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(please gently redirect me if it's a wrong SE and don't question-ban me, I mean no harm)

Paper products have a lag time of 1.5–2.0 years before methane production, a ‘half-life’ of 10–20 years, [...]

(from here, p. 150)

Does it mean that it takes 20–40 years for a piece of paper or cardboard to fully decompose in a landfill? Apparently, not, the two halves are not identical. What is a "full-life" then? Can it be calculated with a "half-life" estimate at hand?

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It is asymptotic. Every 10-20 years you have half of what you had going into that time period. So in 20-40 years you get 1/4. Half of half.. In 30-60 years, you have 1/8th. Half of a half of a half. If 100% = full then mathematically it would take infinite time because you can never reach zero. You can certainly calculate how long it would take to reach a set percentage however so there is no such thing as a full-life.

The math is just like that for time constants except instead of e which gives you about 63% per iteration, you use 0.5 for half

$A = A_02^{-t/t_{1/2}}=A_0e^{-t/\tau}$

  • $A$ final amount
  • $A_0$ original amount
  • $t$ time elapsed
  • $\tau$ time constant (mean lifetime), $t_{1/2}$ half-life

Every $t$ that is a multiple of half-life $t_{1/2}$ multiplies another factor $1/2$ to the original value.

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  • $\begingroup$ It's interesting because I would never imagine that paper can decompose infinitely. Suppose, the percentage is 99% (it's as good as 100%). What would it come to? $\endgroup$
    – Sergey Zolotarev
    Commented Dec 21, 2021 at 3:06
  • $\begingroup$ @SergeyZolotarev Well it's mathematically like that. In reality there are only finite number of molecules but there are a great number of them so you can keep subdividing very far so it would take a very long time until you get 4, then 2, then 1, then zero. $\endgroup$
    – DKNguyen
    Commented Dec 21, 2021 at 3:07
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    $\begingroup$ The behaviour of large numbers of particles is predictable (through statistical processes such as the central limit theorem). Conversely, the behaviour of one particle is totally random, so it's impossible to predict when that last particle will decay to zero. $\endgroup$
    – Toby Mak
    Commented Dec 21, 2021 at 3:10
  • $\begingroup$ @TobyMak Yes, half-life is statistical. 3.32 half-life time constants for 90% reduction. $\endgroup$
    – DKNguyen
    Commented Dec 21, 2021 at 3:11
  • $\begingroup$ For the mathematical details: Central Limit Theorem for exponential distribution $\endgroup$
    – Toby Mak
    Commented Dec 21, 2021 at 3:13

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