# Half-life

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For any reactant subject to first-order decomposition, the amount of time needed for one half of the substance to decay is referred to as the half-life of that compound. Although the term is often associated with radioactive decay, it also applies equally to chemical decomposition, such as the decomposition of azomethane (CH3N=NCH3) into methane and nitrogen gas. Many compounds decay so slowly that it is impractical to wait for half of the material to decay to determine the half-life. In such cases, a convenient fact is that the half-life is 693 times the amount of time required for 0.1% of the substance to decay. Using the value of the half-life of a compound, one can predict both future and past quantities.

Note: The approximation ${\displaystyle \ \ln(2)\approx 0.693\ }$ is used in this article.

## Mathematics

The future concentration of a substance, C1, after some passage of time ${\displaystyle \Delta t}$, can easily be calculated if the present concentration C0 and the half-life th are known:

${\displaystyle C_{1}=C_{0}\left({\frac {1}{2}}\right)^{\frac {\Delta t}{t_{h}}}}$

For a reaction is the first-order for a particular reactant A, and first-order overall, the chemical rate constant for the reaction k is related to the half-life by this equation:

${\displaystyle t_{h}={\frac {0.693}{k}}}$

${\displaystyle t_{avg}=0.693\ t_{h}}$.
${\displaystyle C_{1}=C_{0}\ e^{-{\frac {\Delta t}{t_{avg}}}}}$