Calculate a rate constant from initial rates
Substitute a measured initial rate and concentrations into an experimentally determined rate equation to find k.
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
How do you isolate k in rate = k[A]²?
What you will learn
- Use the rate equation with an initial-rate measurement
- Rearrange for k
- State units based on the overall order
Substitute a trial after determining orders
Once the rate equation is known, k = rate/([A]ᵐ[B]ⁿ). Use a single experiment's initial rate and initial concentrations, with consistent units.
The rate equation orders must first be deduced from experimental evidence. A numerical k cannot validate an assumed equation if the orders are unknown.
Units of k depend on the overall order
Rate units are concentration per time. For an overall first-order reaction k has units time⁻¹; for second order it has concentration⁻¹ time⁻¹, such as dm³ mol⁻¹ s⁻¹. Derive the units from the equation rather than memorising a single unit.
Check that calculated k is positive and that its dimensions make the rate equation return concentration per time.
Pause and explain
What are the units of k for an overall second-order rate equation?
Worked example
For rate = k[A]², an initial rate is 2.0×10⁻⁴ mol dm⁻³ s⁻¹ at [A]=0.20 mol dm⁻³. Calculate k.
Show the worked solution
- Use k = rate/[A]².
- Substitute rate and concentration.
- Derive units from rate divided by concentration squared.
Answer k = (2.0×10⁻⁴)/(0.20)² = 5.0×10⁻³ dm³ mol⁻¹ s⁻¹.
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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.
For rate = k[A][B], rate = 3.0×10⁻⁴ mol dm⁻³ s⁻¹, [A]=0.10 mol dm⁻³ and [B]=0.20 mol dm⁻³. Find k.
Give me a hint
Divide rate by the product of the concentrations.
Compare my reasoning
- [A][B]=0.020 mol² dm⁻⁶.
- k = 3.0×10⁻⁴/0.020.
- Use second-order units.
k = 0.015 dm³ mol⁻¹ s⁻¹.
Look for these in your work
- Concentration product calculated
- Rate divided by product
- Units derived
For rate = k[A], rate = 0.0060 mol dm⁻³ s⁻¹ at [A]=0.30 mol dm⁻³. Calculate k and its units.
Give me a hint
Use k=rate/[A].
Compare my reasoning
- Divide 0.0060 by 0.30.
- Cancel concentration units.
- State reciprocal seconds.
k = 0.020 s⁻¹.
Look for these in your work
- Rearrangement correct
- Value calculated
- First-order units stated
A learner calculates k=rate×[A] for a first-order reaction. Explain the algebra and units error.
Give me a hint
Start from rate=k[A] and isolate k.
Compare my reasoning
- Divide both sides by [A].
- Use k=rate/[A].
- Check units give time⁻¹.
The concentration belongs in the denominator. k = rate/[A], whose units are (mol dm⁻³ s⁻¹)/(mol dm⁻³)=s⁻¹.
Look for these in your work
- Equation rearranged
- Concentration placement corrected
- Units checked
Common mistakes
- Using an average rate when an initial rate is specified
- Using concentrations inconsistent with the trial
- Forgetting concentration powers
- Giving k the units of rate
What can you explain now?
How do you isolate k in rate = k[A]²?
Compare with the explanation
k = rate/[A]²
Divide both sides by [A]².
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Make it stick
After an order is known, how do an initial rate and concentrations give a dimensionally consistent k?
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.