Rate equations relate reaction rate to reactant concentrations

A rate equation is a mathematical expression that shows how the rate of a chemical reaction depends on the concentrations of the reactants.

For a general reaction: A + B ➔ C + D

The rate equation takes the form:

Rate = k[A]^m[B]ⁿ

Where:

• Rate is the reaction rate (mol dm^-3 s^-1).
• k is the rate constant.
• [A] and [B] are the concentrations of reactants A and B (mol dm^-3).
• m and n are the orders of reaction with respect to A and B.

Reaction orders show how reactant concentrations affect the rate

The values of m and n in the rate equation are the reaction orders with respect to each reactant.

They indicate how changing the concentration of a reactant influences the reaction rate.

1. Zero order (m or n = 0) - The rate is independent of the reactant concentration. Doubling or tripling the concentration has no effect on the rate. [A]⁰ = 1, so zero order terms are often written without the concentration term.

2. First order (m or n = 1) - The rate is directly proportional to the reactant concentration. Doubling the concentration doubles the rate, tripling the concentration triples the rate.

3. Second order (m or n = 2) - The rate is proportional to the square of the reactant concentration. Doubling the concentration quadruples the rate (2² = 4), tripling the concentration increases the rate ninefold (3² = 9).

The overall order of the reaction is the sum of m and n.

Importantly, reaction orders can only be determined experimentally, not from balanced chemical equations.

The rate constant k relates reactant concentrations to rate at a given temperature

The rate constant, k, is a proportionality constant that relates the rate of a reaction to the concentrations of the reactants at a specific temperature.

• A larger k value indicates a faster rate of reaction.
• k remains constant for a given reaction at a fixed temperature.
• Increasing the temperature exponentially increases the value of k, as collisions between reactant molecules are more frequent and more energetic.

The units of k depend on the overall order of the reaction.

Calculating reaction rate from the rate constant and orders

If the rate constant and orders of a reaction are known, the rate can be calculated.

Worked example 1 - Calculating the rate of an acid-catalysed reaction

Calculate the rate of the acid-catalysed reaction between propanone and iodine, given that the reaction is first order with respect to propanone, zero order with respect to iodine, and first order with respect to H⁺.

CH₃COCH₃ + I₂ ➔ CH₃COCH₂I + HI

The rate constant (k) at a certain temperature is 630 mol^-1 dm³ s^-1 and the concentrations of propanone, iodine, and H⁺ are each $2.00\times {10}^{-3}{\text{ mol dm}}^{-3}$. Give your answer to 3 significant figures.

Step 1: Write the rate equation

$\text{Rate }={\text{ k[CH}}_3{\text{COCH}}_3{]}^1{[{\text{I}}_2]}^0{[{\text{H}}^{+}]}^1={\text{ k[CH}}_3{\text{COCH}}_3][{\text{H}}^{+}]$

Step 2: Substitution and correct evaluation

$\text{Rate }=630\times (2.00\times {10}^{-3})\times (2.00\times {10}^{-3})=2.52\times {10}^{-3}{\text{ mol dm}}^{-3}{\text{ s}}^{-1}$

Thus, at this temperature, the rate of the acid-catalysed reaction between propanone and iodine is $2.52\times {10}^{-3}{\text{ mol dm}}^{-3}{\text{ s}}^{-1}$.

Calculating the rate constant from experimental data

If the rate and orders of a reaction are known from experiments, the rate constant k can be calculated at a given temperature.

Worked example 2 - Calculating the rate constant for a gas-phase reaction

The following reaction is second order with respect to NO and zero order with respect to CO and O_2:

NO_(g) + CO_(g) + O_2(g) ➔ NO_2(g) + CO_2(g)

At a certain temperature, the reaction rate is $4.00\times {10}^{-3}$ mol dm^-3 s^-1, with the concentrations of NO, CO, and O₂ each at $4.50\times {10}^{-3}$ mol dm^-3.

Calculate the rate constant (k) for the reaction. Give your answer to 3 significant figures.

Step 1: Write the rate equation

$\text{Rate }=\text{ k[NO}{]}^2{[\text{CO}]}^0{[{\text{O}}_2]}^0=\text{ k[NO}{]}^2$

Step 2: Rearrange rate equation

$\text{k }=\frac{\text{Rate}}{{\text{[NO]}}^2}$

Step 3: Substitution and correct evaluation

$\text{k }=\frac{4.00\times {10}^{-3}}{{(4.50\times {10}^{-3})}^2}=198$

Step 4: Determine the units of k

Given that the rate is expressed in mol dm^-3 s^-1 and the concentration squared in (mol dm^-3)², solving for k in the equation results in units of mol^-1 dm³ s^-1.

Thus, at this temperature, the rate constant is 198 mol^-1 dm³ s^-1 for the given gas-phase reaction.