Half-Life from Rate Constant Calculator

A rate constant is a compressed description, and unpacking it is not this page job. To produce one, somebody had to decide which pool of material was being counted, over which stretch of time, and by which route it was leaving. Those choices are baked into the number before it ever reaches a calculator, and dividing ln two by it cannot undo any of them. The arithmetic here is honest about the limit: it takes k at face value and returns the time at which the modelled quantity would halve, which is the only thing a number of that shape can mean.

It follows that k and the half-life are equally fundamental and equally conditional. Neither is measured in a way that settles the question. One is convenient for a spreadsheet and the other is convenient in conversation, and each is a summary of a curve rather than a description of its shape. The guide to first-order elimination mathematics sets out why that curve is a reasonable first approximation and where the reasonable part stops.

The function is a single division, so every pair of inputs maps to exactly one half-life and that output is the only thing it can produce. In particular there is no error band, no sensitivity figure and no allowance for the possibility that k itself was measured badly. A careless k produces a confident half-life out the other side, which is a property of arithmetic rather than a defect of this page. The numbers are exactly as good as the inputs, and the page cannot tell you how good those were.

A useful discipline is to ask what the answer would have to be for the input to be wrong. A rate constant per hour that produces a half-life of a small fraction of an hour is either a genuinely very fast process or a value that belongs to a different time unit. Both are possible and neither can be resolved from inside the calculator, which is exactly why the round trip through the elimination rate constant calculator is worth doing. It makes the implied half-life explicit so it can be checked against whatever the source actually said.

There is no unit selector on this page, and that is a consequence of the relationship rather than an oversight. The unit of the output is not chosen; it is whatever time unit cancels the per-time unit you typed. A rate constant per day therefore returns a half-life in days. That is correct, and it is also the reason the answer so often surprises someone who entered a number they were thinking of as hours.

The fix is to write the unit beside the answer before doing anything else with it. Dividing a dimensionless number by a per-hour number and then reporting the quotient without a unit turns a working calculation into a guess. The time unit converter handles the conversions themselves, and fixing one unit at the start of a chain of work is always cheaper than tracing a factor of sixty through three later pages.

A half-life is not a property of a substance in isolation. It is a property of a substance in a particular system: a particular route of administration, a particular distribution into and out of tissues, particular binding, and in any real system particular variation between individuals and between days. Change any of those and the figure moves. The number most often quoted is usually measured late in a decline, which is not the same quantity as the early decline, and reporting one figure for both is an approximation that nobody flags because it is so widely done.

This page carries no half-life for any substance and does not intend to. There is no average that would be safe to offer, and inventing one would be worse than leaving the gap, because a plausible-looking wrong half-life travels silently into every calculation that uses it. If a value is needed, take it from a labelled source and record where it came from. The single-dose version of the same mathematics sits on the original half-life page , and the reason this site declines to approximate these figures is set out in the site disclaimer page .