Rate Laws from Experiments

Introduction

A balanced equation identifies what is consumed and produced, but usually does not tell how fast the reaction proceeds at different concentrations. A rate law is an experimentally supported relationship such as rate = k[A]ᵐ[B]ⁿ. Its exponents describe observed concentration dependence under stated conditions and can reveal clues about mechanism.

Core explanation

For rate = k[A]ᵐ[B]ⁿ, m is order with respect to A, n order with respect to B and m+n the overall order when this power-law form applies. The rate constant k depends on temperature, catalyst and reaction environment. It is constant across concentration changes only while those conditions and the kinetic regime remain fixed. A change in temperature usually changes k.

The exponents are found from experiments, often by comparing initial rates while varying one concentration at a time. If doubling [A] doubles rate with [B] fixed, m=1. If doubling [B] quadruples rate with [A] fixed, n=2. The resulting empirical law is rate = k[A][B]². It need not match coefficients of the balanced overall equation, because the reaction may occur through several elementary steps.

An elementary step is different: its rate expression can often be inferred from its molecularity in a simple mass-action model. But an overall equation is usually a net result, not one collision of every reactant simultaneously. Deriving exponents from it without evidence is a common error. For example, a reaction written A + 2B → products may be zero order in A under catalyst-saturation conditions and first order in B; the equation alone cannot decide.

Zero order in a species means rate is independent of that species concentration over the measured range, not that the species is absent from the mechanism. It may be present in large excess or a surface may be saturated. Fractional and negative apparent orders can also occur in complex mechanisms, such as inhibition. A measured rate law is valid over the tested conditions, not necessarily for all concentrations.

Rates are positive for forward progress by convention, with units concentration/time. Dimensional analysis therefore gives k units depending on overall order. For rate = k[A][B]², concentration power is three, so k has units M⁻² s⁻¹ if rate is M s⁻¹. Units are a useful consistency check but do not replace experiment in determining exponents.

A rate law can test a proposed mechanism. If a candidate mechanism predicts rate proportional to [A]² but experiments show first order in A, that mechanism is incomplete or wrong under those conditions. Agreement is supportive, not final proof, because different mechanisms can produce the same simple rate expression. Further evidence from intermediates, isotope effects or temperature dependence may be needed.

Step-by-step reasoning

1. Write a tentative rate = k[A]ᵐ[B]ⁿ. 2. Use experiments that change one concentration while holding the other fixed. 3. Solve each exponent from observed rate ratios. 4. Determine overall order and k units. 5. State temperature and concentration range for which the law applies.

Visual explanation

Draw a small table with [A] doubled and [B] fixed, then [B] doubled and [A] fixed. Put rate multipliers 2 and 4 respectively. Arrows label m=1 and n=2, leading to rate = k[A][B]². Place the balanced equation in a separate box to prevent coefficient confusion.

Real-world analogy

A factory recipe lists how many parts go into a product, but its hourly output depends on the slow machines and staffing. Stoichiometry is the recipe; the empirical rate law records how production pace responds to supply changes.

Real-world example

In a catalytic reaction, increasing reactant concentration may initially speed production but eventually stop helping when all active surface sites are occupied. The observed order can change from positive to near zero across concentration ranges.

Why?

Why must rate-law exponents be measured for an overall reaction? The overall equation hides intermediate steps and the bottleneck. Its coefficients count net atoms, not necessarily the particles involved in the rate-controlling event.

Common misconception

“The coefficient 2 before B in the balanced equation means second order in B.” That inference is valid only for an identified elementary step under the appropriate simple model, not for an arbitrary overall reaction.

Worked example

Experiments show rate = 0.010 M s⁻¹ at [A]=0.10 M and [B]=0.10 M. Doubling A alone doubles rate; doubling B alone quadruples it. Thus rate = k[A][B]². Solve k = 0.010/(0.10×0.10²) = 10 M⁻² s⁻¹. The unit follows from M s⁻¹ divided by M³. The numerical k belongs to the stated temperature and medium.

Quick check

1. What is overall order for rate = k[A][B]²? Answer: Three: one in A plus two in B.

Exam focus

Use rate ratios to infer exponents, then calculate k with units. Do not copy stoichiometric coefficients into the rate law unless an elementary-step justification is explicitly given.

Advanced insight

An apparent rate law can emerge after eliminating intermediate concentrations from a mechanism. It may be a rational expression rather than a pure power law, and a simple order can be only a limiting approximation.

Summary

Rate laws are empirical concentration-rate relationships under stated conditions. Partial orders come from measurements, and k changes with temperature or environment. Overall stoichiometric coefficients do not generally dictate the exponents.

Practice questions

1. If rate is unchanged when [A] doubles, what is apparent order in A over that range? Answer: Zero, assuming other conditions are held fixed. 2. If rate = k[A]²[B], what is overall order? Answer: Three. 3. Why does matching a measured rate law not prove one mechanism uniquely? Answer: Different mechanisms or limiting approximations can produce the same concentration dependence.