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Formulas for half-life in exponential decay
Main article: Exponential decay
An exponential decay process can be described by any of the following three equivalent formulas:
where
N0 is the initial quantity of the substance that will decay (this quantity may be measured in grams, moles, number of atoms, etc.),
N(t) is the quantity that still remains and has not yet decayed after a time t,
t1 / 2 is the half-life of the decaying quantity,
τ is a positive number called the mean lifetime of the decaying quantity,
λ is a positive number called the decay constant of the decaying quantity.
The three parameters t1 / 2, τ, and λ are all directly related in the following way:
where ln(2) is the natural logarithm of 2 (approximately 0.693).
[show]Click "show" to see a detailed derivation of the relationship between half-life, decay time, and decay constant.
By plugging in and manipulating these relationships, we get all of the following equivalent descriptions of exponential decay, in terms of the half-life:
Regardless of how it's written, we can plug into the formula to get
N(0) = N0 as expected (this is the definition of "initial quantity")
as expected (this is the definition of half-life)
, i.e. amount approaches zero as t approaches infinity as expected (the longer we wait, the less remains).
[edit]Decay by two or more processes
Some quantities decay by two exponential-decay processes simultaneously. In this case, the actual half-life T1/2 can be related to the half-lives t1 and t2 that the quantity would have if each of the decay processes acted in isolation:
For three or more processes, the analogous formula is:
For a proof of these formulas, see Decay by two or more processes.
[edit]Examples
Main article: Exponential decay--Applications and examples
There is a half-life describing any exponential-decay process. For example:
The current flowing through an RC circuit or RL circuit decays with a half-life of RCln(2) or ln(2)L / R, respectively.
In a first-order chemical reaction, the half-life of the reactant is ln(2) / λ, where λ is the reaction rate constant.
In radioactive decay, the half-life is the length of time after which there is a 50% chance that an atom will have undergone nuclear decay. It varies depending on the atom type and isotope, and is usually determined experimentally.
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