Chemistry Acid Base Equilibrium From Jcost

Acid-Base Equilibrium

Acid-base equilibrium is the balance between an acid and its conjugate base in solution. The Henderson-Hasselbalch equation, pH = pKa + log([A-]/[HA]), connects the pH of a buffer to its pKa and the ratio of the concentrations of the deprotonated form [A-] to the protonated form [HA]. When the two concentrations are equal, the ratio is 1, the log term vanishes, and pH equals pKa. This is the point of maximum buffering capacity, where the solution resists pH changes most effectively.

The equation was derived in 1908 by Lawrence Joseph Henderson and further developed by Karl Albert Hasselbalch in 1917. It is an approximation that works well for weak acids and bases in their buffering range, typically within one pH unit of the pKa. Outside that range, the assumption that the acid and base concentrations remain close to their formal concentrations breaks down, and the equation becomes less accurate.

In Recognition Science, the framework models this balance by defining a cost function, which assigns a number to the mismatch between a measured value and an expected value. The framework defines domainCost(m, e) = Jcost(m / e), where Jcost(x) = (x + 1/x)/2 - 1. This cost is zero when the measured and expected values are equal, and it is always nonnegative for positive inputs. At the buffer optimum, where [A-]/[HA] = 1, the cost is exactly zero, matching the classical result that pH = pKa.

The framework also defines a canonical threshold of phi - 3/2, which is approximately 0.118. The research note suggests that buffering capacity drops to zero when the cost exceeds this threshold. However, the formal proofs in the module establish only the general properties of the cost function: it vanishes at equilibrium, it is nonnegative, and the threshold is positive. The module does not prove anything specific to chemistry, because the definition of domainCost does not reference acid-base chemistry at all.

What the module actually establishes is three general facts about the cost function. First, domainCost(r, r) = 0 for any nonzero r, meaning the cost is zero when measured equals expected. Second, domainCost(m, e) is nonnegative for positive m and e, meaning the cost never goes negative. Third, the canonical threshold phi - 3/2 is positive. These are proved in the machine-checked library of formal theorems. The connection to acid-base chemistry is a research note, not a proved result.

def domainCost (measured expected : ℝ) : ℝ := Jcost (measured / expected)

theorem domainCost_at_equilibrium (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]

What this page does not claim

The module proves any chemistry-specific result about acid-base equilibrium. The threshold phi - 3/2 is empirically validated as the point where buffering capacity drops to zero. The Henderson-Hasselbalch equation is derived from 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/AcidBaseEquilibriumFromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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