# 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? • 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 $t_{1/2}=\ln(2)/k$. Aug 23, 2022 at 13:33
• @porphyrin So they are the same except for a factor of $\ln{2}$, right?
– Karsten
Aug 23, 2022 at 13:45
• 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 $\approx 37$ %. Aug 23, 2022 at 13:51

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: 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).

• 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? Aug 25, 2022 at 9:48
• And yes, there are definitely tricky terms all around. Aug 25, 2022 at 9:52
• 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. Aug 25, 2022 at 9:55