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How can half-life be calculated?

1 Answer

Nov 1, 2015

If the decay is exponential, i.e. $x = x_{0} e^{- k t}$ $x=x_{0}e^{-kt}$ , then the half-life is given by $\frac{ln (2)}{k}$ $ln(2)/k$ .

Explanation:

Let $τ$ $tau$ be the half-life. $τ$ $tau$ is the time taken for $x_{0}$ $x_0$ to reach $\frac{x_{0}}{2}$ $x_0/2$

$\frac{x_{0}}{2} = x_{0} e^{- k τ}$ $x_0/2=x_{0}e^{-ktau}$

$\frac{1}{2} = e^{- k τ}$ $1/2=e^{-ktau}$

$2 = e^{k τ}$ $2=e^{ktau}$

$ln (2) = k τ$ $ln(2)=ktau$

$τ = \frac{ln (2)}{k}$ $tau=ln(2)/k$

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