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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}
Mean lifetime:
\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
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: \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.
* 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!
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}
Mean lifetime:
\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
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: \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.
* 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!