Chemistry Solvent Extraction From Jcost

Solvent extraction and the cost function

Liquid-liquid extraction is a workhorse separation technique. A chemist shakes a mixture of two immiscible liquids, usually water and an organic solvent, and an analyte distributes itself between the layers. The distribution ratio D is the concentration of the analyte in the organic layer divided by its concentration in the aqueous layer. A high D means the analyte prefers the solvent; a low D means it stays in the water. The practical goal is to choose a solvent system that pushes D well above 1, so that a single extraction removes most of the analyte.

The classical theory of extraction is thermodynamic. The Nernst distribution law, stated by Walther Nernst in 1891, says that at a given temperature the ratio of concentrations in the two phases is a constant, independent of the total amount of analyte. That constant is the partition coefficient. Real systems deviate from the ideal law when the analyte dimerizes, ionizes, or complexes in one phase, and chemists account for those effects with correction terms. The subject has a rich history: the Manhattan Project relied on solvent extraction to purify uranium and plutonium, and the technique remains central to hydrometallurgy, pharmaceutical purification, and nuclear fuel reprocessing.

In Recognition Science, the framework models the extraction process with its cost function J(x) = (x + 1/x)/2 - 1, where x is the ratio m/e of two positive quantities. The framework's library proves three general facts about this function: it vanishes when m = e, it is nonnegative for positive inputs, and the quantity phi - 3/2 is positive, where phi is the golden ratio. The first two facts say that the cost is zero when the two quantities are equal and never negative otherwise. The third fact, that phi is greater than 1.5, is a numerical threshold that the framework uses as a marker.

The framework's research note connects these facts to extraction by identifying m and e with concentrations in the organic and aqueous phases. At the golden ratio D = phi, approximately 1.618, the cost function takes the value about 0.118, which the note interprets as 11.8 percent of the analyte remaining in the aqueous phase and 88.2 percent extracted. The framework labels this the optimal distribution ratio. This is a research note, not a proved theorem. The formal module proves the three general facts about the cost function, but it does not prove that the golden ratio is the optimal extraction ratio, because the module defines the cost as J(m/e) without specifying what m and e mean in chemical terms.

The distinction matters. The machine-checked library proves that the cost function has certain properties for any positive inputs. What would turn those properties into a theorem about extraction is a definition of m and e in the chemistry's own terms, a definition the module does not provide. The framework's contribution here is a template: it shows what a cost-based account of extraction would look like, and it proves the mathematical facts that such an account would rely on. The chemistry itself remains a research program, not a formal result.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The golden ratio is the measured optimal distribution ratio for any real solvent system. The framework proves that extraction is efficient at D = phi. The module establishes any chemical fact beyond the three general properties of the cost function.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Chemistry/SolventExtractionFromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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