5.2.3 Deriving Rate Equations

Understanding How Rate Equations are Derived

A rate equation describes the relationship between the concentrations of reactants and the rate of a chemical reaction. The powers to which the concentrations are raised, known as the orders of reaction, indicate how each reactant concentration affects the rate. Here's an example that breaks down the steps involved in deriving a rate equation.

Example Reaction

Consider a reaction:

$$\text{B} + \text{C} + \text{D} \rightarrow \text{E} + \text{F}$$

The rate equation for this reaction could be:

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

where:

• The order of reaction with respect to $B$ is 2.
• The order of reaction with respect to $D$ is 1.
• The order of reaction with respect to $C$ is 0 (since it does not appear in the rate equation).

Explanation of Orders

Order of Reaction with Respect to Each Reactant:

• Order 2 for $B$: The rate depends on the concentration of $B$ raised to the power of 2. Doubling the concentration of $B$ will increase the rate by a factor of $2^2 = 4$
• Order 1 for $D$: The rate depends on the concentration of $D$ raised to the power of 1. Doubling the concentration of $D$ will double the rate.
• Order 0 for $C$: The concentration of $C$ does not affect the rate. In mathematics, anything raised to the power of zero equals 1, so $[\text{C}]^0 = 1$, meaning changes in $C's$ concentration have no effect on the rate.

Overall Order of Reaction:

• The overall order of reaction is the sum of the individual orders: $2 + 1 + 0 = 3$
• The overall order helps predict the combined effect of changing multiple reactant concentrations.

Using the Rate Equation

From this example rate equation, we can determine how changes in concentration affect the reaction rate:

• Doubling $[B]$: Increases the rate by a factor of 4 (since $2^2 = 4$).
• Doubling $[D]$: Increases the rate by a factor of 2 (since $2^1 = 2$).
• Doubling $[C]$: Has no effect on the rate, as $C$ has an order of 0.
• Doubling $[B]$, $[C]$, and $[D]$ Together: Increases the rate by a factor of 8 (since $2^3 = 8$), which corresponds to the overall order of 3.

Importance of Deriving Rate Equations

Deriving rate equations from experimental data allows chemists to:

• Identify which reactants influence the rate.
• Understand the mechanism of the reaction.
• Predict how changes in concentration will affect reaction speed.