3' CHEMICAL REACTION KINETICS

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3. CHEMICAL REACTION KINETICS

• One must wait until the evening to see how
  splendid the day has been.
•                                  
                    -Sophocles

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Contents

• 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

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3.1 LAW OF MASS ACTION

• 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

• chemical concentration (activity) in
  solution
• If the reaction proceeds to chemical equilibrium,
  the rate of the forward reaction becomes equal to
  the reverse reaction

• 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.

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3.2 RATE CONSTANTS AND TEMPERATURE

• 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)

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Figure 3.1 Diagram for transition reaction A BC
? AB C. The free activation energy ?G is
necessary to form the activated complex ABC,
which is in equilibrium with the reactants. The
products ABC 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

• 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

• If both temperatures are known in chemical
  reaction, the equations for k equated, and the
  constant A drops out as follows

• Generally chemical reactions occur in the
  temperature range from 0 to 35 ? ? Eact/RT1T2
  constant, and the equation simplifies to

• ? is a constant temperature coefficient gt 1.0 and
  usually within the range 1.0-1.10, and k20 is the
  rate constant at the reference temperature 20 ?

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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.
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Example 3.1 Effect of Temperature on Reaction
Rate Constants

• 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 eqs proposed above,

• Solving, we find

• Solving for ?, we find

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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.

• 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.

Table 3.2 Catalysts in Selected Aquatic Chemical
Reactions
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3.3 REACTION ORDER AND TESTING REACTION RATE
EXPRESSIONS

• 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.

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3.3.1 Zero-Order Reactions

• 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 .

• 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

• FOR the reaction rate is proportional to the
  concentration of the reactant to the first power

• Solving the above equation for A by separation of
  varlables and integrating

• 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,

• The solution for exponential growth reaction

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Figure 3.4 Summary of simple reaction kinetics
from batch reactor
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• Examples of FOR
• 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.

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3.3.3 Second-Order Reactions
Autocatalytic
One-reactant
Two-reactants

• For the second-order reaction with one reactant

• Nonlinear ordinary differential equation

• 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)

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Figure 3.4 Summary of simple reaction kinetics
from batch reactor
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3.3.4 Other Reaction Orders

• N-order reaction

• If the reaction is not elementary but multi-step
  one, It may be fractional order (0 lt n lt 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 .

• ?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

• 0ltmlt1, mnxz

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3.3.5 Michaelis-Menton Enzyme Kinetics

• Enzyme kinetics often result in rather
  complicated rate expressions. The classic case of
  Michaelis-Menton enzyme kinetics follows a
  two-step reaction mechanism as follows.

• 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

• Rate of formation of products is first order in
  the ES complex.  

• Steady state dES/dt 0, k3 ltlt 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 k3ET 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
  (Sltlt KM), it is second order overall. At high
  substrate concentrations (SgtgtKM) it is first
  order overall and represents a log-growth phase.

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• The growth rate is a maximum when SgtgtKM (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.

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Figure 3.5 Michaelis-Menton enzyme kinetics
showing maximum growth rate µmax and
half-saturation constant (Michaelis constant) KM.
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Figure 3.6 Lineweaver-Burk plot to linearize
data using Michaelis-Menton enzyme kinetics to
obtain the parameters µmax and KM. It is a
double-reciprocal plot of growth rate and
substrate concentration.
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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

• 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.

• 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

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• 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 equilibrium with the atmosphere, and
D.O. is the dissolved oxygen concentration. Csat
depends on temperature and salinity of the water
body.

• To solve the above simultaneous equations, we
  must start from ammonia-nitrogen equation.

• 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

• yy0 at t0

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• The above equation can be the classic D.O. sag
  curve of Streeter-Phelps.

• 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

• k1 can be obtained from a semilogarithmic plot ln
  A versus t.
• k2 can be estimated by using experimental data in
  nonlinear least-squares fit. It also can be found
  through the following equation.

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Figure 3.7 Concentration of ammonia-nitrogen,
nitrite-nitrogen, nitrate-nitrogen versus time in
nitrification reaction
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3.5 REVERSIBLE REACTIONS

• 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-depuratio
  n

The total concentration of chemical is constant
throughout time.

• Reaction rate

• It is solvable by the integration factor method
  or by the use of integration tables and
  separation of variables.

• 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 8

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Figure 3.8 Reversible reaction showing the
mixture of products and reactants at chemical
equilibrium (t 8)
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3.6 PARALLEL REACTIONS, CYCLES, AND FOOD WEBS

• 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.

• 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

• Sulfur cycle five state variables and eight
  reactions.

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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.

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Figure 3.10 Food web of an ecosystem
demonstrates the interconnectedness and cycling
of elements in natural waters.
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3.7 TRANSITION STATE THEORY

• 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).

• Formation of activated complex

• Dissociating into products, irreversibly

• 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
  Boltzmanns constatn (1.38x10-23 K-1), h is
  Plancks 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

• Using this constant, the reaction rate is

• The standard free energy of activation is defined
  as

• The rate constant

• From thermodynamics

• ?H is the standard enthalpy of activation, and
  ?S is the standard entropy of activation.

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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

• Linear free energy relation in terms of
  thermodynamics

• ?G2 and ?G1 - free activation energies, ?G2o
  and ?G1o free energies of the related
  reactions. For a series of i reactants, the final
  linear free energy relationships are

• a the slope of the linear plot, ß the
  intercept.

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• Figure 3.11 Linear free-energy 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

• Three points representing the rate constant
  versus the equilibrium constant for the reactions
  are shown. The rate constant in each case is
  defined by

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Assignments

• Derive the analytical solution of 0, 1, 2,
  catalyst, and nth order reactions.
• Derive the solution of three simultaneous
  equations of nitrification.
• Explain the D. O. Sag Curve using the above
  solution.
• Explain the theory of transition state and the
  relation of linear free energy.
• Explain the activation energy in terms of
  enthalpy and entropy.
• Explain all the models in my web site.
• Explain p7, p8, and p9 in groundwater textbook.
• Make the English table for composite multiphase
  groundwater model.