# Thoughts on serial dilutions

An adventurer has been bitten by a poisonous creature and has sought refuge in a shack. The adventurer is too weak to move from the shack and will die unless an anti-toxoid is administered. Luckily for the adventurer, the shack seems to be someone’s old abandoned home lab. One of the vials in the lab is precisely the quantity of anti-toxoid that the adventurer needs. Unfortunately, the anti-toxoid is dissolved together with a chemical preservative which the adventurer is highly allergic to. This preservative will kill the adventurer unless its levels are reduced by at least 150 times.

Besides 5 ml of anti-toxoid, the adventurer has also found reels of dialysis tubing and 25 ml of water. The adventurer thinks to himself: “I will arbitrarily divide the 25 ml of water into r portions and then use these portions to serially dialyze the anti-toxoid. If r is large enough, I will surely bring the lethal chemical preservative down by at least 150 times.” Is the adventurer correct?

Let us generalize with a starting sample volume Vs and a fixed dialysis buffer volume Vd with which to perform dialysis. If Vs is dialyzed against Vd, we get a dilution factor D where

If we split the dialysis buffer into r portions of Vd/r each, we will get:
Vs = sample volume
Vd = total volume of dialysis buffer
r = number of dialysis rounds each performed with Vd/r volume of dialysis buffer
Does our dilution factor improve indefinitely with increasing r or is there a theoretical limit? We define the ratio
giving us
We notice at this point that the above equation is similar to the derivation of the mathematical constant e. Consistent with intuition, the theoretical limit as r goes to infinity is given by
In short, most unfortunately for the adventurer, the best dilution factor he can hope for is e5 which works out to ~148.4, just below the 150-fold dilution required and this is assuming that he can infinitely divide his water for dialysis.
Addendum: My wife was upset at the implied demise of the adventurer. What she did not know was that a magical unicorn teleported from a parallel universe and used its healing spittle to save the adventurer.
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