Choose a starting point, a half-life, or two measured values, and this calculator derives both the continuous decay constant k and the equivalent periodic decay rate r.
| Decay Rate r | Decay Constant k = −ln(1−r) |
|---|---|
| 2% | 0.0202 |
| 5% | 0.0513 |
| 10% | 0.1054 |
| 20% | 0.2231 |
What the table shows: at a 2% decay rate, k = 0.0202, barely different from r itself. By 20%, k = 0.2231, noticeably larger than 0.20. The two numbers start almost identical at low rates and pull apart as the rate grows, because r measures loss over one whole period while k measures loss applied continuously within that same period.
A decay constant (k) is the continuous-time rate parameter in x(t) = x0e−kt, related to the periodic decay rate r by k = −ln(1 − r) and, in reverse, r = 1 − e−k.
Every exponential decay has two equally valid ways to describe how fast it shrinks. One is the periodic decay rate r, the plain percentage lost each period, the number a spreadsheet or a step-by-step calculator uses directly. The other is the decay constant k, the rate that belongs inside an exponential function when the loss is treated as happening continuously rather than in discrete jumps.
The two are always close in value but rarely identical. A 5% periodic decay rate does not correspond to k = 0.05; it corresponds to k = −ln(0.95) = 0.0513. The gap exists because continuous compounding needs a slightly smaller instantaneous rate to land on the same one-period result as a single discrete 5% cut. At small rates the gap is tiny; at large rates it becomes significant.
k is what shows up in physics and calculus-based treatments of decay, radioactive isotopes, capacitor discharge, cooling curves, because those processes are naturally continuous rather than stepwise. r is what shows up in period-based calculators, financial depreciation schedules, and anywhere else the loss is naturally counted in whole periods. Neither form is more "correct"; they are two coordinate systems for the same underlying curve.
A decay rate converts to a decay constant with k = −ln(1 − r), and a decay constant converts back to a decay rate with r = 1 − e−k. Each formula is simply the inverse of the other.
Start here when you already know the periodic percentage lost per period and need the continuous rate for a physics- or calculus-based model.
Start here when a source hands you a decay constant and you need the plain percentage-per-period figure for a step-based calculator instead.
| Symbol | Name | What it represents | Example |
|---|---|---|---|
| r | Periodic decay rate | The fraction lost per whole period, as a decimal. | 0.10 (that is 10%) |
| k | Decay constant | The continuous-time rate used in x(t) = x₀e−kt. | 0.1054 |
| h | Half-life | Time for the quantity to fall to exactly half its value. | 8 time units |
| x0 | Initial value | A measured or assumed value at t = 0. | 900 units |
| x(t) | Later value | A measured value after elapsed time t. | 600 units |
| t | Elapsed time | Time between the two measurements. | 5 time units |
What the table shows: only six symbols cover every path through this page's calculator. Give it r and it returns k; give it h and it returns both; give it x0, x(t), and t and it returns both again. Every route lands on the same two numbers, just approached from different starting information.
To find a decay rate from a half-life, divide ln(2) by the half-life to get the decay constant k, then convert k to the periodic rate with r = 1 − e−k.
Identify h, the time it takes the quantity to fall to exactly half its starting value.
Compute k = ln(2) / h. ln(2) is a fixed constant, approximately 0.6931.
Compute r = 1 − e−k to get the periodic decay rate as a decimal.
r applies per one unit of whatever time unit the half-life was measured in, such as days, hours, or cycles.
A chemical reagent's potency has a measured half-life of 15 days under storage conditions. Finding the daily decay rate tells a lab how much of the reagent's strength is lost per day of shelf time.
The reagent's potency has a continuous decay constant of 0.0462 and loses 4.52% of its remaining strength per day, a small daily figure that still halves the reagent's potency after 15 days, exactly matching the half-life it was derived from.
Two forms of the decay rate exist because decay gets modeled two different ways: period by period, or continuously. The discrete rate r fits calculators and schedules built around whole periods, such as a year, a week, or a dosing interval, where the natural question is "what percentage is lost each period?"
The continuous decay constant k fits a different kind of tool: one built on calculus, where the rate of loss is defined at every instant rather than counted in steps. This is the form that shows up in the differential equation dx/dt = −kx, the equation underlying radioactive decay, capacitor discharge, and Newton's law of cooling. The step-based exponential decay calculator on this site's decay page uses r directly, because its inputs, such as a starting value, a rate, and a number of periods, are already period-shaped.
Neither form is an approximation of the other; both describe the identical curve exactly, just parameterized differently. Anyone who has used the main calculator on this site for the growth side of the same mathematics has already seen r used this way. A negative rate there produces exactly the decay this page's k and r both describe, just entered through the discrete door instead of the continuous one.
Knowing which form a source hands you matters. A textbook chapter on nuclear physics will quote k. A financial depreciation schedule will quote r. Converting between the two, rather than mixing them up, is the entire purpose of this calculator.
A decay constant (k) is the continuous-time rate parameter in x(t) = x₀e−kt, describing how fast a quantity shrinks when the loss is applied smoothly rather than in discrete steps. It relates to the periodic decay rate r through k = −ln(1 − r).
A larger k means faster decay: k = 0.10 shrinks a quantity noticeably quicker than k = 0.01 over the same stretch of time.
You convert a periodic decay rate r to a continuous decay constant k with k = −ln(1 − r), and you convert back with r = 1 − e−k. A 10% periodic decay rate becomes a decay constant of −ln(0.90) = 0.1054.
The calculator on this page performs both directions instantly, and also derives either one from a half-life or from two measured values.
You find a decay rate from a half-life by first computing the decay constant k = ln(2) / h, then converting it to the periodic rate with r = 1 − e−k. A half-life of 8 time units gives k = ln(2) / 8 = 0.0866 and r = 1 − e−0.0866 = 8.30% per unit.
Use "From Half-Life" mode above to run this conversion for any half-life value.
You find a decay constant from two measurements with k = −ln(x(t) / x₀) / t, where x₀ and x(t) are the values at the start and end of the elapsed time t. Measuring 900 units falling to 600 over 5 time units gives k = −ln(600 / 900) / 5 = 0.0811.
This is the only method that needs no assumed rate at all, just two readings and the time between them.
The decay rate r and decay constant k describe the same decay through two different mechanics, r is the fraction lost per whole period, while k is an instantaneous rate applied continuously, so their numeric values are close but not identical. A 10% periodic rate corresponds to a decay constant of 0.1054, not 0.10, because continuous compounding needs a slightly smaller instantaneous rate to match the same one-period result.
The gap grows with the size of the rate: at small rates like 2% the two numbers are nearly interchangeable, but at 50% they diverge sharply.
No. A 5% decay rate is not the same number as its decay constant; r = 0.05 corresponds to k = −ln(0.95) = 0.0513, a slightly larger figure. Treating the two as interchangeable is a common source of small errors in half-life and dosing-style calculations.
The difference is under 3% at this rate, but it compounds over many periods and grows larger at higher rates, so the two symbols are worth keeping distinct.
The continuous decay constant is used wherever decay is modeled with calculus rather than counted in discrete steps. Radioactive decay physics, pharmacokinetics, and differential-equation models of cooling or discharge all use k. Periodic rate calculators, by contrast, use r because their inputs already arrive as whole periods: a year, a week, a billing cycle.
Either form describes the same underlying curve; the choice is about which unit the surrounding calculation is already built on.