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Here stands for concentration in molarity (mol · L −1), for time, and for the reaction rate constant. The half-life of a first-order reaction is often expressed as t 1/2 = 0.693/k (as ln(2)≈0.693).
In this situation it is generally uncommon to talk about half-life in the first place, but sometimes people will describe the decay in terms of its "first half-life", "second half-life", etc., where the first half-life is defined as the time required for decay from the initial value to 50%, the second half-life is from 50% to 25%, and so on. [7]
As useful rules of thumb, a first-order reaction with a rate constant of 10 −4 s −1 will have a half-life (t 1/2) of approximately 2 hours. For a one-step process taking place at room temperature, the corresponding Gibbs free energy of activation (Δ G ‡ ) is approximately 23 kcal/mol.
where / is the combined or total half-life for the process, and are so-named partial half-lives of corresponding processes. Terms "partial half-life" and "partial mean life" denote quantities derived from a decay constant as if the given decay mode were the only decay mode for the quantity.
Since the reaction rate determines the reaction timescale, the exact formula for the Damköhler number varies according to the rate law equation. For a general chemical reaction A → B following the Power law kinetics of n-th order, the Damköhler number for a convective flow system is defined as:
Using the Eyring equation, there is a straightforward relationship between ΔG ‡, first-order rate constants, and reaction half-life at a given temperature. At 298 K, a reaction with ΔG ‡ = 23 kcal/mol has a rate constant of k ≈ 8.4 × 10 −5 s −1 and a half life of t 1/2 ≈ 2.3 hours, figures that are often rounded to k ~ 10 −4 s ...
The rate of a first-order reaction depends only on the concentration and the properties of the involved substance, and the reaction itself can be described with a characteristic half-life. More than one time constant is needed when describing reactions of higher order.
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