Calculate an initial rate from concentration data
Use concentration changes over short time intervals and tangent gradients to estimate initial rate.
Section 26 · Reaction kinetics
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 sign convention gives a positive rate of disappearance for a reactant?
What you will learn
- Calculate an average rate from concentration and time data
- Estimate initial rate from an early interval or initial tangent
- Use reactant/product sign conventions and units
Rate is concentration change per time
For a reactant A, rate of disappearance is −Δ[A]/Δt so the reported reaction rate is positive. For a product, rate of formation is +Δ[P]/Δt.
The average rate over an interval uses the secant gradient. The initial rate is the limiting rate at the start and is estimated from a very short early interval or the tangent to the concentration–time curve at t=0.
Keep concentration and time units consistent
If concentration is in mol dm⁻³ and time in seconds, rate is mol dm⁻³ s⁻¹. For a graph, calculate gradient as vertical change divided by horizontal change and include axis units.
Use the correct sign for a reactant decrease. A negative concentration gradient for a reactant corresponds to a positive rate of disappearance.
Pause and explain
What graph gradient estimates an initial rate?
Worked example
[A] falls from 0.100 to 0.092 mol dm⁻³ in the first 4.0 s. Estimate its average disappearance rate over this early interval.
Show the worked solution
- Δ[A] = 0.092−0.100 = −0.008 mol dm⁻³.
- Use −Δ[A]/Δt for disappearance.
- Divide by 4.0 s.
Answer Average disappearance rate = 0.008/4.0 = 2.0 × 10⁻³ mol dm⁻³ s⁻¹.
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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.
A product concentration rises from 0.012 to 0.018 mol dm⁻³ in 3.0 s. Calculate its average formation rate.
Give me a hint
Use +Δ[product]/Δt.
Compare my reasoning
- Find Δ[P] = 0.006 mol dm⁻³.
- Divide by 3.0 s.
- State rate units.
Rate = 2.0 × 10⁻³ mol dm⁻³ s⁻¹.
Look for these in your work
- Product change correct
- Time division correct
- Units included
A tangent at the start of an [A]–time graph falls by 0.015 mol dm⁻³ over a horizontal run of 5.0 s. Estimate the initial disappearance rate.
Give me a hint
Use the magnitude of the negative tangent slope.
Compare my reasoning
- Gradient = −0.015/5.0.
- Use its magnitude for disappearance rate.
- Attach concentration per time units.
Initial disappearance rate ≈ 3.0 × 10⁻³ mol dm⁻³ s⁻¹.
Look for these in your work
- Gradient calculated
- Reactant sign interpreted
- Units stated
A learner reports −0.004 mol dm⁻³ s⁻¹ as the rate of disappearance of A. Explain the sign issue.
Give me a hint
A negative gradient shows [A] falls; rate of disappearance is reported positive.
Compare my reasoning
- Recognise Δ[A]/Δt is negative.
- Apply −Δ[A]/Δt for disappearance.
- State the positive rate magnitude.
The negative sign belongs to the concentration gradient. Rate of disappearance is −Δ[A]/Δt, so the reported rate is +0.004 mol dm⁻³ s⁻¹.
Look for these in your work
- Gradient and rate distinguished
- Sign convention applied
- Positive rate reported
Common mistakes
- Reporting a negative disappearance rate
- Using time change divided by concentration change
- Calling an interval average the exact instantaneous initial rate
- Omitting time units from the rate
What can you explain now?
What sign convention gives a positive rate of disappearance for a reactant?
Compare with the explanation
−Δ[reactant]/Δt
The reactant concentration falls, so the leading minus sign gives a positive rate.
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Make it stick
How does a concentration–time gradient become a positive initial reaction rate?
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.