Simple Rate Equations, Orders of Reaction and Rate Constants

Full Notes

Rate equations and Rate Determining Steps have been outlined in more detail here.
This is just what you need to know for CIE A-level Chemistry :)

Key Terms

• Rate Equation:
  Shows how the reaction rate depends on reactant concentrations.
  
• Order of Reaction:
  The power to which the concentration of a reactant is raised in the rate equation.
• Overall Order of Reaction:
  The sum of all individual orders (m + n).
• Rate Constant (k):
  A proportionality constant linking rate to concentrations; depends on temperature, activation energy, and molecular orientation.
• Half-Life (t½):
  The time for the concentration of a reactant to halve during a reaction.
• Rate-Determining Step:
  The slowest step in a multi-step mechanism that controls the overall reaction rate.
• Intermediate:
  A species formed in one step and used in a later step; it does not appear in the overall reaction equation.

Rate Equations and Determining Reaction Orders

Orders of reaction link how changes in reactant concentrations affect the rate. The order "with respect to" a reactant tells us how the rate depends on that particular reactant:

• Zero Order: Changing [A] does not affect the rate.
• First Order: Rate ∝ [A] (if [A] doubles, rate doubles).
• Second Order: Rate ∝ [A]² (if [A] doubles, rate quadruples).

Key Point – Orders must be determined experimentally, they cannot be predicted just from the balanced equation.

Experimental Methods to Determine Orders of Reaction

1. Initial Rate Method

We can measure how the initial reaction rate of a reaction changes when reactant concentrations are varied:

• Carry out the reaction multiple times, varying one reactant concentration at a time.
• Measure the initial rate for each experiment.
• Compare how the rate changes to determine reaction order.

Graphs of rate vs. concentration can be used to identify order with respect to each reactant:

2. Continuous Monitoring Method

We can measure concentration at different times during a reaction. This is useful when it’s difficult to measure initial rates directly. Graphs of concentration vs. time are then used to determine the order.

Graphical Interpretation of Orders:

• Zero Order: Linear concentration vs. time graph.
• First Order: Exponential decay.
• Second Order: Steeper exponential decay.

Rate Equation Calculations

Rate constants can be calculated using experimental data and a given rate equation. The units of k can vary for different reactions and depend on the overall order of reaction.

Matt’s exam tip

You can check your units of k are correct by balancing the units in the rate equation. Rate always has the units mol dm⁻³ s⁻¹. Meaning your units for k multiplied by the concentrations in the rate equation should always give mol dm⁻³ s⁻¹.

Half-Life and First Order Reactions

A key feature of a first-order reaction is a constant half-life: the time for the concentration to halve is the same regardless of starting concentration.

This can be useful for confirming a reaction is first-order as:
k = 0.693 / t_½
where t_½ is the half-life.
Note - this is only valid for first-order reactions.

Multi-Step Mechanisms and Rate-Determining Step

Most reactions do not happen in one simple step, but in a series of steps. These steps do not always occur at the same rate. When looking at how fast a reaction can happen, it's the slowest step that determines the overall rate of the reaction.

The slowest step in a reaction mechanism is the Rate-Determining Step (RDS). Only species involved in the RDS appear in the rate equation.

Example:
Reaction: NO₂ + CO → NO + CO₂
Observed rate equation: rate = k [NO₂]²
Since CO does not appear in the rate equation, it can't be involved in the RDS.
Possible Mechanism:

• NO₂ + NO₂ → NO₃ + NO (slow — RDS)
• NO₃ + CO → NO₂ + CO₂ (fast)

Matt’s exam tip

Mechanisms are only proposed based on experimental data. There may be more than one valid mechanism for a given rate equation!

Effect of Temperature

The rate constant, k, increases with temperature.

This means the rate of a reaction also increases with temperature as rate = k [A]^m [B]ⁿ (a larger k value means a larger rate).

This is explained in detail by the Arrhenius Equation (this isn’t required for CIE study however has been outlined in more detail here).

Summary

• Rate equation: rate = k [A]^m [B]ⁿ
• Orders of reaction show how concentration affects rate (zero, first, second).
• Overall order = sum of all individual orders.
• Rate constant k depends on temperature and activation energy.
• Orders must be found experimentally (initial rates, concentration–time graphs, rate–concentration graphs).
• First-order reactions have a constant half-life; k = 0.693/t½.
• Rate-determining step is the slowest step; only species in the RDS appear in the rate equation.
• k can be calculated from experimental data or half-life.
• Increasing temperature increases k, therefore increases reaction rate.