Rates of Reaction: Orders, Initial-Rate Method, and Mechanisms

11A.3 DETERMINING ORDERS OF REACTION

• Learning objective: determine how rate depends on each reactant to obtain the rate equation.
• General rate form (rate law):
  $\text{rate} = k [A]^m [B]^n \dots$
  where m, n are the orders with respect to each reactant.
• Key orders (per reactant): zero, first, or second order.
  • Zero order in a reactant: $\text{rate} = k$
  • First order in a reactant: $\text{rate} = k [X]$
  • Second order in a reactant: $\text{rate} = k [X]^2$
• Initial-rate method: determine rate from the very start of the reaction and compare how rates change when concentrations are varied.
  • The initial rate is the gradient (slope) of the concentration–time graph at t = 0 (instantaneous rate at t = 0).
  • If doubling a reactant while holding others constant doubles the initial rate, the reaction is first order in that reactant.
  • If doubling a reactant quadruples the initial rate, the reaction is second order in that reactant.
• Example deduction (from the transcript):
  • Experiments show that when [A] doubles, the rate doubles; hence first order w.r.t A.
  • When [B] doubles and the rate doubles; hence first order w.r.t B.
  • Therefore the rate law is: $\text{rate} = k [A][B]$ and the overall order is 2.
• DETERMINING ORDERS FROM A RATE-CONCENTRATION GRAPH
  • Initial-rate plots give approximate initial rates; many reactions have integer orders.
  • If a plot of rate against concentration of a reactant is not linear, test a plot against the square of the concentration.
  • If plotting rate versus [A]^2 gives a straight line through the origin, the reaction is second order with respect to A.
  • If the rate is independent of [A], the reaction is zero order in A (flat line).
  • Example: a plot of rate against [A]^2 that is linear through the origin indicates second order in A.
• CHECKPOINT (concepts tested)
  • Define half-life of a reaction.
  • Use a graph to show the order of reaction (e.g., first order shows a constant half-life regardless of initial concentration).
  • Effect of doubling initial concentration on the half-life (typical result: for first order, t1/2 is independent of initial [A]).
  • Calculate the rate constant using $k = \frac{0.693}{t_{1/2}}$ for a first-order process, with units depending on the overall order.
  • Write the corresponding rate equation for the reaction.
  • Determine reactant concentrations or rate at a given time from graphs and rate laws.
  • Explain how to obtain the rate at a given time directly from the graph (e.g., by drawing a tangent).

11A.4 RATE EQUATIONS AND MECHANISMS

• Core idea: a rate equation is an experimentally determined expression; the mechanism must be consistent with it.

• Elementary vs non-elementary reactions:

  • Elementary reaction: a single collision between reactant particles; rate law can be deduced directly from the stoichiometry.
  • Example: NO(g) + O(g) → NO2(g) (elementary) => $\text{rate} = k [\mathrm{NO}][\mathrm{O}]$
  • Non-elementary reaction: the rate equation cannot be deduced simply from the overall stoichiometry; it proceeds via a mechanism of steps.
  • Example: a reaction with more complex steps may have a rate law not matching the overall equation.

• Mechanisms and rate-determining step (RDS):

  • The overall rate is governed by the slowest step (the rate-determining step) in the mechanism.
  • The order of the rate equation equals the number of reacting particles in the rate-determining step.
  • The steps preceding and following the RDS involve intermediates, which are not present in the overall equation.

• Examples of common mechanisms:

  • SN2 (Substitution Nucleophilic Bimolecular): bimolecular RDS, rate = $k [\text{nucleophile}][\text{substrate}]$, single-step overall process.
  • SN1 (Substitution Nucleophilic Unimolecular): RDS is unimolecular (formation of carbocation), rate = $k [\text{substrate}]$, subsequent fast attack by nucleophile.

• Hydrolysis of halogenoalkanes (alkaline conditions):

  • For primary halogenoalkanes, the rate often follows SN2 characteristics: rate = $k [\text{R-CH}_2\text{Cl}][\text{OH}^-]$; with excess OH⁻, the reaction can appear pseudo-first-order in the substrate.
  • For tertiary halogenoalkanes, the rate-determining step is ionisation to form a carbocation (SN1): rate = $k [\text{R-CR'R''-C}]$ (i.e., first order in the halogenoalkane, zero order in OH⁻).

• Iodination of propanone (practical skills example): acid-catalysed iodination in aqueous solution

  • Observed order: zero order in I₂; first order in propanone and in H⁺.
  • Rate law: $\text{rate} = k [\text{CH}<em>3\text{COCH}</em>3][\mathrm{H}^+]$
  • Proposed mechanism (enol mechanism): rate-determining step involves formation of the enol/enol-like intermediate; iodination occurs in a fast subsequent step.
  • How to test a mechanism:
  • Use a wider range of concentrations to see if the rate law changes.
  • Use instrumental analysis (e.g., NMR) to detect proposed intermediates.
  • Use deuterated substrates (e.g., CD₃COCD₃) to observe isotopic effects on the rate-determining step.

• Hydrolysis of halogenoalkanes – examples to understand mechanisms:

  • 2-chloromethyl (tertiary) hydrolysis (alkaline): rate = first order in substrate, zero order in OH⁻; slow ionisation (SN1) followed by fast capture by OH⁻.
  • Primary halogenoalkane hydrolysis: rate often shows SN2 behavior with rate = $k [\text{R-CH}_2\text{Cl}][\text{OH}^-]$ (pseudo-first-order in substrate when [OH⁻] is large).

• Catalysis concepts:

  • Catalysts provide alternative reaction pathways with lower activation energy; can be homogeneous (in the same phase) or heterogeneous (different phase).
  • An example route for Br₂ formation involves sequential steps with HBr/HBrO₂ and intermediate species; catalysts may route the reaction via different steps.
  • The rate law for a catalyzed reaction is determined by the slow step of the catalytic cycle (which may differ from the uncatalysed pathway).

• Activation energy (conceptual):

  • Activation energy is the energy barrier that reactants must overcome to reach the transition state of the rate-determining step.
  • Temperature dependence and catalysis influence the rate by altering this barrier.

• Practical takeaway: use rate laws and mechanistic reasoning together to deduce which step is rate-determining and what species participate in that step; use experiments to test proposed mechanisms.

• Note: The transcript includes additional examples and details (e.g., specific experimental setups, checks, and diagrams) that reinforce the same principles above. The essential ideas are: determine orders from rate data, connect rate laws to mechanisms via the rate-determining step, and distinguish elementary from non-elementary processes through experimental tests and mechanism proposals.