# Half-Life

The half-life is the time put by a substance (Médicament, radioactive core, or others) to lose half of its pharmacological, physiological or radioactive activity. In particular, the half-life is the time necessary so that a radioactive element loses half of its activity by natural Désintégration.

## In biology and pharmacology

In Pharmacology, the half-life indicates by extension the time necessary so that the quantity of a substance contained in a biological system is decreased by half of its initial value (for example content of a drug in the Blood plasma).

This parameter slightly varies from one individual to another, according to the process of elimination and relative operation at the individual.

In practice, it is considered that a Médicament does not have any more a pharmacological Effet after five to seven half-lives.

## In nuclear physics

In Nuclear physics, the half-life, sometimes called radioactive Half-life, for a radioactive Isotope , is the duration during which its radioactive Activité decrease of half for a Mode of disintegration given. The half-life term does not mean that the activity of radioactive isotope is null at the end of a time equal to 2 half-life, since the activity is then only reduced to 25% of the initial activity (see the table of decrease of the activity). Actually, the activity has is worth, after N half lives, $A = \left\{A_0 \ over 2^n\right\}$, so that the activity is never mathematically null.

It is a property Statistique: lasted at the conclusion which the core of a radioactive Atome would have a chance on two to disintegrate according to the mode of disintegration concerned if this mode were alone. This property on an atomic nucleus scale does not depend on the environmental conditions, such as temperature, pressure, fields, but only of the Isotope and the mode of disintegration considered.

The half-life can vary considerably from one isotope to another, since a fraction of a second to million or billion years (see figure opposite).

The Activité of a given number of atoms of a radioactive isotope is proportional to this number and inversely proportional to the half-life of the isotope.

The radioactive decrease is a process of Poisson. The probability of disintegration independent of passed and the future. For the derivation of the law of probability it is necessary to introduce a scale of time proportional to the half-life. For that one introduces the cumulative probability.
U (T) =Prob {T>t}

Probability of a Disintegration after a time T. Since disintegration is independent of the moment T, U (T) is the conditional probability that there is a disintegration at the moment t+s knowing that there is no disintegration at the moment T U (t+s)/(U (S)). Thus cumulative probability satisfied this equation:

U (t+s) =U (T) U (S)

In the case of a measurable function the single solution is the exponential function. That is to say a unit made up of NR elements of which the number decrease with time according to a rate of decrease noted λ. The equation of this dynamic Système (cf law of exponential Decay) is written:

$\ frac \left\{DNN\right\} \left\{dt\right\} = - \ lambda NR$

where λ is a positive number, with an initial quantity $N \left(t=0\right) =N_0$.

If one carries out a Résolution of the differential equations with constant coefficients, then the solution of such an equation is the function defined by:

$NR \left(T\right) = N_0 \ cdot e^ \left\{- \ lambda T\right\} \,$

This decreasing function reaches a value equal to half of the initial quantity $N_0$ at the end of a certain noted duration $t_ \left\{1/2\right\}$. While simplifying, one obtains then:

$e^ \left\{- \ lambda t_ \left\{1/2\right\}\right\} = \ frac \left\{N_0\right\} \left\{2N_0\right\} = \ frac \left\{1\right\} \left\{2\right\}$
from where one deduces easily
$t_ \left\{1/2\right\} = \ frac \left\{\ ln 2\right\} \left\{\ lambda\right\}$
This duration $t_ \left\{1/2\right\}$ is called the half-life elements of the unit.

### Remarks

In the Radionucléide S where particles are transformed by Radioactivité into another particle, the number of initial particles decrease exponentially according to time.

It is frequent that a radioactive isotope comprises several modes of disintegration, or although it belongs to a radioactive Chain decay. For these cases, the simple exponential law of radioactive decrease does not apply any more, and the decrease of the activity of the substance is then even slower.

## See too

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