Kinetics of Homogeneous Reactions - Detailed Notes

Kinetics of Homogeneous Reactions

Introduction

• Industrial chemical processes aim to economically produce desired products from raw materials.

• These processes involve unit operations and unit processes in a correct sequence.

• The chemical step, like oxidation, sulphonation, or nitration, converts raw materials into desired products and significantly impacts the overall process economics.

• A chemical reactor is equipment where reactants are converted into a useful product.

• Chemical reaction engineering focuses on designing chemical reactors to determine the required reactor volume for a given task.

• This design incorporates knowledge from thermodynamics, chemical kinetics, fluid flow, heat transfer, mass transfer, and economics.

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Chemical Reactions

• Chemical reactions involve the formation of new molecules through rearrangement or redistribution of atoms.

• Species lose their identity and transform via decomposition, combination, or isomerization.

• For instance, methanol synthesis involves carbon dioxide combining with hydrogen to form methanol: $CO + 2H<em>2 \rightarrow CH</em>3OH$

• Chemical engineers need to know:

  • Whether a reaction is feasible (can it go?).

  • How long the reaction will take (kinetics).

• Thermodynamics addresses feasibility, while chemical kinetics addresses the reaction rate.

Chemical Kinetics

• It studies the rates of chemical reactions and the influence of temperature, pressure, and reactant concentration/composition on these rates.

• It provides insights into the reaction mechanism, reaction speed, and the appropriate rate equation for reactor design.

Thermodynamics

• It provides information on the feasibility of a reaction, including whether it will occur under specific conditions.

• It also covers heat absorption or liberation during the reaction (heat of reaction) and the maximum possible extent of the reaction.

• Both thermodynamics and chemical kinetics are crucial for chemical reactor design.

Classification of Chemical Reactions

Based on the number of phases involved:

• Homogeneous Reactions: Occur in a single phase where all reacting materials, products, and any catalysts are present in the same phase.

  • Example: Oxidation of nitrogen oxide to nitrogen dioxide in the gas phase: $NO + \frac{1}{2}O<em>2 \rightarrow NO</em>2$

  • Example: Ester formation from organic acids and alcohols in the liquid phase with sulfuric acid as a catalyst: $C<em>2H</em>5OH + CH<em>3COOH \rightarrow CH</em>3COOC<em>2H</em>5 + H_2O$

• Heterogeneous Reactions: Involve multiple phases, where at least one reactant, catalyst, or product exists in a different phase from the rest of the reacting system.

  • Example: Oxidation of sulfur dioxide to sulfur trioxide using vanadium pentaoxide ($V<em>2O</em>5$) as a solid catalyst: $SO<em>2 + O</em>2 \xrightarrow{V<em>2O</em>5} SO_3$

Based on the presence of catalyst:

• Catalytic Reactions: Reactions that utilize a catalyst to increase the reaction rate.

  • Example: Hydrogenation of ethylene using nickel catalyst: $C<em>2H</em>4 + H<em>2 \xrightarrow{Ni} C</em>2H_6$

• Non-Catalytic Reactions: Reactions that proceed without the use of a catalyst.

  • Example: Oxidation of $NO$ to $NO<em>2$: $NO + \frac{1}{2} O</em>2 \rightarrow NO_2$

Based on molecularity:

• Molecularity defines the number of molecules involved in the rate-determining step of a reaction.

  • Unimolecular Reactions: Involve a single molecule in the rate-determining step.

    • Example: Decomposition of cyclobutane: $C<em>4H</em>8 \rightarrow 2C<em>2H</em>4$

  • Bimolecular Reactions: Involve the collision of two molecules.

    • Example: Decomposition of hydrogen iodide: $2HI \rightarrow H<em>2 + I</em>2$

  • Termolecular Reactions: Involve the collision of three molecules.

    • Example: Oxidation of $NO$ to $NO<em>2$: $2NO + O</em>2 \rightarrow 2NO_2$

Based on heat effect:

• Exothermic Reactions: Reactions that release heat to the surroundings.

  • Example: Synthesis of methanol from carbon monoxide and hydrogen: $CO + 2H<em>2 \rightarrow CH</em>3OH + Heat$

• Endothermic Reactions: Reactions that absorb heat from the surroundings.

  • Example: Dehydration of ethyl alcohol to produce ethylene: $C<em>2H</em>5OH \rightarrow C<em>2H</em>4 + H_2O - Heat$

Based on the order of reaction:

• First-Order Reaction: The overall order of the reaction is one.

  • Example: Decomposition of nitrogen pentaoxide: $N<em>2O</em>5 \rightarrow NO<em>2 + \frac{1}{2}O</em>2$

• Second-Order Reaction: The overall order of the reaction is two.

  • Example: Saponification of ester: $CH<em>3COOC</em>2H<em>5 + NaOH \rightarrow CH</em>3COONa + C<em>2H</em>5OH$

• Third-Order Reaction: The overall order of the reaction is three.

  • Example: $2NO(g) + H<em>2(g) \rightarrow N</em>2O(g) + H_2O(g)$

Based on direction:

• Reversible Reactions: Reactions that proceed in both forward and reverse directions simultaneously.

  • Example: Esterification reaction: $C<em>2H</em>5OH + CH<em>3COOH \rightleftharpoons CH</em>3COOC<em>2H</em>5 + H_2O$

• Irreversible Reactions: Reactions that proceed in only one direction until the reactants are fully consumed.

  • Example: Nitration of benzene: $C<em>6H</em>6 + HNO<em>3 \rightarrow C</em>6H<em>5NO</em>2 + H_2O$

Rate of a Chemical Reaction

• Reaction rates vary widely, from instantaneous to practically zero.

• Ionic reactions are very fast, while some reactions, like the combination of hydrogen and oxygen at room temperature without a catalyst, are immeasurably slow.

• Most industrial reactions occur at intermediate rates.

• The rate of reaction can be defined based on:

  • Unit volume of reacting fluid (homogeneous system).

  • Unit mass of solid (fluid-solid system).

• Rate is expressed as the rate of disappearance of reacting component A.

Rate Definitions

• Based on unit volume of reacting fluid:
  $-r<em>A = -\frac{1}{V} \frac{dN</em>A}{dt}$

• Based on unit mass of solid:
  $-r'<em>A = -\frac{1}{W} \frac{dN</em>A}{dt}$

• Based on unit interfacial surface:
  $-r''<em>A = -\frac{1}{S} \frac{dN</em>A}{dt}$

• Based on unit volume of solid:
  $-r'''<em>A = -\frac{1}{V</em>s} \frac{dN_A}{dt}$

• Based on unit volume of reactor:
  $-r<em>A = -\frac{1}{V</em>r} \frac{dN_A}{dt}$

• In homogeneous systems, $V$ (volume of fluid) and $V_r$ (volume of reactor) are often identical.

Relationship between Rates

• The rates based on different bases are related by:
  $(-r<em>A)V = (-r'</em>A)W = (-r''<em>A)S = (-r'''</em>A)V<em>s = (-r</em>A)V_r$

• For homogeneous systems, the reaction rate is defined per unit volume.

Reaction Stoichiometry and Rate

• Consider an irreversible reaction:
  $aA + bB \rightarrow rR$

• The rate of disappearance of reactant A is:
  $-r<em>A = -\frac{1}{V} \frac{d(N</em>A)}{dt}$

• For constant volume systems:
  $-r<em>A = -\frac{dC</em>A}{dt}$
  where $C<em>A = \frac{N</em>A}{V}$

• The rate of reaction can be expressed as the rate of change in concentration of any reactant or product.

Rate Expressions

• Rate of disappearance of B:
  $-r<em>B = -\frac{1}{b} \frac{dC</em>B}{dt}$

• Rate of formation of R:
  $r<em>R = \frac{1}{r} \frac{dC</em>R}{dt}$

• The negative sign indicates that the reactant concentration decreases over time.

Stoichiometric Relationship

• For the reaction $aA + bB \rightarrow rR$, the rates are related as:
  $\frac{-r<em>A}{a} = \frac{-r</em>B}{b} = \frac{r_R}{r}$

• Example: For $A + 3B \rightarrow 2R$:
  $-r<em>A = \frac{-r</em>B}{3} = \frac{r_R}{2}$

Factors Affecting Reaction Rate

• Nature of reactants.

• Concentration of reactants.

• Temperature.

• Pressure.

• Presence of a catalyst.

• Physical state of reactants (contact area).

• Rates of heat and mass transfer.

• For homogeneous systems, temperature, pressure, and composition (concentration).

Rate Expression

• For homogeneous systems, the rate of reaction is a function of temperature and composition:
  $-r_A = f(temperature, concentration)$

• Example: For $A \rightarrow R$:
  $-r<em>A = kC</em>A^\alpha$

• $k = k_0 e^{-\frac{E}{RT}}$

• Where:

  • $k_0$ is the pre-exponential factor.

  • $E$ is the activation energy.

  • $\alpha$ is the order of the reaction with respect to A.

Concentration-Dependent Term of a Rate Equation

• Single Reaction: Represented by a single stoichiometric equation and rate expression.

• Multiple Reactions: Require more than one stoichiometric equation and kinetic expression.

• For a reaction $aA + bB + … \rightarrow rR + sS + …$, the rate of disappearance of A is written as:
  $-r<em>A = k f(C</em>A, C_B, …)$

• Often, the rate equation is expressed as:
  $-r<em>A = kC</em>A^{n<em>A} C</em>B^{n_B} …$

• Where $n<em>A$, $n</em>B$ are the orders of the reaction with respect to A, B, etc.

Rate Constant

• The rate constant (k) is a measure of the reaction rate when all reactants are at unit concentration.

• The units of k depend on the units of concentration and time.

• For an nth order reaction, the dimensions of k are:
  $(time)^{-1} (concentration)^{1-n}$

• For a first-order reaction (n = 1), the unit of k is $s^{-1}$.

• For a second-order reaction (n = 2), the units of k are $L \cdot mol^{-1} \cdot s^{-1}$.

Determining Units of k

• For the second-order reaction, $-r<em>A = kC</em>A^2$

• $k = \frac{-r<em>A}{C</em>A^2}$

• If $-r<em>A$ is measured in $mol/(L \cdot s)$ and $C</em>A$ in $mol/L$, then k is in $\frac{L}{mol \cdot s}$.

Reaction Mechanism

• It refers to the step-by-step sequence of elementary reactions by which overall chemical change occurs

• Reactions usually occur in a series of steps rather than a single step, which is represented by the overall stoichiometric equation.

Example

• Consider the reaction: $2A + B \rightarrow R + S$

• It might take place in the following steps:

  1. $A + B \rightarrow AB$

  2. $A + AB \rightarrow A_2B$

  3. $A_2B \rightarrow AB + R$

  4. $AB \rightarrow S$

Rate-Determining Step

• The slowest step in the series of steps controls the reaction rate and determines the rate equation.

Elementary and Nonelementary Reactions

• Elementary Reactions: Reactions that occur in a single step, where molecules react exactly as described by the stoichiometric equation.

• Nonelementary Reactions: Reactions that occur through a series of steps. The overall reaction is the result of a complex sequence of elementary reactions.

Order and Stoichiometry

• For elementary reactions, the order with respect to each reactant is equal to its stoichiometric coefficient.

• For nonelementary reactions, there is no direct relationship between the order of reaction and reaction stoichiometry.

Rate Equations

• For the irreversible elementary reaction:
  $aA + bB \rightarrow rR$:
  The rate equation is:
  $-r<em>A = kC</em>A^a C_B^b$

• For the nonelementary reaction:
  $aA + bB \rightarrow rR$:
  The rate equation is given by:
  $-r<em>A = kC</em>A^\alpha C_B^\beta$
  Where $\alpha$ and $\beta$ are determined experimentally and may not be equal to a and b.

Differences between Elementary and Nonelementary Reactions

• Elementary reactions are single-step reactions, whereas nonelementary reactions are multistep.

• Elementary reactions are simple, while nonelementary reactions are complex.

• For elementary reactions, the order of each reactant matches its stoichiometric coefficient; for nonelementary reactions, they don't match.

• Elementary reaction orders must be integers, whereas nonelementary reaction orders may be integers or fractions.

Molecularity of Reaction

• The molecularity of a reaction is the number of reacting species (molecules, atoms, or ions) involved in the rate-limiting step.

Types

• Unimolecular: One reacting species (e.g., $C<em>4H</em>8 \rightarrow 2C<em>2H</em>4$

• Bimolecular: Two reacting species (e.g., $2HI(g) \rightarrow H<em>2(g) + I</em>2(g)$).

• Termolecular: Three reacting species (e.g., $2NO + O<em>2 \rightarrow 2NO</em>2$).

Notes

• Reactions with molecularity greater than three are uncommon due to the low probability of simultaneous collisions.

• Molecularity is concerned with each elementary step and has no meaning for the overall reaction if it is complex.

• The molecularity of a reaction is a theoretical quantity and must be a whole number.

Order of Reaction

• The order of a reaction is the sum of the exponents of the concentrations in the rate expression.

Rate Equation

• For a reaction of the type:
  $aA + bB + … \rightarrow rR + sS + …$
  The rate expression is given by:
  $-r<em>A = kC</em>A^{n<em>1} C</em>B^{n_2} …$

• The overall order of the reaction is $n = n<em>1 + n</em>2 + …$

Molecularity vs. Order

• For elementary reactions, there is an identity between molecularity and order of reaction. However, for complex reaction they are independent of each other.

Difference between Molecularity and Order of Reaction

• Molecularity is concerned with the number of molecules involved in the rate-determining step, whilst reaction order is the power dependence of rate on concentration.

• Molecularity is a theoretical quantity, whilst order is an experimentally determined quantity.

Representation of an Elementary Reaction

• Rate equations can use any measure equivalent to concentration, such as partial pressures for gas-phase reactions.

• The order with respect to the reacting component remains the same.

Example

• $k\newline 2A \rightarrow 2S$
  Represents a bimolecular irreversible reaction with rate $-r<em>A=r</em>S=kC_A^2$

Representation of a Nonelementary Reaction

• A nonelementary reaction is one whose stoichiometry does not match with its kinetics.

Example

• Stoichiometry: $H<em>2 + Br</em>2 \rightarrow 2HBr$

• $r<em>{HBr} = \frac{k</em>1 C<em>{H</em>2} C<em>{Br</em>2}^{1/2}}{k<em>2 + \frac{C</em>{HBr}}{C<em>{Br</em>2}}}$

Kinetic Models for Nonelementary Reactions

• Assume the overall reaction is the result of a series of the elementary reactions that involve the formation and subsequent reaction of the intermediate species.

Types of Intermediates

• Free Radicals

• Ionic Intermediates

• Molecules

• Transition Complexes

Reaction Schemes

• Non-Chain Reactions

• Chain Reactions

Pseudo-Steady-State Approximation

• The pseudo-steady-state approximation assumes that the net rate of formation of intermediates is zero.

Temperature-Dependent Term of a Rate Equation

• The rate of reaction depends upon the temperature.

Arrhenius' Law

• $k = k_0 e^{-E/RT}$

• Where:

  • $k_0$ is the pre-exponential factor.

  • $E$ is the activation energy.

  • $R$ is the gas constant.

  • $T$ is the absolute temperature.

• Taking logarithms of Arrhenius’ law, $ln(k) = ln(k_0) - \frac{E}{RT}$

Temperature Dependency from the Collision Theory

• The rate of the reaction $A+B$ products is given by $-r<em>A = Z</em>{AB}*e^{-E/RT}$

• $k = k_0 T^{1/2}*e^{-E/RT}$

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Temperature Dependency from Transition State Theory

• Rate = [Activated complex] x [Frequency of decomposition of activated complex]

• $k= \frac{k_bT}{h} * e^{\frac{\Delta S}{R}} e^{\frac{-\Delta H}{RT}}$

Comparison of Theories

• Reaction rate predicted from Transition State theory is closer to experimental data than from the collision theory

• Transition State theory is based on Statistical mechanics, whilst collision theory is based on kinetic theory of gasses.

• Arrhenius equation is a good approximation for the temperature dependency of the reaction rate constant compared to both the collision and transition state theories.