First-Order Elimination Calculator

The most useful consequence of first-order elimination is that the starting amount is not in the fraction at all. It scales the answer and does nothing else, which is why a small sample and a large container decline at the same proportional rate under the same assumptions. The percent remaining calculator is this same calculation with the starting amount divided out, and the two pages return one ratio in different clothing.

The corollary is that a half-life is a statement about proportion, so it cannot be made shorter by starting with more and cannot be made longer by starting with less. If you observe a decline that does not follow that rule, the model is what is wrong, not the arithmetic, and working out which of the two is the case is usually more productive than reaching for a more elaborate formula. It begins by asking a question with a mechanical answer: does the amount lost per unit time depend on how much was there?

The exponent in front of e is a product, and a product is only meaningful when its factors cancel. That single requirement is the reason the unit rule in this family is strict. The rate constant carries inverse time, the elapsed time carries time, and their product is a pure number. Attach a unit to the exponent and every figure below it is wrong by the magnitude of that unit, with no error raised anywhere, because the calculator was asked a well-formed numerical question and answered it perfectly.

Doing the dimensional check by hand takes a couple of seconds and catches essentially every silent failure. A rate constant per hour requires a time in hours. If the source gave days, convert before typing. The time unit converter is the quickest route, and the general practice of settling on one unit before starting a chain of peptide arithmetic is covered in the peptide unit conversion guide .

First-order behaviour means the fraction lost per unit time is constant. It holds while removal is proportional to what is present, which is the usual situation away from saturation. It fails when the removal machinery becomes capacity limited, because there the amount falls by roughly the same absolute quantity per unit time regardless of how much is left, and the curve straightens toward a line. A decline that looks linear in its later portion is telling you something about the mechanism rather than about the calculator.

There is a second and subtler failure. Real decline curves usually contain more than one shape, because material is often lost quickly at first and slowly later as a rapid distribution phase gives way to a slower one. A single rate constant fits one of those phases well and the other badly, and which phase a quoted figure describes is precisely the thing that is usually left unsaid. The guide to first-order elimination mathematics works through where a single exponential is defensible and where it is not.

This page returns two figures and they are not redundant. The fraction remaining is the model output proper: it depends only on k and the elapsed time, and it would be identical for any starting amount at all. The amount remaining is that fraction multiplied by the number typed into the first field, which makes it a statement about one specific starting figure and nothing more general.

Keeping the two apart tends to prevent the commonest downstream error, which is treating a derived absolute value as though it carried information the model does not have. Neither output says why the amount is falling or what the loss would mean. It is a curve, not a mechanism, and it holds no absorption, no distribution, no binding, no metabolite and no person. When what you need is a concentration rather than an amount, the conversion to a volume has to be supplied separately, which is what the molarity calculator exists for.