3. CHEMICAL REACTION KINETICS

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

                                                    -Sophocles

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

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 sub

stance 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

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 irr

eversibly

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

The rate constant carries its own units necessary to convert the mass law expression into a reaction rate

But for the 2nd-order reaction, k: L3M-1T-1(L mol-1 s-1 or L mg-1 d-1)

For the first-order decay reaction, the units on k are inverse time (T-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#

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

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

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 ℃

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

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.

The Q10 rule in biology states that for a 10 increase in temperature, the rate o℃f 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,

Example 3.1 Effect of Temperature on Reaction Rate Constants

Solving, we find

Solving for θ, we find

Table 3.1 Effect of Temperature on Reaction Rate Constants

Enzymes are catalysts that speed the rate of reaction but are not consumed in the reaction.

Table 3.2 Catalysts in Selected Aquatic Chemical Reactions

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.

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.

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.

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:

Figure 3.4 Summary of simple reaction kinetics from batch reactor

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.

3.3.3 Second-Order Reactions

For the second-order reaction with one reactant:

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 -k

2(B0 – A0)

Two-reactants Autocatalytic

Figure 3.4 Summary of simple reaction kinetics from batch reactor

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 < 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 .

⊪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⊪

0<m<1, m=nxz

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 : d[ES]/dt = 0, k3 << k2

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.

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 double-reciprocal 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

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

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).

( 자정작용 ). 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

y=y0 at t=0

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.

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

3.5 REVERSIBLE REACTIONS Many physical chemical reactions that occur in nature are results of forward and revers

e reactions coming into a chemical equilibrium. Some examples of reversible reaction are: acid-base reactions, gas transfer, adsorption-desorption, bio concentration-depuration

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 = ∞

The total concentration of chemical is constant throughout time.

Figure 3.8 Reversible reaction showing the mixture of products and reactants at chemical equilibrium (t = ∞)

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.

Figure 3.9 The example of elemental cycle – sulfur cycle. Each reaction has a rate constant and reaction rate expression.

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

Transition state theory considers the free-energy requirements of a chemical reaction. Rate expressions based on transition state theory provide an important bridge between the

rmodynamics (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 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).

From thermodynamics:

ΔH++ is the standard enthalpy of activation, and ΔS++ is the standard entropy of activation.

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:

3.8 LINEAR FREE-ENERGY RELATIONSHIPS Quantitative relation can be established between reaction rate constant and equilibriu

m constant. For two related reactions, the following relationship can be established

Linear free energy relation in terms of thermodynamics:

ΔG2++ and ΔG1++ - free activation energies, ΔG2o and ΔG1

o – free energies of the related reactions. For a series of i reactants, the final linear free energy relationships are

α – the slope of the linear plot, β – the intercept.

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

Figure 3.11 Linear free-energy relationship for the oxidation of various Fe(II) species (Fe2+, FeOH+, and Fe(OH)2

0) 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

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.