Elimination Rate Constant Calculator

A half-life and a rate constant are not two facts about a substance. They are two spellings of one curve, and moving between them is a definition rather than a measurement. If the amount falls as the exponential of minus k times t, then setting t equal to the half-life forces that exponential to equal one half, and taking logarithms of both sides gives k times t-half equal to ln two. No property of any substance is consulted to get there. The 0.6931 is not something about the substance. It is the fixed price of measuring decay in halvings instead of in exponents.

Neither form is the more fundamental one, so it does not matter which you happen to hold. What does not survive a careless move is the unit. The same half-life written in hours and the same half-life written in minutes generate rate constants differing by a factor of sixty, both correct, neither comparable with the other. So when a rate constant arrives from a table, read it as a half-life in disguise and settle the time unit before doing anything else with it. The model behind the constant is set out at more length in the guide to first-order elimination mathematics .

The result is a proportionality constant, not an amount per hour. The instantaneous rate of loss at any moment is k multiplied by the amount present, so the loss per hour shrinks as the amount shrinks. That single feature is what makes the decay exponential rather than linear, and it is why a rate constant and a rate get swapped in conversation. A figure quoted in mg per hour changes as the system changes. A figure quoted per hour that stays fixed whatever the amount present is k.

The constant is also only meaningful relative to a stated model and a stated compartment. It says how fast one assumed pool loses material by one assumed route. Two compartments holding the same substance do not share a rate constant, and the fast early decline and the slow late decline that a real curve usually shows are exactly why a single figure is at best an approximation of the whole shape. That is a limit of the model rather than of the arithmetic. The original half-life page places the same constant inside a larger calculation, where those omissions are easier to see.

Because everything downstream in this family multiplies k by a time, the unit of k carries more weight than the unit of an amount ever does. Give the first-order elimination calculator a rate constant per minute and a time in hours, and the exponent comes out as a large number that should have been dimensionless. Nothing errors, because numbers are unaware of their units. The result is simply wrong by a factor of sixty and looks entirely reasonable while it is.

The remedy is unglamorous: settle the time unit once, at the top of the working, and convert before dividing anything. If a source quotes per minute and the working is in hours, the conversion is a multiplication by sixty. The time unit converter performs that conversion, and the wider habit of holding one unit across a whole chain of peptide arithmetic is discussed in the peptide unit conversion guide .

The cheapest available verification is a round trip. Take the k this page returns, push it back through the half-life from rate constant calculator , and the number you started with should reappear. A result that differs by more than rounding points at a unit error rather than at the formula, since the formula has no adjustable parameters that could be set incorrectly.

If a source offers two rate constants, or a rate constant and a half-life that do not agree, that is not a rounding artefact. It is a sign that material is leaving by more than one route, each with its own constant. Two independent first-order losses acting together add their rates, and the effective half-life calculator performs that combination. This site carries no measured rate constant and no measured half-life for any particular substance, and that gap is deliberate rather than an oversight: a plausible average would be wrong in a way nothing downstream could reveal. Measured values belong to a labelled source, not to a calculator.