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This is the form of the equation that is most commonly used to describe exponential decay. Any one of decay constant, mean lifetime, or half-life is sufficient to characterise the decay. The notation λ for the decay constant is a remnant of the usual notation for an eigenvalue. In this case, λ is the eigenvalue of the negative of the differential operator with N (t) as the corresponding ...
Half-life is constant over the lifetime of an exponentially decaying quantity, and it is a characteristic unit for the exponential decay equation. The accompanying table shows the reduction of a quantity as a function of the number of half-lives elapsed.
Example solution An example solution to the differential equation with initial value V0 and no forcing function is where is the initial value of V. Thus, the response is an exponential decay with time constant τ.
Equations for the approach to steady state Derivation of equations that describe the time course of change for a system with zero- order input and first-order elimination are presented in the articles Exponential decay and Biological half-life, and in scientific literature. [1][7]
The integral solution is described by exponential decay: where N0 is the initial quantity of atoms at time t = 0. Half-life T1/2 is defined as the length of time for half of a given quantity of radioactive atoms to undergo radioactive decay: Taking the natural logarithm of both sides, the half-life is given by
With the decay constant it is possible to calculate the effective half-life using the formula: The biological decay constant is often approximated as it is more difficult to accurately determine than the physical decay constant.
Here the problem with the term partial half-life is evident: after (341+6.60) days almost all the nuclei will have decayed, not only half as one may initially think. Isotopes with significant branching of decay modes include copper-64, arsenic-74, rhodium-102, indium-112, iodine-126 and holmium-164.
A modification, which does not satisfy the general form above, with an exponential cutoff, [10] is In this distribution, the exponential decay term eventually overwhelms the power-law behavior at very large values of .