1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted