Rate Of A Chemical Reaction

Q: What is the rate constant k?

The rate constant k is the proportionality constant in the rate law rate = k[A]^m[B]^n. Numerically, it equals the rate when every reactant concentration is 1 mol/L, so it is also called the specific reaction rate. It depends on temperature and the presence of a catalyst, but not on concentration.

Q: What is the rate-determining step?

The rate-determining step is the slowest elementary step in a multi-step reaction mechanism. The overall rate cannot exceed the rate of this step, so the observed rate law is derived from the molecularity of the slowest step alone.

ChemistryChemical KineticsFor JEE aspirants

The rate of a chemical reaction is the change in concentration of a reactant or product per unit time. For a general reaction $a A + b B \to c C + d D$ , the rate is expressed as $rate = - \frac{1}{a} \frac{d [ A ]}{d t} = - \frac{1}{b} \frac{d [ B ]}{d t} = \frac{1}{c} \frac{d [ C ]}{d t} = \frac{1}{d} \frac{d [ D ]}{d t}$ . The rate depends on the concentration of reactants through the rate law $rate = k [A]^{m} [B]^{n}$ , where $k$ is the rate constant and $m + n$ is the overall order. Two related but distinct ideas govern this chapter: molecularity (a theoretical count from a single-step mechanism) and order (an experimentally determined power of concentration in the rate law).

Key Formulas - Quick Reference

Average rate: $r_{a v g} = - \frac{Δ [ Reactant ]}{Δ t} = + \frac{Δ [ Product ]}{Δ t}$
Instantaneous rate: $r = - \frac{d [ Reactant ]}{d t} = + \frac{d [ Product ]}{d t}$
General reaction $a A + b B \to c C + d D$ : $rate = - \frac{1}{a} \frac{d [ A ]}{d t} = \frac{1}{c} \frac{d [ C ]}{d t}$
Rate law: $rate = k [A]^{m} [B]^{n}$ , overall order $= m + n$
Law of mass action (for elementary reactions): rate $\propto$ product of active masses raised to stoichiometric coefficients
For a reaction with slow (rate-determining) step, the order matches the molecularity of that slow step alone.

1. Introduction

Chemical reactions differ enormously in the time they take. Ionic reactions such as $A g N O_{3} + N a C l \to A g C l ↓ + N a N O_{3}$ are essentially instantaneous. Molecular reactions such as the hydrogenation of ethene are moderate. Reactions such as the rusting of iron or the fermentation of sugar proceed over hours or days. Chemical kinetics is the branch of chemistry that studies how fast a reaction proceeds and by what mechanism.

2. Rate of Reaction

Consider the simple reaction $R \to P$ . As time passes, the concentration of reactant $R$ decreases and that of product $P$ increases. The rate of reaction is defined as the change in concentration of a species (reactant or product) per unit time.

Definition. The rate of a chemical reaction is

- \frac{d [ R ]}{d t} = + \frac{d [ P ]}{d t}

. The minus sign for a reactant makes the rate a positive quantity because

[R]

is decreasing.

2.1 Average rate of reaction

The average rate over a time interval $Δ t = t_{2} - t_{1}$ is

$r_{a v g} = - \frac{[ R ] _{t_{2}} - [ R ] _{t_{1}}}{t _{2} - t _{1}} = - \frac{Δ [ R ]}{Δ t}$

Graphically, the average rate is the slope of the chord joining two points on the concentration-time curve.

2.2 Instantaneous rate

The rate changes continuously during a reaction because the concentrations of reactants keep falling. The instantaneous rate at any moment $t$ is obtained by shrinking $Δ t$ to zero:

$r = Δ t \to 0 lim - \frac{Δ [ R ]}{Δ t} = - \frac{d [ R ]}{d t}$

Graphically, this is the slope of the tangent to the concentration-time curve at that instant. We usually prefer the instantaneous rate because the average rate hides how the rate is actually changing across the interval.

Reactant side

Product side

Concentration–time curves. The chord (deep orange) between two points on the curve gives the average rate; the tangent (blue) at a single point gives the instantaneous rate. Reactant concentration falls; product concentration rises correspondingly.

2.3 Rate expression for a general reaction

For $a A + b B \to c C + d D$ , the different species change at different numerical rates (because of stoichiometry), but the "rate of reaction" is a single quantity if we divide each rate by the corresponding coefficient:

$rate = - \frac{1}{a} \frac{d [ A ]}{d t} = - \frac{1}{b} \frac{d [ B ]}{d t} = + \frac{1}{c} \frac{d [ C ]}{d t} = + \frac{1}{d} \frac{d [ D ]}{d t}$

Solved Example 1

For the reaction

2 I^{-} + H_{2} O_{2} + 2 H^{+} \to I_{2} + 2 H_{2} O

, the concentration of

I^{-}

falls from

0.50 mol/L

0.46 mol/L

10.0

minutes. Find the average rate.

Solution:

$Δ [I^{-}] = 0.46 - 0.50 = - 0.04 mol/L$ , and $Δ t = 10.0 min$ .

$r_{a v g} = - \frac{1}{2} \frac{Δ [ I ^{-} ]}{Δ t} = - \frac{1}{2} \cdot \frac{- 0.04}{10.0} = 2.0 \times 1 0^{- 3} mol L^{- 1} min^{- 1}$ .

Solved Example 2

For the reaction

2 N O (g) + O_{2} (g) \to 2 N O_{2} (g)

, express the rate in terms of the disappearance of

O_{2}

and of

N O

, and in terms of the appearance of

N O_{2}

Solution:

Using the general expression with $a = 2, b = 1, c = 2$ :

$rate = - \frac{1}{2} \frac{d [ N O ]}{d t} = - \frac{d [ O _{2} ]}{d t} = + \frac{1}{2} \frac{d [ N O _{2} ]}{d t}$ .

So $N O$ disappears twice as fast as $O_{2}$ , and $N O_{2}$ appears at the same rate that $N O$ disappears.

Solved Example 3

Why is the instantaneous rate preferred over the average rate?

Solution:

The average rate over an interval only tells us the mean concentration change; it does not reveal how the rate varies within the interval. Since reaction rate generally decreases with time as reactants get consumed, the "true" rate at any moment is best captured by the instantaneous rate, which uses the slope of the tangent at that point. Rate laws are always written for the instantaneous rate.

3. Law of Mass Action

The law of mass action, given by Guldberg and Waage (1867), states that at a given temperature, the rate of a chemical reaction is directly proportional to the product of the molar concentrations of the reactants, each raised to a power equal to its stoichiometric coefficient in the balanced equation.

For $a A + b B \to products$ , the law of mass action gives $rate \propto [A]^{a} [B]^{b}$ , or

$rate = k [A]^{a} [B]^{b}$

where $k$ is the rate constant (or velocity constant). The rate constant is characteristic of the reaction at a given temperature.

Important: The law of mass action in the form above holds strictly only for elementary reactions (single-step reactions). For multi-step reactions the rate law must be determined experimentally and may involve powers different from the stoichiometric coefficients.

3.1 Physical meaning of the rate constant

Set $[A] = [B] = 1 mol/L$ ; then $rate = k$ . So the rate constant is the rate of reaction when all reactant concentrations are unity. It is also called the specific reaction rate.

4. Molecularity

Molecularity is the number of reacting species (atoms, ions, or molecules) that must collide simultaneously in a single elementary step to bring about the chemical change. It is a theoretical concept and is always a small whole number (1, 2, rarely 3).

Elementary reaction	Molecularity	Name
$P C l_{5} \to P C l_{3} + C l_{2}$	1	Unimolecular
$H_{2} + I_{2} \to 2 H I$	2	Bimolecular
$2 N O + O_{2} \to 2 N O_{2}$	3	Termolecular
$2 H_{2} O_{2} \to 2 H_{2} O + O_{2}$	2	Bimolecular

Molecularity greater than 3 is essentially never observed because the probability of four or more particles colliding simultaneously with the right energy and orientation is vanishingly small.

5. Order of Reaction

For an experimentally observed rate law $rate = k [A]^{m} [B]^{n}$ , the order with respect to $A$ is $m$ , the order with respect to $B$ is $n$ , and the overall order is $m + n$ .

Definition. The order of a reaction is the sum of the powers of the concentration terms in the experimentally determined rate law. It is a purely empirical quantity and may be an integer, a fraction, zero, or even negative.

5.1 Order for a multi-step (complex) reaction

When a reaction proceeds through several elementary steps, the overall rate is controlled by the slowest step, called the rate-determining step (RDS). The order of the overall reaction equals the molecularity of the slowest step.

Solved Example 4

For the reaction

N O_{2} + F_{2} \to N O_{2} F + \frac{1}{2} F_{2}

, the mechanism is:
Step 1 (slow):

N O_{2} + F_{2} \to N O_{2} F + F

Step 2 (fast):

N O_{2} + F \to N O_{2} F

Deduce the rate law.

Solution:

The slow step is bimolecular in $N O_{2}$ and $F_{2}$ , so it determines the rate:

$rate = k [N O_{2}] [F_{2}]$ .

Order with respect to $N O_{2}$ = 1, order with respect to $F_{2}$ = 1, overall order = 2.

Solved Example 5

A reaction has the mechanism
Step 1 (slow):

N O_{2} + N O_{2} \to N O + N O_{3}

Step 2 (fast):

N O_{3} + C O \to N O_{2} + C O_{2}

Overall:

N O_{2} + C O \to N O + C O_{2}

. Write the rate law and predict order.

Solution:

The slow step involves two $N O_{2}$ molecules, so:

$rate = k [N O_{2}]^{2}$ .

Note that $[C O]$ does not appear in the rate law even though it appears in the overall equation, because it participates only in the fast step. Overall order = 2, zero order with respect to $C O$ .

6. Difference between Order and Molecularity

Order	Molecularity
Experimental quantity, from the rate law.	Theoretical count from a single elementary step.
Can be zero, fractional, or negative.	Always a small positive whole number (1, 2, or 3).
Applies to the overall reaction.	Applies only to elementary reactions or a single step.
Equal to sum of powers in the rate law.	Equal to number of species colliding in that step.
Determined by the rate-determining step in complex reactions.	Undefined for a multi-step reaction as a whole.

Common Mistakes to Avoid

Watch out

Forgetting the stoichiometric factor $1/ a$ when expressing rate in terms of a specific species. The single "rate of reaction" needs the coefficient normalisation; the "rate of disappearance of $A$ " does not.
Confusing order with molecularity. Molecularity comes from a mechanism; order comes from experiment. They match only for a genuine elementary step.
Assuming the rate law from the balanced equation. This is valid only if the reaction is elementary; for multi-step reactions the rate law must be measured.
Dropping the minus sign for a reactant and reporting a negative rate. Rate is by convention a positive number.
Treating the rate constant $k$ as concentration-independent but temperature-independent. In fact $k$ depends strongly on temperature (via the Arrhenius equation) but not on concentration.

Frequently Asked Questions

Q1. What is the rate of a chemical reaction?

The rate of a chemical reaction is the change in the concentration of a reactant or product per unit time. For a reactant $R$ , rate $= - d [R] / d t$ ; for a product $P$ , rate $= + d [P] / d t$ . The negative sign for a reactant makes the rate a positive number.

Q2. What is the difference between average rate and instantaneous rate?

Average rate is the change in concentration over a finite time interval, equal to the slope of the chord on the concentration-time curve. Instantaneous rate is the rate at a particular instant, equal to the slope of the tangent at that point. Rate laws are always written for the instantaneous rate.

Q3. What is the order of a reaction?

The order is the sum of the powers of the concentration terms in the experimentally determined rate law $rate = k [A]^{m} [B]^{n}$ . Overall order = $m + n$ . Order is an empirical quantity and can be zero, fractional, or negative.

Q4. What is molecularity of a reaction?

Molecularity is the number of reacting species that must collide simultaneously in a single elementary step of a reaction. It is a theoretical concept, always a small positive whole number (usually 1, 2, or occasionally 3), and is only defined for elementary reactions or individual steps of a mechanism.

Q5. How are order and molecularity different?

Order is experimental and applies to the overall reaction. Molecularity is theoretical and applies to a single elementary step. Order can be fractional, negative, or zero; molecularity is always a small positive integer. For an elementary reaction they coincide; for a multi-step reaction they need not.

Q6. What is the rate constant $k$ ?

The rate constant $k$ is the proportionality constant in the rate law $rate = k [A]^{m} [B]^{n}$ . Numerically, it equals the rate when every reactant concentration is $1 mol/L$ , so it is also called the specific reaction rate. It depends on temperature and the presence of a catalyst, but not on concentration.

Q7. Why is molecularity greater than three practically not observed?

A collision of four or more molecules simultaneously with sufficient energy and correct orientation is statistically extremely improbable. Reactions that appear to involve many species always proceed through a sequence of elementary steps, each with molecularity 1, 2, or at most 3.

Q8. What is the rate-determining step?

The rate-determining step (RDS) is the slowest elementary step in a multi-step reaction mechanism. The overall rate cannot exceed the rate of this step, so the observed rate law is derived from the molecularity of the slowest step alone.

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1. Introduction

2. Rate of Reaction

2.1 Average rate of reaction

2.2 Instantaneous rate

2.3 Rate expression for a general reaction

3. Law of Mass Action

3.1 Physical meaning of the rate constant

4. Molecularity

5. Order of Reaction

5.1 Order for a multi-step (complex) reaction

6. Difference between Order and Molecularity

Common Mistakes to Avoid

Frequently Asked Questions

Q1. What is the rate of a chemical reaction?

Q2. What is the difference between average rate and instantaneous rate?

Q3. What is the order of a reaction?

Q4. What is molecularity of a reaction?

Q5. How are order and molecularity different?

Q6. What is the rate constant k?

Q7. Why is molecularity greater than three practically not observed?

Q8. What is the rate-determining step?

Previous year questions on Rate Of A Chemical Reaction

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Q6. What is the rate constant $k$ ?