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What is half-life

  1. Jul 23, 2014 #1

    The half-life, [itex]t_{1/2}[/itex], 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, [itex]{\tau}[/itex]. 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, [itex]{\lambda}[/itex], also referred to as the decay rate, or probability per unit time of decay.


    Inverse exponential process (exponential decay) with decay constant [itex]\lambda[/itex]:

    [tex]A = A_0e^{-\lambda t}[/tex]

    Mean lifetime:

    [tex]\tau\ =\ \frac{1}{\lambda} \ =\ \frac{t_{1/2}}{\log 2}[/tex]

    where [itex]\log[/itex] denotes the natural logarithm.


    [tex]t_{1/2}\ =\ \frac{log2}{\lambda} \ = \ \tau\ \log 2 [/tex]

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

    [tex]\lambda\ =\ \lambda_1\ +\ \cdots\ +\ \lambda_n[/tex]

    [tex]\frac{1}{\tau}\ =\ \frac{1}{\tau_1}\ +\ \cdots\ +\ \frac{1}{\tau_n}[/tex]

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

    Extended explanation

    Radioactive decay:

    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: [itex]\frac{1}{\lambda}\ =\ \tau\ =\ RC[/itex].

    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.

    * This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
  2. jcsd
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