Simple Rate Equations, Orders of Reaction and Rate Constants

· Rate equation shows how rate depends on reactant concentrations: rate = k[A]^m[B]^n.
· m and n are the orders of reaction with respect to A and B; in CIE they are usually 0, 1 or 2.
· Overall order = sum of all individual orders, e.g. rate = k[A]^1[B]^2 → overall order = 3.
· Rate constant, k = proportionality constant in the rate equation; its value changes with temperature.
· Orders must be found experimentally — they cannot be deduced from the balanced equation unless the reaction is a single elementary step.
· Rate-determining step (RDS) = slowest step in a multi-step mechanism; it controls the overall rate.

These diagrams show how reaction rate depends on concentration for different reaction orders. They help students link rate equations to the meaning of 0, 1st and 2nd order. Source

Meaning of Reaction Orders

· Zero order with respect to A: rate = k[A]^0 = k → changing [A] has no effect on rate.
· First order with respect to A: rate ∝ [A] → doubling [A] doubles rate.
· Second order with respect to A: rate ∝ [A]^2 → doubling [A] quadruples rate.
· If rate = k[A][B], the reaction is first order in A, first order in B, and second order overall.
· If a reactant appears in the chemical equation but not in the rate equation, it is zero order with respect to that reactant.
· Exam tip: compare experiments where only one concentration changes and all others are constant.

Initial Rates Method

· Use initial rate data from experiments with different starting concentrations.
· If [A] doubles and rate is unchanged → zero order in A.
· If [A] doubles and rate doubles → first order in A.
· If [A] doubles and rate quadruples → second order in A.
· If [A] triples and rate increases by ×9 → second order in A.
· After finding all orders, write the rate equation, then substitute values into it to calculate k.
· Always include units for k, worked out from: units of k = units of rate ÷ units of concentration terms.

This page supports the method for identifying reaction orders from experimental rate data. It is useful for practising CIE-style questions where students compare pairs of experiments. Source

Rate–Concentration and Concentration–Time Graphs

· Rate–concentration graph for zero order: horizontal line; rate is independent of concentration.
· Rate–concentration graph for first order: straight line through origin; gradient = k if rate = k[A].
· Rate–concentration graph for second order: curve through origin; rate increases faster as concentration increases.
· Concentration–time graph can be used to calculate rate from the gradient of a tangent.
· Initial rate = gradient of the tangent at t = 0 on a concentration–time graph.
· As reactant concentration decreases, the curve usually becomes less steep because the reaction slows down.
· For graphs where concentration decreases, rate is often taken as positive, so use the magnitude of the gradient.

These graphs compare the shapes expected for zero, first and second order reactions. They are helpful for recognising reaction order from graphical data in exams. Source

Half-Life Method

· Half-life, t₁/₂ = time taken for the concentration of a reactant to fall to half its original value.
· For a first-order reaction, half-life is constant and independent of concentration.
· If successive half-lives from a concentration–time graph are equal, the reaction is likely first order.
· For a first-order reaction: k = 0.693 / t₁/₂.
· Units of k for a first-order reaction are usually s⁻¹ or min⁻¹, depending on the time unit used.
· Exam tip: use the same units for t₁/₂ and k; if t₁/₂ is in seconds, k is in s⁻¹.

This graph shows that the time for concentration to halve is constant in a first-order reaction. It is useful for interpreting half-life directly from concentration–time data. Source

Calculating the Rate Constant, k

· From the rate equation: rearrange rate = k[A]^m[B]^n to give k = rate ÷ [A]^m[B]^n.
· Substitute initial rate and initial concentrations into the rate equation.
· Units of rate are usually mol dm⁻³ s⁻¹.
· Common units of k:
· Zero order: mol dm⁻³ s⁻¹.
· First order: s⁻¹.
· Second order: dm³ mol⁻¹ s⁻¹.
· Third order: dm⁶ mol⁻² s⁻¹.
· For first order only: k = 0.693 / t₁/₂.
· A larger k means a faster reaction at the same concentrations.

Mechanisms, Intermediates and the Rate-Determining Step

· A reaction mechanism is a sequence of elementary steps showing how reactants form products.
· The rate-determining step is the slowest step and controls the overall rate.
· Species in the rate equation usually appear in or before the rate-determining step.
· An intermediate is formed in one step and used up in a later step; it does not appear in the overall equation.
· A catalyst is used in one step and regenerated later; it does not appear in the overall equation.
· To check a mechanism, add all steps together and cancel species appearing on both sides.
· The mechanism must be consistent with both the overall equation and the experimental rate equation.
· If the slow step contains 1 molecule of A and 1 molecule of B, the predicted rate equation is usually rate = k[A][B].
· If a reactant is only involved after the slow step, it is often not present in the rate equation.

This resource links the experimental rate equation to a possible reaction mechanism. It is useful for learning how to identify the rate-determining step, intermediates and species affecting rate. Source

Effect of Temperature on k

· Increasing temperature usually increases k, so the rate increases at the same concentrations.
· Higher temperature means more particles have energy equal to or greater than activation energy.
· More successful collisions occur per second, so the reaction is faster.
· In rate equations, temperature affects the rate constant k, not the reaction orders.
· Reaction orders are determined experimentally and do not usually change unless the mechanism changes.