Calculate a rate constant from half-life
Find a first-order rate constant from its half-life and convert the result to the required time unit.
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 is k related to first-order half-life?
What you will learn
- Rearrange t₁/₂ = 0.693/k
- Calculate k from a measured first-order half-life
- Keep time units consistent
The first-order half-life gives k directly
For a first-order reaction, t₁/₂ = 0.693/k, so k = 0.693/t₁/₂. The half-life must be measured under the same temperature conditions as the rate constant being reported.
If half-life is given in minutes, k first comes out in min⁻¹. Convert only after calculation if the answer is required in s⁻¹.
Check the reciprocal-time unit
A shorter first-order half-life corresponds to a larger k. The units must be reciprocal to the time unit used in the denominator.
Do not use this relation for an arbitrary order unless first-order behaviour has been established.
Pause and explain
A half-life is measured in minutes. What are the direct units of k?
Worked example
A first-order reaction has t₁/₂ = 40.0 s. Calculate k.
Show the worked solution
- Use k = 0.693/t₁/₂.
- Substitute 40.0 s.
- State reciprocal seconds.
Answer k = 0.693/40.0 = 0.0173 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.
Calculate k for a first-order reaction with t₁/₂ = 25.0 min.
Give me a hint
Divide 0.693 by the half-life in minutes.
Compare my reasoning
- k = 0.693/25.0.
- Calculate the quotient.
- Attach min⁻¹.
k = 0.0277 min⁻¹.
Look for these in your work
- Equation rearranged
- Value calculated
- Unit reciprocal to minutes
A first-order reaction has t₁/₂ = 120 s. Convert k to s⁻¹.
Give me a hint
Use the given seconds directly.
Compare my reasoning
- k=0.693/120 s.
- Calculate the value.
- State s⁻¹.
k = 5.78 × 10⁻³ s⁻¹.
Look for these in your work
- Half-life used
- Division correct
- Seconds reciprocal stated
Two first-order reactions have half-lives 10 s and 40 s. Which has the larger k and by what factor?
Give me a hint
Since k is inversely proportional to half-life, compare 1/t₁/₂.
Compare my reasoning
- k₁=0.693/10.
- k₂=0.693/40.
- Compare the two values.
The 10 s reaction has the larger k, by a factor of four.
Look for these in your work
- Inverse relation used
- Reaction identified
- Factor calculated
Common mistakes
- Using k=t₁/₂/0.693
- Applying the relation before identifying first order
- Reporting s instead of s⁻¹
- Mixing minutes and seconds
What can you explain now?
How is k related to first-order half-life?
Compare with the explanation
k = 0.693/t₁/₂
Rearrange t₁/₂=0.693/k.
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Make it stick
What does a measured first-order half-life tell you about k, including the unit?
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.