Terra Somnia

Abstract — When a well-mixed vessel is repeatedly partially emptied and refilled with a pure diluent, the fraction of the original component decays geometrically. This note derives the closed form for the remaining fraction after N cycles and shows how it generalizes to non-constant emptying fractions. The result matches the classic remove-and-replace / serial-dilution formula used in mixing problems.

Model and derivation — Let p_0 \in [0,1] be the initial fraction of component B in a perfectly mixed tank. In each cycle you remove a fixed fraction r of the current mixture and top up with pure A. The fraction of B you retain each time is f = 1 – r. Because the mixture is uniform, every cycle multiplies the fraction of B by f, giving the recurrence p_{n+1} = f\,p_n and the closed form:

p_N = p_0 (1 – r)^N

This is a geometric progression with common ratio f. For example, with p_0 = 0.5 and r = 0.5 (remove half, refill with A), we get p_1 = 0.25 and p_2 = 0.125, i.e. one eighth of the original fraction. With p_0 = 0.10 and r = 0.90 (leave 10%), we get p_1 = 0.01, i.e. one hundredth of the original fraction. These are the standard outcomes of successive remove-and-replace steps / serial dilutions.

Extensions — If the removed fraction varies by cycle, (r_1, r_2, \dots, r_N), the retained fractions are f_k = 1 – r_k and

p_N = p_0 \prod_{k=1}^{N} f_k

This is still a multiplicative (geometric) decay and the discrete analogue of the serial-dilution ladder, where equal dilution factors yield logarithmic spacing of concentration.

Conclusion — Repeated partial emptying with perfect mixing is governed by a one-line law: fraction left after N cycles equals initial fraction times the retained fraction to the N^{\text{th}} power. It covers constant- and variable-step protocols and aligns with textbook remove-and-replace / serial-dilution calculations.

Author: Terra somnia (SPBG)