Chemical Kinetics: Reaction Rates and Mechanisms

• Chemical kinetics is the study of reaction rates, including explosions, medications, rusting, and erosion.
• Reaction rate refers to the speed at which chemical reactions occur.
• This chapter will cover factors affecting reaction rate, rate laws, activation energy, mechanisms, and catalysis.

Reaction Rates

• The reaction rate is the change in concentration of reactants or products per unit of time, measured in molarity per time (M/time).
• Four factors influence reaction rate: the physical state of reactants, reactant concentrations, reaction temperature, and the presence of a catalyst.

Physical State of the Reactants

• The physical state of reactants influences collision frequency and phase.
• Reactions proceed more rapidly when reactants collide more readily.
• Homogeneous reactions, where all reactants are gases or liquids, tend to be faster.
• Heterogeneous reactions, which involve solids, are generally slower.

Reactant Concentrations

• Increased reactant concentration typically leads to faster reactions.
• Higher concentrations result in more molecules and, consequently, more collisions.
• An increased surface area enhances the reaction rate; fine powders or thin fibers react faster than pellets or tablets.

Reaction Temperature

• Generally, reaction rates increase with higher temperatures.
• Temperature is related to the kinetic energy of molecules.
• At elevated temperatures, molecules move faster, increasing both the number of collisions and the energy of these collisions.
• More molecules can overcome the activation energy barrier at higher temperatures.

Presence of a Catalyst

• Catalysts accelerate reactions without being consumed in the process.
• Catalysts do not appear in the overall balanced equation.
• They alter the types of collisions and often change the reaction mechanism
• Catalysts are crucial in many biological (enzymes) and industrial reactions.

Reaction Rate

• It is defined as the change in concentration over a time period represented as Δ[ ]/Δt.
• Δ represents "change in."
• indicates molar concentration.
• t represents time.
• Reaction rates are conventionally expressed as positive values.
• Understanding average and instantaneous rates helps to analyze a reaction.

Average Rate

• It can measure the increase in product concentration over time.
• It can also measure the decrease in reactant concentration over time.
• A visual representation can be used to illustrate reaction progress over time with changes in color to represent the reactants turning into products.

Following a Reaction Rate

• In the example: C₄H₉Cl(aq) + H₂O(l) → C₄H₉OH(aq) + HCl(aq), [C₄H₉Cl] is measured to determine the average rate using the formula: Rate = -Δ[C₄H₉Cl]/Δt.

Instantaneous Rate

• Rate data for [C₄H₉Cl] can be plotted to determine initial rate.
• Initial rate is the rate at time zero.
• Measuring concentration at time zero is difficult so a slope from a later time (t) can be used.
• Instantaneous rate is the slope of the curve at one point in time.
• Reactions typically slow down over time.

Reaction Rates and Stoichiometry

• Rates can be measured using either reactants or products.
• Rates may be the same for each reactant and product for 1:1 reactions.
• For the disappearance of C₄H₉Cl and the appearance of C₄H₉OH, the magnitudes are equal but have opposite signs (one decreasing, one increasing).
• Rate = -Δ[C₄H₉Cl]/Δt = Δ[C₄H₉OH]/Δt

Average Rates Depend on Stoichiometric Coefficients

• If the reaction is not 1:1 the rate must be normalized.
• If a stoichiometric coefficient is 2, a factor of 1/2 should be added when setting up the rate equality.
• For 2 O₃(g) → 3 O₂(g), Rate = -1/2(Δ[O₃]/Δt) = 1/3(Δ[O₂]/Δt)

Rate Laws and Rate Constants: The Method of Initial Rates

• Rate laws are mathematical expressions determined experimentally for each reaction.
• For A → B, Rate = k[A]^x[B]^y, here k = specific rate constant, and x, y refer to reactant order.
• If Rate = k[NH₄⁺][NO₂⁻], the order with respect to each reactant is 1, thus first order in NH₄⁺ and NO₂⁻.
• So the reaction order is second order (1 + 1 = 2), you add up all of the reactants orders to get the overall reaction order.

Order ≠ Stoichiometry

• Reaction order is determined experimentally and is not related to the balanced equation.
• Examples include:
  • 2 N₂O₅(g) → 4 NO₂(g) + O₂(g), Rate = k[N₂O₅] Eq. [14.9]
  • H₂(g) + I₂(g) → 2 HI(g), Rate = k[H₂][I₂] Eq. [14.10]
  • CHCl₃(g) + Cl₂(g) → CCl₄(g) + HCl(g), Rate = k[CHCl₃][Cl₂]^½ Eq. [14.11]

Magnitudes and Units of k

• Reactions with k ~ 10^9 or higher are fast.
• Reactions with k less than 10 are slow.
• M/s is the Rate, units of k: the rate is zero order - Ms⁻¹, first order - s⁻¹, second order - M⁻¹s⁻¹

Method of Initial Rates

• To determine the rate law, choose two experiments where one reactant is constant.
• Make the ratio of rates versus reactant concentrations: conc ratio^exponent = rate ratio
• Placing the larger rate on top is useful.
• If NO₂⁻ and NH₄⁺ are reactants rate = k [NO₂⁻]^x[NH₄⁺]^y and exponents need to be determined.

An Example of How Concentration Affects Rate

• Experiments 1–3 illustrate how [NH₄⁺] affects rate.
• Experiments 4–6 show how [NO₂⁻] affects rate.
• To find y: In experiments 1-3, the NO₂⁻ concentration is constant. Any change in rate is due to [NH₄⁺]: Experiment 2 (0.0200)^y / Experiment 1 / 0.0100 = 10.8 x 10⁻⁷ / 5.4 x 10⁻⁷
  • 2^y = 2, so y = 1 and the order with respect to NH₄⁺ is 1. As [NH₄⁺] doubles, the rate doubles.
• To find x: In experiments 4–6, the NH₄⁺ concentration is constant. Any change in rate is due to [NO₂⁻]: Experiment 5 / Experiment 4 =(0.0400)^x / 0.0202 = 21.6 x 10⁻⁷ / 10.8 x 10⁻⁷
  • 2^x = 2, and x = 1 so [NO₂⁻] is 1
• The rate law is shown with the relationship between rate and concentration for all reactants: Rate = k[NH₄⁺][NO₂⁻].
• To find k: k = Rate/[NH₄⁺][NO₂⁻] = 5.4 x 10⁻⁷ M/s / (0.0100 M)(0.200 M) = 2.7 x 10⁻⁴ M⁻¹s⁻¹

Integrated Rate Laws

• The initial rates can often lead to errors so using a graphical treatment is preferred.
• Rate Laws show the relationship between rate (M/time), rate constant (k), and [M] of reactants.
• Integrated Rate Laws show the relationship between time and concentration (M).

First-Order Reactions

• Conversion of methyl isonitrile to acetonitrile: CH₃NC → CH₃CN
• If the rate law is: Rate = k[A]
• The average rate is: rate = -Δ[A]/Δt, then k[A] = -Δ[A]/Δt

More First-Order Reactions

• Rearrange to: Δ[A]/[A] = -kΔt Integrate: ln([A]/[A]₀) = -kt
• To rearrange: ln[A] = -kt + ln[A]₀
• The previous equations follow the equation of a line: y = mx + b
• The plot of ln[A] versus t is linear.

Evidence That Conversion of Methyl Isonitrile to Acetonitrile Is First-Order

• The plot of ln[CH₃CN] versus time is linear and follows the Rate = k[CH₃NC].

Finding the Rate Constant, k

• Find rate constant by using the plot of ln[A] versus t and use: ln[A] = -kt + ln[A]₀.
• The slope of a line is -k.

Second-Order Reactions

• It is when rates depend on a reactant to the second power, so Rate = k[A]², where rate = -Δ[A]/Δt and k[A]² = -Δ[A]/Δt
• When rearranged Δ[A]/[A]² = -kΔt
• Use calculus: 1/[A] = 1/[A]₀ + kt and y = b + mx

An Example of a Second-Order Reaction: Decomposition of NO₂

• In the reaction NO₂ → NO+½O₂, a plot showing NO₂ decomposition reveals that it must be second order because it is linear.
• In [NO₂] is not linear, but 1/[NO₂] is linear.

Zero-Order Reactions

• This is when rate is independent of the concentration of the reactant: Rate = k
• These reactions are linear in concentration.
• Equation: [A]t = -kt + [A]₀, where y = mx + b

Half-Life for First-Order Reaction

• Half-life is the amount of time a reactant takes to be used up in a chemical reaction.
• First-Order Reaction equations include:
  • ln [A] = - kt + ln [A]₀
  • -ln ([A]₀/2) = -kt½ + ln[A]₀
  • -ln ([A]₀/2) +ln[A]₀ = kt½

More Half-Life for First-Order Reaction

 - ln ([A]₀/[A]₀/2) = kt½
 - ln2 = kt½ or t½ = 0.693/k

Half-Life and Second-Order Reactions

Using the integrated rate law, half-life is shown: - 1/[A] = 1/[A]₀ + kt - 1/([A]₀/2) = 1/[A]₀ + kt½ - 2/[A]₀ = 1/[A]₀ + kt½

More Half-Life and Second-Order Reactions

• Solve: t½ = 1/ (k[A]₀)
• For second-order reactions, half-life is a concentration-dependent quantity.

Temperature and Rate: Activation Energy and the Arrhenius Equation

• Rates of reaction increase as temperature increases.
• Look at a rate law: Rate = k[A]^x, so the reaction must change with temperature.
• More factors: collision frequency, orientation of molecules, and energy needed for the reaction.

Temperature and Rate

• Generally, as temperature increases, rate increases.
• The rate constant is temperature dependent: It increases as temperature increases.
• The rate constant doubles with every 10°C rise.

Collision Model

• Based on the kinetic molecular theory.
• Molecules must collide to react
  • The more collisions, the more reactions occur
  • If there are more molecules, the reaction rate is faster
• Chemical reactions happen when bonds are broken and formed.
• Molecules can only react if they collide and align properly

Orientation Factor

• There are shortcomings to describe why reactions do not occur such as not explaining the minimum energy requirement.
• There is no direct method of calculating the accurate orientation.

Activation Energy Model

• Activation energy is the minimum energy needed for a reaction to take place.
• An energy barrier must be overcome for a reaction to take place, i.e. reactant to product.

Transition State (Activated Complex)

• Reactants gain energy as the reaction proceeds until particles reach the maximum energy state.
• Organization of atoms at the highest energy state is called the transition state, or activated complex.
• Activation energy is the energy needed to form the state (Eₐ).
• Energy difference is between the reactants and the highest energy along the reaction pathway.

Reaction Progress

• Plots show the energy possessed in the particles as the reaction.
• At the highest energy state, the transition state is formed.
• Reactions can proceed to a product or return to a reactant.
• The rate constant (k) depends on the magnitude of Eₐ.

Effect of Temperature on the Distribution of the Energy of Molecules

• Molecules have an average temperature, but each individual molecule has its own energy.
• At higher energies, more molecules possess the energy needed for the reaction to occur.

Determining Activation Energy

• Arrhenius found a relationship: k = Ae^(Eₐ/RT)
• This can be reorganized: ln k = -Eₐ/RT + ln A
• Plotting ln k vs 1/T gives Eₐ.
• At new temperature, use: ln (k₁/k₂) = Eₐ/R x (1/T₂ - 1/T₁)

More Determining Activation Energy

• Eₐ can also be determined by plotting reaction data at various temperatures.

Reaction Mechanisms

• Describes how reactions happen.
  • Occurs by a single step known as an elementary reaction
  • Can also occur through several discrete steps
• The molecularity of an elementary reaction tells how many molecules involved in the mechanism.
  • Unimolecular: involves a single molecule
  • Bimolecular: two molecules collide
  • Termolecular: three molecule collide

Molecularity

• Table 14.3 shows the relationship between elementary reactions and rate law.

Termolecular Possibility

• Termolecular steps need three molecules that simultaneously collide using the correct orientation.
• These steps are rare if they even occur.
• They are slower than unimolecular to bimolecular steps
• Basically all mechanisms use unimolecular or bimolecular reactions.

What Limits the Rate?

• The overall reaction cannot occur faster than the slowest reaction in the mechanism.
• What determines rate is we call the rate-determining step.

Requirements of a Plausible Mechanism

• The rate law must be derived from a rate-determining step.
• The rate-determining step is the slow step.
• Each step is balanced.
• Intermediates are all made and used up.
• The stoichiometry is obtained when all steps are added up.
• Any catalyst is used and regenerated.

A Mechanism with a Slow Initial Step

• Overall reaction: NO₂ + CO → NO + CO₂
• Rate law: Rate = k[NO₂]²
  • Propose a mechanism is used as a first step
  • Rate determination as the slow step

More of A Mechanism with a Slow Initial Step

• NO₂ + NO₂ → NO + NO₃ (slow)
• NO₃ + CO → NO₂ + CO₂ (fast)
• NO₃ is called an intermediate: made and used in the reaction
• If first step is the slowest step, then it gives the rate law: Rate = k[NO₂]²
  • If you add steps you obtain a balanced chemical equation
  • NO₂ + NO₂ → NO + NO₃
  • NO₃ + CO → NO₂ + CO₂
  • NO₂ + CO → NO + CO₂
• Mechanism is plausible

Intermediates

• An intermediate that is a reactant or product, is not the transition state.
• Intermediates are stable
• They occur along the reaction pathway.
• It is possible that they are isolated or identified.

A Mechanism with a Fast Initial Step

• For the termolecular step: 2 NO + Br₂ → 2 NOBr
• Rate law for this reaction: Rate = k[NO]²[Br₂]
  • NO + Br₂ → NOBr₂ (fast)
  • NOBr₂ + NO → 2 NOBr (slow)

More of A Mechanism with a Fast Initial Step

• NOBr₂ has two reactions
  • NO to form NOBr
  • Decomposition to reform NO and Br₂
• Reactants and products are at an equilibrium.
• Rate subscript f = rate subscript r

Even More of A Mechanism with a Fast Initial Step

• k₁[NO][Br₂] = k-₁[NOBr₂]
• Solve for [NOBr₂] and substitute into law,
  • Rate equals k₂ (k₁/k-₁) [NO][Br₂][NO]
• This gives the observed rate law and possible mechanism
  • Rate = k-[NO]²[Br₂]
• Equilibrium is must be used when the first step is not the slowest.

Catalysis

• A substance that changes rate of a chemical reaction without any permanent change itself.
• The catalyzed reaction is faster.

Homogeneous Catalysts

• When reactants and catalyst are in the same phase
• When reactants and catalyst are dissolved in the same solvent

Heterogeneous Catalysts

• Catalyst and reactants are in different phases.
• When gases are passed over a solid catalyst.
• The adsorption of reactants is often the rate-determining step.

Enzymes

• Biological catalysts
• Takes place at specific locations; active sites.
• Substances are known as substrates.

Lock-and-Key Model

• A simple explanation for an enzyme.
• Current mode recognize that an active site flexes and that MA changes shape when forming on enzyme-substrate complex.