# What is half-life

1. Jul 23, 2014

### Greg Bernhardt

Definition/Summary

The half-life, $t_{1/2}$, of an inverse exponential process (an exponential decay) is the time taken for the amount to reduce by one-half. It is constant.

Processes with a half-life include radioactive decay, first-order chemical reactions, and current flowing through an RC electrical circuit.

The half-life divided by the (natural) logarithm of 2 is the mean lifetime, ${\tau}$. It is the time taken for the amount to reduce by a factor e (ie 2.718...). It is the inverse of the decay constant, ${\lambda}$, also referred to as the decay rate, or probability per unit time of decay.

Equations

Inverse exponential process (exponential decay) with decay constant $\lambda$:

$$A = A_0e^{-\lambda t}$$

$$\tau\ =\ \frac{1}{\lambda} \ =\ \frac{t_{1/2}}{\log 2}$$

where $\log$ denotes the natural logarithm.

Half-life:

$$t_{1/2}\ =\ \frac{log2}{\lambda} \ = \ \tau\ \log 2$$

For decay of the same population by two or more simultaneous inverse exponential processes with decay constants $\lambda_1,\cdots,\lambda_n$:

$$\lambda\ =\ \lambda_1\ +\ \cdots\ +\ \lambda_n$$

$$\frac{1}{\tau}\ =\ \frac{1}{\tau_1}\ +\ \cdots\ +\ \frac{1}{\tau_n}$$

$$\frac{1}{t_{1/2}}\ =\ \frac{1}{\left(t_1\right)_{1/2}}\ +\ \cdots\ +\ \frac{1}{\left(t_n\right)_{1/2}}$$

Extended explanation

The quantity which reduces is the expectation value of the quantity of radioactive material.

RC circuits:

The flow of current discharged from a capacitor through a resistor (an RC circuit) is an inverse exponential process with mean lifetime (time constant) equal to the resistance times the capacitance: $\frac{1}{\lambda}\ =\ \tau\ =\ RC$.

Other meanings:

Technically, a half-life could be defined for any process, at each stage of that process, but it would not be constant …

it is only for an inverse exponential process that the half-life is the same at each stage …

and so it is only for an inverse exponential process that a half-life for a process can be defined.

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