Construct a rate equation from initial rates
Determine individual orders from controlled experiments and combine them into an empirical rate equation.
A Level; Cambridge International Chemistry 9701, 2028–2030 syllabus, Topic 26. Rate laws are empirical unless an elementary step is explicitly stated. Written practice is self-checked
Before you begin
- Use AS collision theory, concentration, graph gradients, and balanced reaction equations.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should a controlled initial-rate comparison change to find the order in A?
What you will learn
- Use separate experiment pairs to find each order
- Combine the orders into a rate equation
- Check that the equation predicts the measured rate factors
Find one order at a time
For rate = k[A]ᵐ[B]ⁿ, compare trials where [B] is constant to find m. Then compare trials where [A] is constant to find n. Use the ratio of rates and concentrations to solve the relevant power relationship.
Once orders are known, write the rate equation and determine k if needed. The equation is empirical and applies to the conditions and concentration range tested.
Check the proposed equation against all trials
Substitute each trial's concentrations or compare predicted rate factors with the measured factors. An order assignment should fit every relevant comparison, not just one pair.
Orders may be zero, one or two in the simple rate equations assessed here. Use the data to decide, rather than imposing an order from the balanced equation.
Pause and explain
Doubling [A] doubles the rate when [B] is fixed. What order in A fits?
Worked example
Experiment 1: [A]=0.10, [B]=0.10, rate=2.0×10⁻³. Experiment 2 doubles A only and rate becomes 4.0×10⁻³. Experiment 3 doubles B only from experiment 1 and rate becomes 8.0×10⁻³. Construct the rate equation.
Show the worked solution
- A doubling doubles rate, so order in A is 1.
- B doubling quadruples rate, so order in B is 2.
- Write the equation with k.
Answer rate = k[A][B]²; the overall order is 3.
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
When [A] doubles with [B] fixed, rate quadruples. When [B] doubles with [A] fixed, rate doubles. Write the rate equation.
Give me a hint
Translate each factor into an exponent.
Compare my reasoning
- A order is 2 because 2²=4.
- B order is 1 because 2¹=2.
- Combine the concentration terms.
rate = k[A]²[B].
Look for these in your work
- Orders deduced separately
- Rate factors matched
- Equation written
For a reaction, doubling [A] changes rate by no measurable amount; doubling [B] increases rate by four. Construct a simple rate equation.
Give me a hint
No rate response indicates order zero for that species.
Compare my reasoning
- Order in A = 0.
- Order in B = 2.
- Write rate = k[A]⁰[B]² and simplify if desired.
rate = k[B]²; the reaction is zero order in A and second order in B over the tested range.
Look for these in your work
- Zero order inferred
- B order inferred
- Range stated
A proposed rate equation fits trials 1 and 2 but predicts a rate factor of 9 for trial 3, where the measured factor is 3. What should be checked?
Give me a hint
An empirical equation must fit every controlled comparison.
Compare my reasoning
- Recheck the changed concentrations in trial 3.
- Recalculate the relevant order from the rate ratio.
- Revise the equation if the evidence does not support it.
The proposed order or the trial comparison is inconsistent. Recheck which concentrations changed and use the measured rate factor to revise the relevant order; do not keep an equation that fails trial 3.
Look for these in your work
- All trial data considered
- Concentration comparison rechecked
- Empirical nature respected
Common mistakes
- Changing multiple reactants in a comparison
- Assigning an order before taking rate ratios
- Using balanced-equation coefficients as rate orders
- Failing to check the equation against every trial
What can you explain now?
What should a controlled initial-rate comparison change to find the order in A?
Compare with the explanation
Change [A] while keeping other relevant concentrations fixed
This isolates the effect of A on rate.
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Make it stick
How do separate controlled comparisons combine to produce and check an empirical rate equation?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.