Kinetics Practice

Introduction

Kinetics problems often tempt a student to copy overall equation coefficients into a rate law. That shortcut is justified only for a truly elementary step, not for a net reaction whose path is unknown. A better approach uses controlled experiments, units and temperature dependence to infer a rate law, then treats a mechanism as a hypothesis tested against several observations. The exercises here require distinguishing measured rate behavior from a plausible microscopic story.

Core explanation

For a rate law r = k[A]^m[B]^n under a specified range, partial orders m and n describe how rate changes as each concentration changes with other conditions fixed. If doubling [A] doubles rate, m = 1 for that range; if doubling [B] multiplies rate by four, n = 2 . Overall order is m + n . Rate units for concentration per time are mol L⁻¹ s⁻¹. The units of k must make the equation dimensionally consistent: first-order k is s⁻¹, second-order k is L mol⁻¹ s⁻¹, and third-order k is L² mol⁻² s⁻¹ in these concentration units.

An Arrhenius relation k = A exp(−Eₐ/RT) describes a rate constant's temperature dependence over a suitable range. Taking logarithms gives ln k = ln A − ( Eₐ/R )(1/ T ). A plot of ln k against reciprocal absolute temperature has slope − Eₐ/R if the model fits. Temperatures must be in kelvin, and Eₐ and R must have consistent energy units. An apparent activation energy can summarize several elementary processes; it is not always one literal transition-state height.

A mechanism should reproduce the observed overall stoichiometry and rate behavior and be consistent with selectivity, isotope effects, intermediate evidence and other data. An observed zero order in one reactant could mean saturation of a catalyst site or another limiting process, not that the reactant is absent from chemistry. The rate-determining-step shorthand can be useful, but a multi-step catalytic cycle may not have one permanently slow step under all conditions. A catalyst speeds a pathway without changing the thermodynamic equilibrium constant of the same net reaction.

Step-by-step reasoning

1. Write the measured initial-rate table with one variable changed at a time. 2. Form rate ratios to solve for exponents without assuming net coefficients. 3. Derive k and its units from one experiment. 4. Check temperature data in kelvin and use Arrhenius only over its supported range. 5. Test any mechanism against all independent observations, not merely one rate law.

Visual explanation

Draw three experiments as rows: [A] doubles in the second, [B] doubles in the third. An arrow to measured rate ratios reveals exponents. Beside it, a straight-line Arrhenius plot shows slope − Eₐ/R . A separate reaction-coordinate sketch has a lower catalytic pathway but the same start and end states, reminding the solver not to infer equilibrium changes from rate changes.

Real-world analogy

A checkout line can be limited by the number of cashiers even if customers continue to arrive; this resembles saturation-like zero-order behavior over a range. The analogy cannot determine a chemical mechanism, because adsorption, diffusion and molecular events must be measured directly or inferred from chemical evidence.

Real-world example

An enzyme assay shows rate rising with substrate concentration at low substrate but approaching a plateau at high substrate. A single first-order law in substrate cannot describe the entire range. The plateau suggests a limiting catalytic capacity under that model, although product inhibition or transport limits could also contribute. Reporting “the reaction is zero order” without its substrate range and conditions would hide the change in behavior.

Why?

Why measure initial rates? Early in a reaction, reactant concentrations are close to their prepared values and product buildup is limited, so comparison across experiments is simpler. Later data can be valuable, but reverse reaction, catalyst deactivation and concentration changes may make a single simple ratio misleading.

Common misconception

“Overall coefficients are always rate orders.” Net mechanisms can have different exponents. “A large equilibrium constant means a large rate constant.” Thermodynamic position and kinetic speed differ. “The Arrhenius activation energy is always one exact barrier.” It may be an apparent parameter. “A matching rate law proves a unique mechanism.” Different mechanisms can fit limited data.

Worked example

Suppose three initial-rate experiments at one temperature give: experiment 1, [A] = 0.10 M, [B] = 0.10 M, rate = 2.0 × 10⁻³ M s⁻¹; experiment 2, [A] = 0.20 M, [B] = 0.10 M, rate = 4.0 × 10⁻³ M s⁻¹; experiment 3, [A] = 0.10 M, [B] = 0.20 M, rate = 8.0 × 10⁻³ M s⁻¹. Doubling A doubles rate, so first order in A. Doubling B quadruples rate, so second order in B. Thus r = k[A][B]² . From experiment 1, k = 0.0020/(0.10 × 0.10²) = 2.0 L² mol⁻² s⁻¹ . This empirical law alone does not tell how many elementary steps occur.

Quick check

1. If doubling [A] at fixed [B] leaves rate unchanged, what is the observed order in A? Answer: Zero over the tested range. 2. Does a catalyst necessarily change the same net reaction's equilibrium constant? Answer: No. It changes the approach rate and pathway, not the fixed-temperature equilibrium constant.

Exam focus

Use rate ratios from experiments that change one variable at a time. Derive k with dimensional units. Convert Celsius to kelvin for Arrhenius work. State the tested concentration and temperature range. Treat mechanisms as evidence-based proposals and explain what further observation could distinguish alternatives.

Advanced insight

An apparent order can change with concentration if catalyst sites saturate or inhibition occurs. Diffusion control can make a measured rate reflect transport rather than intrinsic chemistry. Arrhenius plots may curve if heat capacities, mechanism or catalyst state changes with temperature. Such deviations are data, not merely nuisances; they can reveal where a simple model no longer applies.

Summary

Rate laws are inferred from controlled kinetic data, activation energy describes temperature sensitivity under an Arrhenius model, and mechanisms require corroborating evidence. Units, range and experimental design decide how strong a kinetic conclusion is.

Practice questions

1. If rate quadruples when [A] doubles alone, what order in A is indicated? Answer: Second order in A over that range. 2. Give units of k for r = k[A]² with rate in mol L⁻¹ s⁻¹. Answer: L mol⁻¹ s⁻¹. 3. What is the slope of ln k versus 1/ T under the simple Arrhenius model? Answer: − Eₐ/R . 4. Why is an observed rate law insufficient to prove one mechanism? Answer: More than one sequence of steps can produce the same concentration dependence.