Rate law Equation, Derivation & Rearrangements

Core idea

Overview

The rate law mathematically relates the speed of a chemical reaction to the molar concentrations of its reactants. It utilizes a proportionality constant called the rate constant, k, and reactant orders, m and n, which indicate how sensitive the rate is to changes in each substance's concentration.

When to use: Apply this equation when you need to calculate the instantaneous speed of a reaction or determine the reaction order from experimental kinetic data. It is valid under conditions where temperature is held constant, as the rate constant k is temperature-dependent.

Why it matters: This formula is fundamental for designing safe chemical reactors and predicting the shelf-life of pharmaceuticals. By identifying the reaction order, chemists can deduce the molecular mechanism and sequence of steps occurring at the atomic level.

Symbols

Variables

k = Rate Constant, [A] = Concentration of A, [B] = Concentration of B, m = Order wrt A, n = Order wrt B

k

Rate Constant

units

[A]

Concentration of A

m o l / d m^{3}

[B]

Concentration of B

m o l / d m^{3}

m

Order wrt A

Variable

n

Order wrt B

Variable

rate

Rate

m o l / d m^{3} / s

Walkthrough

Derivation

Formula: Rate Law

Links reaction rate to reactant concentrations (or partial pressures) using experimentally determined orders and a temperature-dependent rate constant.

Reaction orders are determined experimentally (e.g., initial rates), not from overall stoichiometry.
Temperature is constant while measuring the rate constant k.

1

State the General Form:

Rate depends on reactant concentrations raised to their orders m and n; overall order is m+n.

Rate = k [A]^{m} [B]^{n}

2

Interpret the Rate Constant:

k is a constant for a given reaction at a given temperature (it changes with temperature).

k

Result

k

Source: AQA A-Level Chemistry — Kinetics

Free formulas

Rearrangements

Solve for rate

Make rate the subject

r a t e = k [A]^{m} [B]^{n}

rate is already the subject of the formula.

Difficulty: 1/5

Solve for $k$

Make k the subject

k = \frac{r a t e}{[ A ] ^{m} [ B ] ^{n}}

To make the rate constant (k) the subject of the Rate law equation, divide both sides by the concentration terms [A]^m[B]^n.

Difficulty: 2/5

Solve for [A]

Make [A] the subject

[A] = (\frac{r a t e}{k [ B ] ^{n}})^{\frac{1}{m}}

Start from the Rate law, rate = k[A]^m[B]^n. To make [A] the subject, first divide both sides by k[B]^n, then raise both sides to the power of 1/m.

Difficulty: 2/5

Solve for [B]

Make [B] the subject

[B] = (\frac{r a t e}{k [ A ] ^{m}})^{\frac{1}{n}}

To make [B] the subject of the rate law equation, first isolate [B]^n by dividing by k[A]^m, then raise both sides to the power of $\frac{1}{n}$ .

Difficulty: 2/5

Solve for $m$

Make m the subject

m = \frac{ln ( \frac{r a t e}{k [ B ] ^{n}} )}{ln [ A ]}

Start from the Rate law equation. Isolate the term containing the exponent $m$ by dividing both sides by $k [B]^{n}$ . Take the natural logarithm of both sides to move $m$ from the exponent to a multiplier.

Difficulty: 2/5

Solve for $n$

Make n the subject

n = \frac{ln ( \frac{r a t e}{k [ A ] ^{m}} )}{ln [ B ]}

Start from Rate law. To make n the subject, isolate the power term, take natural logs, then divide by ln[B].

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph follows a power law curve where the rate increases alongside the concentration of A, curving upward if the exponent m is greater than one or downward if it is between zero and one. For a chemistry student, this shape illustrates that at low concentrations the reaction proceeds slowly, while higher concentrations significantly accelerate the rate of product formation depending on the reaction order. The most important feature of this curve is the steepness of the slope, which reveals how sensitive the overall reaction rate is to changes in the amount of reactant A present in the system.

Graph type: power_law

Why it behaves this way

Intuition

The rate law describes the reaction rate as a statistical outcome of molecular collisions, where the frequency of effective collisions is proportional to reactant concentrations, with their individual influences weighted

rate

The instantaneous speed at which reactants are converted into products or vice versa.

How fast the reaction is progressing at a specific moment; a higher rate means the reaction finishes more quickly.

k

The rate constant, a proportionality factor unique to a given reaction at a specific temperature, reflecting its intrinsic speed.

A higher 'k' means the reaction is inherently faster, even with the same reactant concentrations, due to factors like lower activation energy or more frequent effective collisions.

[A]

Molar concentration of reactant A, representing the amount of substance per unit volume.

The more concentrated a reactant, the more likely its molecules are to collide and react, generally leading to a faster rate.

[B]

Molar concentration of reactant B, representing the amount of substance per unit volume.

The more concentrated a reactant, the more likely its molecules are to collide and react, generally leading to a faster rate.

m

The reaction order with respect to reactant A, an experimentally determined exponent indicating how the rate depends on that reactant's concentration.

If m=1, doubling [A] doubles the rate. If m=2, doubling [A] quadruples the rate. If m=0, changing [A] has no effect on the rate.

n

The reaction order with respect to reactant B, an experimentally determined exponent indicating how the rate depends on that reactant's concentration.

If n=1, doubling [B] doubles the rate. If n=2, doubling [B] quadruples the rate. If n=0, changing [B] has no effect on the rate.

Signs and relationships

^m: The exponent 'm' (reaction order) quantifies the non-linear sensitivity of the reaction rate to changes in the concentration of reactant A, empirically determined and reflecting the molecularity of the rate-determining
^n: The exponent 'n' (reaction order) quantifies the non-linear sensitivity of the reaction rate to changes in the concentration of reactant B, empirically determined and reflecting the molecularity of the rate-determining

Free study cues

Insight

Canonical usage

The reaction rate is typically expressed in molarity per second (mol L-1 s-1), with reactant concentrations in molarity (mol L-1), and the rate constant 'k' having units that ensure dimensional consistency based on the

Common confusion

A common mistake is assuming a fixed unit for the rate constant 'k'. Its units are variable and must be derived from the overall reaction order to ensure the rate law equation is dimensionally consistent.

Unit systems

ratemol L-1 s-1 · Can also be mol dm-3 s-1 or mol m-3 s-1. The unit of time (e.g., min, h) should be consistent with the rate constant's time unit.

$k$ Varies by overall reaction order · For a 0th order reaction, k is mol L-1 s-1. For a 1st order, k is s-1. For a 2nd order, k is L mol-1 s-1. For a 3rd order, k is L2 mol-2 s-1.

[A]mol L-1 · Molar concentration of reactant A. Often denoted as M.

[B]mol L-1 · Molar concentration of reactant B. Often denoted as M.

$m$ dimensionless · Reaction order with respect to reactant A. Determined experimentally, not from stoichiometry.

$n$ dimensionless · Reaction order with respect to reactant B. Determined experimentally, not from stoichiometry.

One free problem

Practice Problem

A reaction has the rate law: rate = k[A][B]^2. The rate constant k = 0.015 $d m^{6}$ mol^-2 s^-1. If [A] = 0.3 mol/ $d m^{3}$ and [B] = 0.2 mol/ $d m^{3}$ , calculate the reaction rate.

Rate Constant0.015 units

Concentration of A0.3 mol/dm^3

Concentration of B0.2 mol/dm^3

Order wrt A1

Order wrt B2

Solve for: rate

Hint: rate = k[A]^m[B]^n. Square [B] first, then multiply all terms.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

When predicting how doubling reactant concentration affects rate, Rate law is used to calculate Rate from Rate Constant, Concentration of A, and Concentration of B. The result matters because it helps connect measured amounts to reaction yield, concentration, energy change, rate, or equilibrium.

Study smarter

Tips

The exponents m and n must be determined experimentally; they are not necessarily the coefficients from the balanced equation.
The units of k change depending on the overall order (m + n) to ensure the rate is always in M/s.
Reactants with an order of zero do not affect the rate, regardless of how much their concentration changes.

Avoid these traps

Common Mistakes

Using stoichiometric coefficients as orders.
Forgetting units of k depend on order.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Links reaction rate to reactant concentrations (or partial pressures) using experimentally determined orders and a temperature-dependent rate constant.

Apply this equation when you need to calculate the instantaneous speed of a reaction or determine the reaction order from experimental kinetic data. It is valid under conditions where temperature is held constant, as the rate constant k is temperature-dependent.

This formula is fundamental for designing safe chemical reactors and predicting the shelf-life of pharmaceuticals. By identifying the reaction order, chemists can deduce the molecular mechanism and sequence of steps occurring at the atomic level.

Using stoichiometric coefficients as orders. Forgetting units of k depend on order.

When predicting how doubling reactant concentration affects rate, Rate law is used to calculate Rate from Rate Constant, Concentration of A, and Concentration of B. The result matters because it helps connect measured amounts to reaction yield, concentration, energy change, rate, or equilibrium.

The exponents m and n must be determined experimentally; they are not necessarily the coefficients from the balanced equation. The units of k change depending on the overall order (m + n) to ensure the rate is always in M/s. Reactants with an order of zero do not affect the rate, regardless of how much their concentration changes.

References

Sources

Atkins' Physical Chemistry
Wikipedia: Rate law
Atkins' Physical Chemistry, 11th Edition
IUPAC Gold Book (Reaction rate, Rate constant, Order of reaction)
Bird, Stewart, Lightfoot - Transport Phenomena, 2nd Edition
Atkins' Physical Chemistry, 11th Edition, Peter Atkins, Julio de Paula, James Keeler
IUPAC Gold Book (Compendium of Chemical Terminology)
Wikipedia: Rate equation

Rate law

Overview

Variables

Derivation

State the General Form:

Interpret the Rate Constant:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Arrhenius equation

Frequently Asked Questions

Sources