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First-Order Rate Constant Calculator

Convert a half-life into a rate constant.

How it is calculated

For a first-order process the half-life fixes everything else:

k = ln 2 / t½, τ = 1 / k and t90 = ln 10 / k

t½ can be in any unit of time (seconds, hours, days). The rate constant k then comes out in 1 per that unit and is printed in scientific notation with four decimals, so 8.6643 × 10⁻² means 0.086643. The mean lifetime τ and the time to 90 % decay t90 are in the same unit you entered and are shown with two decimals. The half-life must be above zero. The results only apply when the process really is first order, such as radioactive decay or simple exponential elimination.

Worked example

A first-order process with a half-life of 8 hours.

  1. k = 0.6931 / 8 = 8.6643 × 10⁻² per hour
  2. τ = 1 / 0.086643 = 11.54 hours
  3. t90 = 2.3026 / 0.086643 = 26.58 hours

After about 26.58 hours, 90 % of the starting amount has gone and 10 % remains.

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Common mistakes

  • Confusing the mean lifetime with the half-life: τ is about 1.44 times longer than t½, so 11.54 is not 8.
  • Mixing time units: if the half-life is in days, k is per day and τ and t90 are in days, so do not read them as hours.
  • Reading the time for 90 % decay as the time for 90 % remaining: t90 leaves only 10 % behind.
  • Applying the result to second-order or zero-order reactions, where the half-life depends on the starting concentration.

About this calculator

For a first-order reaction, where the rate depends linearly on a single reactant's concentration (−d[A]/dt = k[A]), the rate constant relates directly to half-life through k = ln 2 / t½ ≈ 0.693 / t½. This calculator also derives the mean lifetime τ = 1/k = t½/ln 2 (the average time a molecule survives before reacting) and the time needed for 90% decay, t₉₀ = ln 10 / k.

This math is identical to radioactive decay kinetics, so chemists, nuclear physicists, and pharmacologists all use it — chemists for reaction-rate studies, pharmacologists to convert a drug's elimination half-life into a decay rate for dosing schedules, and chemical engineers sizing reactors where a first-order rate law governs conversion.

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Comments (1)

  • Editorial team

    A quick check on the output: t90 is always about 3.32 times the half-life, because 90 percent decay takes log base 2 of 10 half-lives. If your t90 is not roughly 3.3 times the number you entered, something is off. Remember also that k carries the inverse of your time unit, so convert before comparing it with a literature value. Which process and which time unit are you working in?

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