HesapHanem

Exponential Decay Calculator

Find how much remains after exponential decay.

How it is calculated

The tool evaluates N = N0 × e−k×t, where N0 is the starting amount (any unit, must be above zero), k is the decay constant per unit of time and t is the elapsed time in that same unit. If k is in per-hour units, t must be in hours. It reports the remaining amount to four decimals and the percentage left, N / N0 × 100, to two decimals. If you know the half-life instead of k, use k = ln(2) / half-life, about 0.693 divided by the half-life. The tool does not reject a negative k; it then returns a value above 100%, which is exponential growth.

Worked example

Start with 200 units and a decay constant of 0.0693 per day over 10 days. Exponent: −0.0693 × 10 = −0.693. e−0.693 is about 0.50005. Remaining: 200 × 0.50005 = 100.0147. As a share of the start that is 50.01%. A constant of 0.0693 corresponds to a half-life of about 10 days, so roughly half remains.

Try this example →

Common mistakes

  • Mixing time units: a constant per day with a time in hours or years gives an answer that is wrong by a large factor.
  • Entering the half-life in the rate field. The field wants k, which is ln(2) divided by the half-life, not the half-life itself.
  • Entering the decay rate as a percentage (5 for 5% per period). Use 0.05, or note that a 5% drop per period corresponds to k = −ln(0.95), about 0.0513.
  • Entering a minus sign on k and expecting decay. The formula already has the minus built in, so a negative k makes the value grow.
  • Reading the second result as the amount lost. It is the percentage still remaining.

About this calculator

Exponential decay describes a quantity that shrinks at a rate proportional to its current value, following N = N₀·e^(−kt), where N₀ is the initial amount, k is the decay constant, and t is elapsed time. This same equation underlies radioactive decay (where k relates to a substance's half-life via k = ln(2)/t½), drug elimination from the bloodstream in pharmacokinetics, capacitor discharge in RC circuits, and cooling of a hot object toward ambient temperature (Newton's law of cooling).

Physicists, chemists, pharmacologists, and engineers use this calculation to find how much of a radioactive isotope remains after a given time (used in carbon dating and nuclear medicine dosing), how much medication remains active in the body, or how a signal or voltage fades over time.

Was this helpful?

You can pick up to 3 reactions.

Comments (1)

  • Editorial team

    When you know the half-life rather than k, how many digits you keep for k matters a little. Using 0.0693 for a half-life of 10 time units leaves 50.01 percent after 10 units instead of exactly 50, while 0.693147 gives 50.00. Enter k with four or more significant digits, or the small error carries into every later period. What process are you modelling, and would a half-life input be more convenient?

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