Half-life vs. relaxation time vs. average lifetime

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?


[![](https://i.stack.imgur.com/dQ4O0.jpg)](https://i.stack.imgur.com/dQ4O0.jpg)




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](http://hyperphysics.phy-astr.gsu.edu/hbase/Nuclear/halfli2.html) 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](https://i.stack.imgur.com/wkkPk.gif)](https://i.stack.imgur.com/wkkPk.gif)
> 
> 
> 


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](https://edu.gcfglobal.org/es/estadistica-basica/media-mediana-y-moda/1/) because they look very similar (in both languages).


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