ConverterHub

First-Order Rate Constant Calculator

Convert a half-life into a rate constant.

Result

8.6643e-2
11.54
26.58

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 8.6643e-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 = 8.6643e-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.

Try this example →

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. Enter a known half-life to get k, τ, and t₉₀ instantly.

Was this helpful?

You can pick up to 3 reactions.

Comments (0)

  • Be the first to comment.

Popular calculators

Loan Calculator

Monthly payment, total interest and full amortization schedule.

Percentage Calculator

Percent change, percent of, and ratio calculations.

Discount Calculator

Calculate final price, savings, and discount amount.

BMI Calculator

Body mass index from height and weight (metric or imperial).

Age Calculator

Compute your exact age in years, months and days from a birth date.

Compound Interest Calculator

Calculate the final amount and interest earned with compounding.

All Calculators