R2.2.10 Order of a reaction (Higher Level Only)

Reaction Order and Graphical Representations of Reaction Kinetics

You’re in a relay race where each runner’s speed depends on the number of teammates cheering them on.
Some runners might speed up when the crowd gets louder, while others might stick to their usual pace regardless of the noise.

Chemical reactions behave in a similar way. The "speed" of a reaction—the rate at which reactants turn into products—depends on the concentration of the reactants, but this dependence varies for different reactions.

Reaction Order: What Does It Mean?

Definition

Order of a reaction

The order of a reaction with respect to a reactant is the power to which the concentration of that reactant is raised in the rate equation.

The general form of the rate equation is:

$v = k [A]^{n} [B]^{m}$

Where:

$v$ is the rate of reaction,
$k$ is the rate constant (a value specific to the reaction and its temperature),
$[A]$ and $[B]$ are the concentrations of the reactants,
$n$ and $m$ are the reaction orders with respect to $A$ and $B$ , respectively.

Tip

The overall reaction order is the sum of the individual orders: $n + m$ .

Examples of Reaction Orders

Zero Order ( $n = 0$ ): The rate is independent of the concentration of the reactant. Doubling $[A]$ has no effect on $v$ .
$v = k$
First Order ( $n = 1$ ): The rate is directly proportional to the concentration of the reactant. Doubling $[A]$ doubles $v$ .
$v = k [A]$
Second Order ( $n = 2$ ): The rate is proportional to the square of the reactant concentration. Doubling $[A]$ increases $v$ by a factor of four.
$v = k [A]^{2}$

Common Mistake

Reaction orders cannot be deduced from the stoichiometric coefficients of the balanced chemical equation. They must be determined experimentally.

Reaction Mechanisms and the Rate-Determining Step

The reaction order provides critical clues about the rate-determining step (RDS), which is the slowest step in a reaction mechanism. For instance:

If a reaction is first order with respect to reactant $A$ , the RDS likely involves a single molecule of $A$ .
If it’s second order with respect to $A$ , the RDS might involve the collision of two $A$ molecules.

Theory of Knowledge

Why do scientists refer to reaction mechanisms as "possible mechanisms"?
How does this reflect the iterative nature of scientific inquiry?

Graphical Representations of Reaction Kinetics

Graphs are essential tools for analyzing reaction kinetics.
Two key types of graphs are rate–concentration graphs and concentration–time graphs, which help determine the reaction order and rate constant.

Rate–Concentration Graphs

These graphs plot the reaction rate ( $v$ ) against the concentration of a reactant ( $[A]$ ).
The shape of the graph reveals the reaction order:
1. Zero Order: A horizontal line, because $v$ is constant and independent of $[A]$ .
  $v = k$
2. First Order: A straight line passing through the origin, as $v \propto [A]$ .
  $v = k [A]$
3. Second Order: A curve (parabola), since $v \propto [A]^{2}$ .
  $v = k [A]^{2}$

Concentration–Time Graphs

These graphs show how the concentration of a reactant ( $[A]$ ) changes over time.
The shape of the graph depends on the reaction order:
1. Zero Order: A straight line with a negative slope. The rate of decrease in $[A]$ is constant.
  $[A] = [A]_{0} - k t$
2. First Order: An exponential decay curve, where the rate of decrease slows over time.
  $[A] = [A]_{0} e^{- k t}$
3. Second Order: A hyperbolic decay curve, steeper than that of a first-order reaction.
  $\frac{1}{[A]} = \frac{1}{[A]_{0}} + k t$

Tip

To distinguish between first and second order reactions, plot $\ln [A]$ vs. time (for first order) or $1 / [A]$ vs. time (for second order). The graph that produces a straight line corresponds to the reaction order.

Summary of Graphical Representations

Zero-order

First-order

Second-order

Experimental Determination of Reaction Order

To determine the order of a reaction experimentally:

Measure Initial Rates: Perform several experiments, varying the initial concentration of one reactant at a time, and record the initial rate of reaction.
Analyze Relationships: Use the data to determine how changes in concentration affect the rate.

Example

If doubling $[A]$ doubles $v$ , the reaction is first order with respect to $A$ .
If doubling $[A]$ quadruples $v$ , the reaction is second order with respect to $A$ .

Example

Consider the reaction $A + B \to C$ . The following data is obtained:

$\begin{array}{cccc} Experiment & [A] ({mol dm}^{- 3}) & [B] ({mol dm}^{- 3}) & v ({mol dm}^{- 3} s^{- 1}) \\ 1 & 0.10 & 0.10 & 0.020 \\ 2 & 0.20 & 0.10 & 0.040 \\ 3 & 0.10 & 0.20 & 0.080 \end{array}$

From this data:

Doubling $[A]$ (Exp. 1 → Exp. 2) doubles the rate. Thus, the reaction is first order with respect to $A$ .
Doubling $[B]$ (Exp. 1 → Exp. 3) quadruples the rate. Thus, the reaction is second order with respect to $B$ .The rate equation is:
$v = k [A]^{1} [B]^{2}$
The overall reaction order is $1 + 2 = 3$ .