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On page 25 of Ángel González Ureña's Cinética química (2001), appear the terms half-life time (of an intermediate) and relaxation time, but it refers to both of them with the letter $\tau$. Why is this?

Also, it defines half-life time as $\tau=\frac{c}{v}$, so when is first order $\tau=\frac{c}{kc}=\frac{1}{k}$, but this is the same value that for relaxation time (the time for the decreasing then initial concentration in $\frac1e$). So, are they essentialy the same? Can that formula be used in both cases? What I'm missing here?

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    $\begingroup$ Your textbook is confusing, if you define the decay as $\sim\exp(-kt)$ then we normally define $\tau=1/k$ where $\tau$ is called the lifetime or relaxation time. The half-life is $t_{1/2}=\ln(2)/k$. $\endgroup$
    – porphyrin
    Aug 23, 2022 at 13:33
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    $\begingroup$ @porphyrin So they are the same except for a factor of $\ln{2}$, right? $\endgroup$
    – Karsten ♦
    Aug 23, 2022 at 13:45
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    $\begingroup$ They do differ by that factor, but they represent different things, $t_{1/2}$ is the time for the population to decease to $1/2$ its initial value , $\tau$ for the population to decrease to $1/e$ or $\approx 37$ %. $\endgroup$
    – porphyrin
    Aug 23, 2022 at 13:51

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The confusion is between the half-life ($t_\frac{1}{2}$) and the average life time ($\tau$). These terms seem to originate from the description of the first order decay of nuclei.

For example, this source states:

Note that the radioactive half-life is not the same as the average lifetime, the half-life being 0.693 times the average lifetime.

And they illustrate this with the following set of expressions:

enter image description here

Why have two separate definitions? One (half-life) is the median time for a particle to decay, the other (average life time) is the average. For a symmetric distribution, these would be the same, but exponential decay is not symmetric.

Here is a juxtaposition of Spanish and English terms to the best of my knowledge:

  • Periodo de semirreacción: half-life of the reaction
  • Tiempo de relajación: relaxation time
  • Tiempo de vida media: average life time (of a particle, intermediate or transition state)

And finally:

  • Media: mean
  • Mediana: median

The last two seem very tricky to me because they look very similar (in both languages).

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  • $\begingroup$ Ok, thanks for the answer. I really appreciate it. But I'm sorry, I continue with the doubt of why the relaxation time equals the average lifetime of a nucleus decay. Are they just different definitions for different concepts but they rather equal in value? $\endgroup$
    – Jorge Bonifaz
    Aug 25, 2022 at 9:48
  • $\begingroup$ And yes, there are definitely tricky terms all around. $\endgroup$
    – Jorge Bonifaz
    Aug 25, 2022 at 9:52
  • $\begingroup$ Maybe they only equal for first order cases, but radioactive decay only makes sense in this order, we can only change the order for relaxation time instead. $\endgroup$
    – Jorge Bonifaz
    Aug 25, 2022 at 9:55

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