification reactions,
Nitrosomonas spp. mediate the first reaction and Nitrobacter spp. mediate the
second reaction. The overall balanced chemical action for nitrification is
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1 mole of ammonia combines with 2 moles of
oxygen to form 1 mole of nitrate, overall. On
a mass basis, 1.0 gram of ammonia-nitrogen
consumes 4.57 grams of oxygen to form 1.0
gram of nitrate-nitrogen.
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A is the ammonia-nitrogen concentration, B is the nitrite-nitrogen
concentration, and C is the nitrate-nitrogen concentration. The above
equations represent a set of three ordinary differential equations that must
be solved simultaneously
The concentration of biodegradable organic material can be measured
using a biochemical oxygen demand test. It measures the concentration
of dissolved oxygen that is consumed via microbial oxidation of the
organics. This process results in a dissolved oxygen deficit in equation
(52). The deficit, in turn, reaerates away due to the absorption of
oxygen from the atmosphere to the stream. Instead of forming a
product, the deficit goes to zero as atmospheric reaeration proceeds to
chemical equilibrium (saturation).
Csat is the saturated concentration of dissolved oxygen in
(자정작용).
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equilibrium with the atmosphere, and D.O. is the
dissolved oxygen concentration. Csat depends on
temperature and salinity of the water body.
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To solve the above simultaneous equations, we must start from
ammonia-nitrogen equation.
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Substitute A concentration into the ammonia-nitrogen
equation
Solving equation for B by integration factor
p(t) is integrating factor, q(t) is
nonhomogeneous forcing function
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y=y0 at t=0
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The above equation can be the classic D.O. sag curve of StreeterPhelps.
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Solution for nitrate can be found by the following equation. NT is the
total moles of species A, B, and C or the sum of their initial
concentrations
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k1 can be obtained from a semilogarithmic plot ln A versus t.
k2 can be estimated by using experimental data in nonlinear leastsquares fit. It also can be found through the following equation.
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Figure 3.7 Concentration of ammonia-nitrogen, nitrite-nitrogen, nitratenitrogen versus time in nitrification reaction
3.5 REVERSIBLE REACTIONS
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Many physical chemical reactions that occur in nature are results of
forward and reverse reactions coming into a chemical equilibrium.
Some examples of reversible reaction are: acid-base reactions, gas
transfer, adsorption-desorption, bio concentration-depuration
The total concentration of chemical is
constant throughout time.
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Reaction rate
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It is solvable by the integration factor method or by the use of
integration tables and separation of variables.
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At steady state the equilibrium is
reached: dA/dt = 0
B^ and A^ are steady-state concentrations and
Keq is the equilibrium constant
We can obtain solution for A at t = ∞
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Figure 3.8 Reversible reaction showing the mixture of
products and reactants at chemical equilibrium (t = ∞)
3.6 PARALLEL REACTIONS, CYCLES, AND FOOD WEBS
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In the nitrification of ammonia, parallel reactions might include the
uptake of ammonia by algae and the stripping of ammonia from the
water body to the atmosphere at high pH.
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The pathway that ammonia disappears from the environment, then,
depends on the relative magnitude of the rate constants k1, k3, and k4. A
rate expression must include all three reactions
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Sulfur cycle: five state variables and eight reactions.
Figure 3.9 The example of elemental cycle – sulfur cycle. Each reaction
has a rate constant and reaction rate expression.
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Heavy metals, nitrogen, carbon, sulfur, and phosphorus are
elemental cycles that can be modeled at the microscale, mesoscale, or
even global scale using chemical reaction kinetics. .
Food webs are similar to elemental cycles for carbon or biomass. In a
lake, one might be interested in modeling the transport and
transformation of a contaminant (e.g., polychlorinated biphenyls or
PCBs) as they move through the aquatic food web, etc.
The entire system is driven by primary production involving
photosynthesis (the sun's energy) and the uptake of carbon dioxide
by algae and rooted plants.
Figure 3.10 Food web of an ecosystem: demonstrates the
interconnectedness and cycling of elements in natural waters.
3.7 TRANSITION STATE THEORY
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Transition state theory considers the free-energy requirements of a
chemical reaction.
Rate expressions based on transition state theory provide an important
bridge between thermodynamics (energetics and equilibrium reactions
of Ch.4) and rates of reactions (kinetics in Ch. 3).
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Formation of activated complex:
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Dissociating into products, irreversibly:
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The higher the activation energy (standard free energy of
activation), the less is the probability that the reaction occurs, and
the smaller is the rate of reaction. kB is Boltzmann’s constatn
(1.38x10-23 K-1), h is Planck’s constant (6.63x10-34 J s-1), T is is the
absolute temperature (K).
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The activated complex, ABC++, is in
equilibrium with the reactants:
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Using this constant, the reaction rate is:
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The standard free energy of
activation is defined as :
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The rate constant:
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From thermodynamics:
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ΔH++ is the standard enthalpy of activation, and ΔS++ is the standard
entropy of activation.
3.8 LINEAR FREE-ENERGY RELATIONSHIPS
Quantitative relation can be established between reaction rate constant
and equilibrium constant. For two related reactions, the following
relationship can be established
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Linear free energy relation in terms of thermodynamics:
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ΔG2++ and ΔG1++ - free activation energies, ΔG2o and ΔG1o – free
energies of the related reactions. For a series of i reactants, the final
linear free energy relationships are
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α – the slope of the linear plot, β – the intercept.
Figure 3.11 Linear freeenergy relationship for the
oxidation of various Fe(II)
species (Fe2+, FeOH+, and
Fe(OH)20) with O2(aq) and the
equlilbrium constant for the
reaction
• the rate expressions follow
the law of mass action as the
product of the Fe(II) species
times the molar oxygen
concentration in solution
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Three points representing the rate
constant versus the equilibrium constant
for the reactions are shown. The rate
constant in each case is defined by
Assignments
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Derive the analytical solution of 0, 1, 2, catalyst, and nth order reactions.
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Derive the solution of three simultaneous equations of nitrification.
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Explain the D. O. Sag Curve using the above solution.
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Explain the theory of transition state and the relation of linear free energy.
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Explain the activation energy in terms of enthalpy and entropy.
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Explain all the models in my web site.
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Explain p7, p8, and p9 in groundwater textbook.
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Make the English table for composite multiphase groundwater model.