Chemistry Reaction Calorimetry3 From Jcost

A calorimeter's sensitivity floor

Reaction calorimetry measures the heat released or absorbed by a chemical reaction. A calorimeter is a vessel that tracks temperature change; its sensitivity is the smallest heat pulse it can distinguish from its own thermal noise. At room temperature, that noise is set by Brownian motion. For a measurement lasting a tenth of a second, the noise power is roughly 1.7e-20 watts. A detector that cannot see a signal smaller than its noise floor is blind to it.

The framework's cost function, a measure of how expensive it is to recognize a difference, applies here. The module defines a cost for a calorimeter as J(m/e), where m is the measured heat and e is the expected or reference heat. The cost is zero when the measured and expected values match, and it is never negative for positive inputs. These are general properties of the cost function, proved in the machine-checked library of formal theorems, and they hold for any positive values of m and e.

The module also defines a canonical threshold, phi - 3/2, which is positive. This threshold, about 0.118, appears in the framework's account of when a signal becomes recognizable. The research note in the module applies this to calorimetry: it estimates the minimum detectable heat as the cost at the golden ratio times the thermal noise, giving about 2e-21 watts. The note records that this is consistent with the sensitivity of a commercial instrument, roughly 0.1 microwatt, after amplification.

What the module actually proves is narrower than the note. The Lean code establishes three facts about the cost function: it vanishes at equality, it is nonnegative, and the threshold is positive. It proves nothing specific to calorimetry, because the module defines m and e without reference to a particular instrument. The calorimetric application is a research note, not a theorem. The general facts, however, are a foundation: any calorimeter model that plugs in its own definitions of m and e inherits these properties.

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]

What this page does not claim

This module proves the calorimetric sensitivity estimate of 2e-21 watts; that is a research note, not a theorem. The module does not define what m and e are for a specific calorimeter instrument. The consistency with ITC sensitivity is an empirical observation, not a derived result.

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/Reaction_Calorimetry3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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