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Chemistry 232. Kinetics of Complex Reactions. The Pre-Equilibrium Approximation. Examine the following process. Pre-Equilibrium (II). B is obviously an intermediate in the above mechanism. Could use SSA. What if the initial equilibrium is fast? Step 2 is the rds!. Pre-Equilibrium (III).

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Chemistry 232 l.jpg

Chemistry 232

Kinetics of Complex Reactions

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The Pre-Equilibrium Approximation

  • Examine the following process

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Pre-Equilibrium (II)

  • B is obviously an intermediate in the above mechanism.

    • Could use SSA.

  • What if the initial equilibrium is fast?

    • Step 2 is the rds!

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Pre-Equilibrium (III)

  • We now have a simple expression for the [B]; hence

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Lindemann-Hinshelwood Mechanism

  • An early attempt to explain the kinetics of complex reactions.


Rate Laws

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The ‘Activated’ Intermediate

  • Formation of the product depends directly on the [A*].

  • Apply the SSA to the net rate of formation of the intermediate [A*]

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Is That Your ‘Final Answer’?

  • Substituting and rearranging

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The ‘Apparent Rate Constant’ Depends on Pressure

  • The rate laws for the Lindemann-Hinshelwood Mechanism are pressure dependent.

High Pressure Case

Low Pressure Case

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The Pressure Dependence of k’

  • In the Lindemann-Hinshelwoood Mechanism, the rate constant is pressure dependent.

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  • So far, we have considered one way of speeding up a reaction (i.e. increasing T usually increases k). Another way is by the use of a catalyst.

  • A catalyst - a substance that speeds up the rate of the reaction without being consumed in the overall reaction.

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Types of Catalyst

  • We will briefly discuss three types of catalysts. The type of catalyst depends on the phase of the catalyst and the reacting species.

    • Homogeneous

    • Heterogeneous

    • Enzyme

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Homogeneous Catalysis

  • The catalyst and the reactants are in the same phase

  • e.g. Oxidation of SO2 (g)to SO3 (g)

    2 SO2(g) + O2(g) ® 2 SO3 (g) SLOW

  • Presence of NO (g), the following occurs.

    NO (g) + O2 (g) ® NO2 (g)

    NO2 (g) + SO2 (g) ® SO3 (g) + NO (g) FAST

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  • SO3 (g) is a potent acid rain gas

    H2O (l) + SO3 (g)  H2SO4 (aq)

  • Note the rate of NO2(g) oxidizing SO2(g) to SO3(g) is faster than the direct oxidation.

  • NOx(g) are produced from burning fossil fuels such as gasoline, coal, oil!!

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Heterogeneous Catalysis

  • The catalyst and the reactants are in different phases

    • adsorption the binding of molecules on a surface.

  • Adsorption on the surface occurs on active sites

    • Places where reacting molecules are adsorbed and physically bond to the metal surface.

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  • The hydrogenation of ethene (C2H4 (g)) to ethane

    C2H4 (g) + H2(g)  C2H6 (g)

  • Reaction is energetically favourable

    • rxnH = -136.98 kJ/mole of ethane.

  • With a finely divided metal such as Ni (s), Pt (s), or Pd(s), the reaction goes very quickly .

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  • There are four main steps in the process

    • the molecules approach the surface;

    • H2 (g) and C2H4 (g) adsorb on the surface;

    • H2 dissociates to form H(g) on the surface; the adsorbed H atoms migrate to the adsorbed C2H4 and react to form the product (C2H6) on the surface

    • the product desorbs from the surface and diffuses back to the gas phase

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Simplified Model for Enzyme Catalysis

  • E º enzyme; S º substrate; P º product

    E + S ® ES

    ES ® P + E

    rate = k [ES]

  • The reaction rate depends directly on the concentration of the substrate.

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Enzyme Catalysis

  • Enzymes - proteins (M > 10000 g/mol)

  • High degree of specificity (i.e., they will react with one substance and one substance primarily

  • Living cell > 3000 different enzymes

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The Lock and Key Hypothesis

  • Enzymes are large, usually floppy molecules. Being proteins, they are folded into fixed configuration.

  • According to Fischer, active site is rigid, the substrate’s molecular structure exactly fits the “lock” (hence, the “key”).

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The Michaelis-Menten Mechanism

  • Enzyme kinetics – use the SSA to examine the kinetics of this mechanism.

ES – the enzyme-substrate complex.

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Applying the SSA to the Mechanism

  • Note that the formation of the product depends directly on the [ES]

  • What is the net rate of formation of [ES]?

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ES – The Intermediate

  • Apply the SSA to the equation for d[ES]/dt = 0

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Working Out the Details

  • Let [E]o = [E] + [ES]

Complex concentration

Initial enzyme concentration

Free enzyme concentration

Note that [E] = [E]o - [ES]

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The Final Equation

  • Substituting into the rate law vp.

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The Michaelis Constant and the Turnover Number

  • The Michaelis Constant is defined as

  • The rate constant for product formation, k2, is the turnover number for the catalyst.

  • Ratio of k2 / KM – indication of catalytic efficiency.

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The Maximum Velocity

  • As [S]o gets very large.

Note – Vmax is the maximum velocity for the reaction. The limiting value of the reaction rate high initial substrate concentrations.

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Lineweaver-Burk Equation

  • Plot the inverse of the reaction rate vs. the inverse of the initial substrate concentration.

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Chain Reactions

  • Classifying steps in a chain reaction.

    • Initiation

      • C2H6 (g) 2 CH3•

    • Propagation Steps

      • C2H6 + •CH3  •C2H5 + CH4

    • Branching Steps

      • H2O + •O•  2 •OH

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Chain Reactions (Cont’d)

  • Retardation Step

    • HBr + H•  H2 + Br•

  • Terminations Steps

    • 2 CH3CH2•  CH3CH2CH2CH3

  • Inhibition Steps

    • R• + CH3•  RCH3

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The H2 + Br2 Reaction

  • The overall rate for the reaction was established in 1906 by Bodenstein and Lind

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The Mechanism

  • The mechanism was proposed independently by Christiansen and Herzfeld and by Michael Polyani.

Rate Laws


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Using the SSA

  • Using the SSA on the rates of formation of Br• and H•

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Hydrogenation of Ethane

  • The Rice-Herzfeld Mechanism


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Rate Laws for the Rice-Herzfeld Mechanism

  • The rate laws for the elementary reactions are as follows.

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  • Thermal explosions

    • Rapid increase in the reactions rate with temperature.

  • Chain branching explosions

    • chain branching steps in the mechanism lead to a rapid (exponential) increase in the number of chain carriers in the system.

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Photochemical Reactions

  • Many reactions are initiated by the absorption of light.

  • Stark-Einstein Law – one photon is absorbed by each molecule responsible for the primary photochemical process.

I = Intensity of the absorbed radiation

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Primary Quantum Yield

  • Define the primary quantum yield, 

  • Define the overall quantum yield, 

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  • Transfer of excitation energy from one molecule (the photosensitizer) to another nonabsorbing species during a collision..

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Polymerization Kinetics

  • Chain polymerization

    • Activated monomer attacks another monomer, chemically bonds to the monomer, and then the whole unit proceeds to attack another monomer.

  • Stepwise polymerization

    • A reaction in which a small molecule (e.g., H2O) is eliminated in each step.

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Chain Polymerization

  • The overall polymerization rate is first order in monomer and ½ order in initiator.

  • The kinetic chain length, kcl

    • Measure of the efficiency of the chain propagation reaction.

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  • Initiation

    I  2 R•


    M + R•  M1 •

  • Propagation

    M + M1•  M2 •

    M + M2•  M3 •

    M + M3•  M4 •


Rate Laws

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Mechanism (Cont’d)

  • Termination

    M + M3•  M4 •

Note – Not all the initiator molecules produce chains

Define  = fraction of initiator molecules that produce chains

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Return to Kinetic Chain Length

  • We can express the kinetic chain length in terms of kt and kp

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Stepwise Polymerization

  • A classic example of a stepwise polymerization – nylon production.

    NH2-(CH2)6-NH2 + HOOC-(CH2)4COOH 

    NH2-(CH2)6-NHOC-(CH2)4COOH + H2O

  • After many steps


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The Reaction Rate Law

  • Consider the condensation of a generic hydroxyacid


  • Expect the following rate law

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The Reaction Rate Law (Cont’d)

  • Let [A] = [-COOH]

  • A can be taken as any generic end group for the polymer undergoing condensation.

  • Note 1 –OH for each –COOH

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The Reaction Rate Law (Cont’d)

  • If the rate constant is independent of the molar mass of the polymer

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The Fraction of Polymerization

  • Denote p = the fraction of end groups that have polymerized

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Statistics of Polymerization

  • Define Pn = total probability that a polymer is composed of n-monomers

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The Degree of Polymerization

  • Define <n> as the average number of monomers in the chain

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Degree of Polymerization (cont’d)

  • The average polymer length in a stepwise polymerization increases as time increases.

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Molar Masses of Polymers

  • The average molar mass of the polymer also increases with time.

  • Two types of molar mass distributions.

    • <M>n = the number averaged molar mass of the polymer.

    • <M>w = the mass averaged molar mass of the polymer.

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Definitions of <M>n

  • Two definitions!

Mo = molar mass of monomer

n = number of polymers of mass Mn

MJ = molar mass of polymer of length nJ

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Definitions of <M>w

  • <M>w is defined as follows

Note - xn the number of monomer units in a polymer molecule

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The Dispersity of a Polymer Mixture

  • Polymers consists of many molecules of varying sizes.

  • Define the dispersity index () of the mass distribution.

Note – monodisperse sample ideally has <M>w=<M>n

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The Dispersity Index in a Stepwise Polymerization

  • The dispersity index varies as follows in a condensation polymerization

Note – as the polymerization proceeds, the ratio of <M>w/<M>n approaches 2!!!

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Monodisperse Sample

Polydisperse Sample


















Mass Distributions in Polymer Samples

  • For a random polymer sample


Molar mass / (10000 g/mole)