Biological half-life

The half-life of a quantity subject to exponential decay is the time required for the quantity to fall to half of its initial value. The concept originated in the study of radioactive decay, but it now also occurs in many other fields.

The table at right shows the reduction of the quantity in terms of the number of half-lives elapsed.

Derivation
Quantities that are subject to exponential decay are commonly denoted by the symbol N. (This convention suggests a decaying number of discrete items. This interpretation is valid in many, but not all, cases of exponential decay.)  If the quantity is denoted by the symbol N, the value of N at a time t is given by the formula:


 * $$N(t) = N_0 e^{-\lambda t} \,$$

where


 * $$N_0$$ is the initial value of N (at t=0)
 * &lambda; is a positive constant (the decay constant).

When t=0, the exponential is equal to 1, and N(t) is equal to $$N_0$$. As t approaches infinity, the exponential approaches zero.

In particular, there is a time $$t_{1/2} \,$$ such that:


 * $$N(t_{1/2}) = N_0\cdot\frac{1}{2} $$

Substituting into the formula above, we have:


 * $$N_0\cdot\frac{1}{2} = N_0 e^{-\lambda t_{1/2}} \,$$


 * $$e^{-\lambda t_{1/2}} = \frac{1}{2} \,$$


 * $$- \lambda t_{1/2} = \ln \frac{1}{2} = - \ln{2} \,$$


 * $$t_{1/2} = \frac{\ln 2}{\lambda} \,$$

Thus the half-life is 69.3% of the mean lifetime.

Examples


The generalized constant &lambda; can represent many different specific physical quantities, depending on what process is being described.


 * In an RC circuit or RL circuit, &lambda; is the reciprocal of the circuit's time constant &tau;. For simple RC and RL circuits, &lambda; equals $$RC$$ or $$L/R$$, respectively.
 * In first-order chemical reactions, &lambda; is the reaction rate constant.

Decay by two or more processes
Some quantities decay by two processes at once (see Exponential decay). In a fashion similar to the previous section, we can calculate the new total half-life $$T_{1/2}$$ and we'll find it to be:


 * $$T_{1/2} = \frac{\ln 2}{\lambda _1 + \lambda _2} \,$$

or, in terms of the two half-lives


 * $$T_{1/2} = \frac{t _1 t _2}{t _1 + t_2} \,$$

where $$t _1$$ is the half-life of the first process, and $$t _2$$ is the half life of the second process.