Half-life calculator

First-order means the rate of loss is proportional to the amount present, so the quantity falls fastest at the start and slows as it shrinks. Solving that gives an exponential: the fraction remaining after an elapsed time is one half raised to the power of that time divided by the half-life. The half-life therefore fixes the time axis and nothing whatsoever about the shape of the curve. Ten units and ten million units lose the same proportion over the same time, which is the property that makes the model usable without knowing the starting amount in advance, and also the assumption that fails most often in a real system.

There is a second reason the ratio is what matters. Because the exponent is a ratio, the result is dimensionless, so a half-life and an elapsed time in the same unit produce a number that cannot be wrong by a unit conversion. That safety disappears the moment the two are in different units, and a half-life quoted in days entered against a time in hours is the most common fault across this whole family of calculators. It produces no message and a perfectly plausible number. The guide to first-order elimination and half-life arithmetic works through the same model at greater length, and the calculators around it on this site take the fraction as the target instead of the time.

The exponential can be written as one half raised to the power of time over half-life, or as e raised to the power of minus k t. They are the same curve, and the bridge between them is the elimination rate constant, which is the natural log of two divided by the half-life. Neither form is the correct one and the other a mistake; they suit different questions. The power-of-a-half version is easy to check against whole half-lives, because one, two and three halvings give exactly one half, one quarter and one eighth with no approximation at all. The exponential version is easier to differentiate, to integrate and to combine.

The rate constant is the quantity that behaves consistently across substances in a way the half-life does not, which is why it turns up in so many of the calculators on this site. Converting between the two is a single division in either direction. The elimination rate constant calculator goes from a half-life to a rate constant, and the half-life from a rate constant calculator goes back the other way. Reporting a half-life and a rate constant for the same process is not a contradiction in any way, and a disagreement between the two on the same page is a unit error before it is anything else.

A measured half-life belongs to a measured situation and carries everything that situation happened to contain. It belongs to one substance, in one salt form, by one route, in one species, in one matrix, measured by one method, over one interval. It already contains the effect of distribution, of binding and of clearance, folded together into a single number, and separating them apart again is not something a calculator can do. The half-life quoted for one of those conditions is not automatically the half-life for any of the others, which is why reaching for a figure from the wrong context is not a small error.

So be clear about what this page is. It takes a half-life that you supply and manipulates it. It supplies none, for anything, and it holds no table of values it could draw on, because a table of half-lives with their measurement conditions stripped off is precisely the thing that gets misapplied later. If a half-life is needed, the honest place to find one is the source that measured it, cited, and where no citable source exists for the question being asked then the correct answer is that the value is not knowable from here. The effective half-life calculator exists for the case where a rate constant and an interval are in hand, and the site disclaimer is the shortest statement of what all of this is not.

A one-compartment first-order model has no absorption phase, no distribution into tissues, no binding, no saturable pathway, no metabolites and no route of removal other than the one it names. Real concentration time curves after a real administration are usually multi-exponential: a fast phase that reflects distribution away from the site and a slow phase that reflects removal. Fitting a single exponential to that curve produces an apparent terminal half-life which is a property of the fit and of the window chosen to make it, not a constant of the substance, and two analyses of the same data can disagree about it.

The steady state extension carries assumptions of its own. The accumulation factor treats input as instantaneous and repetition as unending, so the peak to trough ratio it produces is a statement about a geometric decay rather than about anything that was measured. Two related tools set the shape out without changing the model: the accumulation factor calculator isolates the factor itself, and the time to steady state calculator answers how many intervals it takes to approach the asymptote. The guide to accumulation and steady state sets the arithmetic against what it does and does not model.