R2.2.9 Rate equations (Higher Level Only)

Rate Equations and Determining Reaction Order

1. Imagine you are observing a reaction where a solution gradually changes color over time.
2. How could you determine the mathematical relationship between the concentration of the reactants and the speed of the reaction?

This relationship is described by the rate equation, a fundamental tool in chemical kinetics for predicting how reaction speed depends on reactant concentrations.

The Rate Equation and Reaction Mechanisms

It takes the form:

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

where:

• Rate: Speed of the reaction, measured in $mold{m}^{-3}$.
• $k$: Rate constant, dependent on temperature and catalysts.
• $[A]$ and $[B]$: Molar concentrations of the reactants.
• $m$ and $n$: The orders of reaction with respect to each reactant.

What Do the Orders of Reaction Mean?

• If $m=0$ changing $[A]$ has no effect on the rate (zero-order).
• If $m=1$, doubling $[A]$ doubles the rate (first-order).
• If $m=2$, doubling $[A]$ quadruples the rate (second-order).

The overall order of a reaction is the sum of the exponents:

$$\text{Overall Order}=m+n$$

Why Must Rate Equations Be Determined Experimentally?

1. The rate equation reflects the rate-determining step in a reaction mechanism—the slowest step that limits the overall reaction speed.
2. Since reaction mechanisms can involve complex steps that aren't obvious from the overall balanced chemical equation, experiments are necessary to determine the correct orders of reaction.

Experimental Methods to Determine the Rate Equation

• The rate equation is typically determined by measuring how the rate changes with varying concentrations of reactants.
• Common methods include:

Method 1: Initial Rates Method

The initial rate of a reaction is measured by varying the concentration of one reactant while keeping others constant.

Steps:

1. Prepare multiple reaction mixtures with different concentrations of reactant A.
2. Measure the initial rate for each mixture.
3. Compare how the rate changes as $[A]$ changes.

Method 2: Graphical Analysis (will be covered in R2.2.10)

By plotting data from concentration and rate measurements, you can identify the reaction order based on the shape of the graph:

• Zero-order: Rate vs. $[A]$ is a flat line.
• First-order: Rate vs. $[A]$ is a straight line.
• Second-order: Rate vs. $[A]$ produces a curve with increasing slope.

Deducing the Rate Equation from Experimental Data

You are given the following data for a reaction:

$$\begin{array}{|cc|}\hline [A]({\text{mol dm}}^{-3})& {\text{Initial Rate (mol dm}}^{-3}{\text{s}}^{-1})\\ 0.10& 0.005\\ 0.20& 0.020\\ 0.40& 0.080\\ \hline\end{array}$$

Step 1: Identify the effect of concentration changes on rate:

• Doubling $[A]$ results in a quadrupling of the rate.
• This suggests second-order behavior.

Step 2: Write the rate equation:
Since the rate quadruples when $[A]$ is doubled:

$$\text{Rate}=k[A{]}^2$$

Step 3: Solve for the rate constant $k$:

Using the first set of data:

$$0.005=k(0.10{)}^2$$

Solving for $k$:

$$k=\frac{0.005}{0.01}=0.5{\text{mol}}^{-1}{\text{dm}}^3{\text{s}}^{-1}$$

Reflection