Reaction Orders and Half-Life

• Graphical approach can determine if a reaction is zeroth, first, or second-order.
• Plot the data to analyze the reaction order and find the rate constant, k.

Reaction Orders

• For the reaction NH4NCO(aq) → (NH2)2CO(aq), the data is plotted for zeroth, first, and second orders to determine reaction order
• [A] versus t plot isn't linear; the reaction isn't zeroth order
• ln[A] versus t plot isn't linear; the reaction isn't first order
• 1/[A] versus t plot is linear; the reaction is determined to be second order
• The second-order rate equation is y = 0.0502x + 10.164
• The slope equals the rate constant, k
• k = 0.0502 mol-1 L min-1

Half-Life

• The half-life (t1/2) is the time for the concentration of A to decrease to one-half of its initial amount: [A] = (1/2)[A]₀ at t = t1/2
• Expressions can be derived for half-lives of zeroth-, first-, and second-order reactions.

Zeroth-Order Reaction Half-Life

• t = t1/2, [A]₀ = [A]₀ - kt1/2
• t1/2 = [A]₀ / 2k
• The half-life is directly proportional to the starting concentration.
• As the concentration of A decreases, so does the half-life.

First-Order Reaction Half-Life

• t = t1/2, ln([A]₀/[A]) = kt1/2
• t1/2 = ln(2)/k
• The half-life is independent of the starting concentration.

Second-Order Reaction Half-Life

• t = t1/2, 1/[A] = 1/[A]₀ + kt1/2
• t1/2 = 1/(k[A]₀)
• The half-life is inversely proportional to the starting concentration.
• As the concentration of A decreases, the half-life increases.
• In all cases, half-life is inversely proportional to k, where a larger k results in a faster reaction and a shorter half-life.

Successive Half-Lives

• Successive half-lives are the lengths of time it takes for the concentration of A to be successively "cut in half"
• First half-life refers to the time it takes for [A] to decrease from [A]₀ to (1/2)[A]₀
• Second half-life refers to the time it takes for [A] to decrease from (1/2)[A]₀ to (1/4)[A]₀
• Third half-life refers to the time it takes for [A] to decrease from (1/4)[A]₀ to (1/8)[A]₀

Graphical Deduction of Half-Lives

• In a zeroth-order reaction, successive half-lives decrease by a factor of two each time
• In a first-order reaction, successive half-lives are equal and do not change
• In a second-order reaction, successive half-lives increase by a factor of two each time

Nuclear Decay

• Nuclear decay follows first-order kinetics
• The nuclear decay of carbon-14 is: ¹⁴C → ¹⁴N + e⁻
• The half-life of carbon-14 is 5730 years
• Living organisms maintain constant levels of ¹⁴C; the amount of ¹⁴C decreases slowly with time after death

Summary of Zero-Order Kinetics

• Differential rate law: Rate = k
• Integrated rate law: [A] = [A]₀ - kt
• Units of k: mol L⁻¹ s⁻¹
• Half-life: t₁/₂ = [A]₀ / 2k
• Successive half-lives: decrease by a factor of two each time
• Graphical: [A] versus t is linear with slope -k

Summary of First-Order Kinetics

• Differential rate law: Rate = k[A]
• Integrated rate law: [A] = [A]₀e^(-kt)
• Units of k: s⁻¹
• Half-life: t₁/₂ = ln(2) / k
• Successive half-lives: are equal/do not change
• Graphical: ln[A] versus t is linear with slope -k

Summary of Second-Order Kinetics

• Differential rate law: Rate = k[A]²
• Integrated rate law: 1/[A] = 1/[A]₀ + kt
• Units of k: mol⁻¹ L s⁻¹
• Half-life: t₁/₂ = 1 / (k[A]₀)
• Successive half-lives: increase by a factor of two each time
• Graphical: 1/[A] versus t is linear with slope +k

Reaction Mechanisms

• The reaction mechanism is a sequence of elementary processes that shows how reactants are converted into products
• An elementary process is a specific collisional event or molecular process

Elementary Steps and Intermediates

• Each elementary step has its own rate constant
• An intermediate appears in the mechanism but not in the overall reaction (e.g., O atom in ozone conversion)
• A reversible step is emphasized with ⇌, while an irreversible step is emphasized with →

Conversion of Ozone

• The mechanism shows that the conversion of O₃ into O₂ involves:
  • Dissociation of O₃ into O₂ + O
  • A reactive collision between O₃ and O
• The overall equation shows the net conversion of 2 molecules of O₃ to 3 molecules of O₂, not a direct collision between O₃ molecules

IUPAC Convention for Rate Constants

• Rate constants for each step are numbered sequentially (k₁, k₂, k₃, etc.)
• The rate constant for the reverse reaction of step "n" is denoted as k₋ₙ

Factors Affecting the Rate of an Elementary Process

• Collision theory is used to predict the rates of elementary processes
• Rate of an elementary process is ∝ ZAB × f × p

Collision Theory Terms

• ZAB = total collision rate ∝ [A][B]
• f = fraction of collisions with sufficient kinetic energy
• p = fraction of collisions with correct alignment
• Typically, ZAB is greater than 10³⁰ collisions per litre per second

Reaction Requirements

• A reaction results only if the reactants collide with sufficient energy and a favorable alignment
• Not all collisions lead to a reaction

Favorable Collision Example

• NO₂Cl + Cl → NO₂ + Cl₂ is a favorable collision forming NO₂ and Cl₂
• The short-lived, high-energy complex formed when kinetic energy is converted into potential energy is called the activated complex
• A collision between NO₂Cl and Cl that is much less favourable for forming NO₂ and Cl₂ occurs with incorrect alignment

Potential Energy Diagrams and Reaction Profiles

• Potential energy (PE) diagrams and reaction profiles illustrate the changes in potential energy during an elementary process
• The transition state is the state of maximum potential energy where the activated complex is present
• The activation energy (Ea) is the energy required to form the activated complex

Activation Energy and Reaction Rates

• Ea is the activation energy
• The activation energies for forward and reverse reactions are related by: ΔH = Efor - Erev
• The activated complex can either go towards products or revert back to reactants
• Reactant molecules must collide with sufficient kinetic energy (KE ≥ Efor) to form products
• The higher the activation energy for an elementary process, the slower the rate
• The greater the structural complexity of the reactants, the slower the rate

Rate Laws for Elementary Processes

• Collision theory states the rate law for an elementary process is determined by the number of reactant molecules involved
• For two reactants A and B: Rate ∝ ZAB × f × p

Kinetic Theory

• The collision frequency is proportional to [A] and [B]
• Given the constant "k", Rate = constant × [A][B] × e^(-Ea/RT) × p

Reaction Classifications

• A → products is a unimolecular process with Rate = k[A], following a first-order rate law
• A + A → products is a bimolecular process with Rate = k[A]²
• A + B → products is a bimolecular process with Rate = k[A][B]
• Bimolecular processes have second-order rate laws
• A + A + A → products is a termolecular process with Rate = k[A]³
• 2A + B → products is a termolecular process with Rate = k[A]²[B]
• A + B + C → products is a termolecular process with Rate = k[A][B][C]
• Termolecular processes have third-order rate laws
• The rules apply only when the process occurs in a single step

Multi-step Reactions

• Involve two or more elementary processes, the reaction profile includes two or more transition states, one for each step
• The rate-determining step corresponds to the elementary step with the transition state of highest energy
• Having highest activation energy does not necessarily make a step rate-determining

Mechanisms

• Step 1 has the highest activation energy but is not the slowest step
• Once B is formed, it will be converted back into A more rapidly than it will be converted into C
• As a result, the concentration of B is very small, and the rate of conversion of B into C is very slow
• The rate of production of C is ultimately determined by the rate of conversion of B into C
• In the given reaction profile, step 2 is the rate-determining step, even though step 1 has the smallest rate constant

Determining Rate Law

• Combine the rate laws for individual elementary processes to deduce the rate law for the overall reaction
• To ensure consistency with the observed rate law, derive the rate law for the mechanism
• Two important cases to consider; whether or not we know which step is the rate-determining step

Case One

• Case #1: Deriving the rate law when the rate-determining step is known
• If which step is the slowest is assumed, the slowest step determines how fast products will be formed
• Rate of the overall reaction ≈ Rate of the slowest step

Case Two

• Case #2: Deriving the rate law when the rate-determining step is not known
• Deriving the rate law if the energies of the various transition states in a multistep reaction are approximately equal
• The following strategy can be employed:
  • Pick any reactant or product appearing in the overall reaction
  • Identify steps in the mechanism that involve that particular reactant or product
  • Combine the rate laws for those steps algebraically to obtain an expression for the rate of consumption or the rate of production of that particular reactant or product
  • Simplify the resulting expression to obtain the rate law for the overall reaction (cannot include the concentration of any intermediate species)

Steady State Approximation

• The steady-state approximation assumes that the rate of production of an intermediate is approximately equal to the rate of its consumption

Justifying Steady State Approximation

• Step 1: NO₂Cl ⇌ NO₂ + Cl
• Step 2: NO₂Cl + Cl → NO₂ + Cl₂
• Overall: 2 NO₂Cl → 2 NO₂ + Cl₂
• Rate of producing Cl = k[NO₂Cl]
• Rate of consuming Cl = k₋₁[NO₂][Cl] + k₂[NO₂Cl][Cl] = {k₋₁[NO₂] + k₂[NO₂Cl]}[Cl]

Changing Rates

• The rate of producing Cl is large initially because [NO₂Cl] is large in the initial stages
• However, the rate of production decreases as NO₂Cl is consumed
• The rate of consuming Cl is zero initially because [Cl] = 0 at time t = 0
• Initially, Cl is being produced faster than it is being consumed, and as [Cl] increases, so too does the rate of consumption of Cl
• Eventually, the rate of production and the rate of consumption become equal, [Cl] remains constant until the later stages of the reaction
• In later stages, [NO₂Cl] becomes very small, and the rate of production of Cl decreases

Arrhenius Equation

• k = A * e^(-Ea/RT) where the fraction of molecules, f, that has KE ≥ Ea, is given by f≈ e^(-Ea/RT)

Temperature

• According to collision theory, the rate of an elementary reaction depends on these factors:
  • the collision frequency
  • the fraction of collisions with sufficient kinetic energy to lead to a reaction
  • the fraction of collisions that have a favorable alignment for a given reaction to occur
• Rate ∝ ZAB × f × p, where ZAB is the collision rate, f is the collision kinetic energy, and p is the fraction for correct alignment
• A relatively small decrease in activation energy can cause a significant increase in the value of the rate constant

Steric Factor

• p represents steric factor to show that the steric factor decreases as the the structural complexity the reactants increase

Catalysts

• Catalyst ↔ provides a new mechanism with a lower activation energy
• A catalyst lowers the energy requirements which allows a greater fraction of collisions and changes the mechanism
• The uncatalyzed has Ea = 75.3 kJ mol⁻¹, but the number of steps in the mechanism is not known, so its impossible to be completely specified completely
• Catalyzed reaction proceeds via a two-step mechanism