Energy Cohesion Model (ECM): An Energy-Balance Framework for Extracting Molecular Cohesion Energies, Cross–Interaction Energies, and Unknown Species Contributions in Gas Mixtures

Energy-Based Reconstruction of Binding Contributions in Unknown Gas Mixtures Jae Un Kim [email protected] Abstract Only two macroscopic quantities are typically accessible in gasmixture experiments: the injected energy Ein and the released energy Eout. All internal mechanisms—including binding, cross-species influence, and equipment-related dissipation—collapse into the measurable residual: Eres =Ein −Eout. This work develops a general framework (ECM) that reconstructs or estimates the binding contribution of a target species using only this residual quantity. The analysis clarifies which internal components are fundamentally identifiable from macroscopic energy behavior, and how the available information determines the degree of reconstruction. The method requires no spectral, chemical, or molecular-level access, making it applicable across a wide class of experimental systems. Expanded theoretical discussion and structural examples are provided for all information states (Case 0–4). 1 1 Introduction Measuring internal binding contributions inside a gas mixture is a fundamentally indirect problem: the relevant physical processes cannot be observed in isolation. Instead, all microscopic events manifest collectively in the difference between injected and outgoing energy. This motivates the residual definition: Eres =Ein −Eout. The central difficulty is that Eres is not a single mechanism but a composite: Eres =Ebind +Ecross +Eloss. A key contribution of this paper is to formalize how much of this composite quantity can be disentangled under varying levels of information. Even though ECM is based solely on externally measurable energy, it can recover binding contributions to the greatest extent allowed by the underlying structure. What is identifiable—and what is fundamentally not— becomes mathematically transparent. This expanded version includes deeper context: why energy-only approaches are valuable, how structural decomposition aligns with physical intuition, and why certain Cases allow exact reconstruction whereas others necessarily yield only estimates. 2 Notation •Ein — injected energy •Eout — outgoing energy •Eres =Ein −Eout — residual •Ebind — intrinsic binding contribution 2 •Ecross — cross-species influence •Eloss — equipment dissipation •xi— mole fraction (composition) of species i •Bi— binding coefficient of species i 3 Framework: Residual Decomposition Every measurable residual can be decomposed conceptually as: Eres =Ebind +Ecross +Eloss. If composition is known, binding forms a weighted sum: Ebind =X i xiBi. If composition is unknown but species count nis given, xiand Bimust be inferred jointly. When nis not known, structural patterns of residual differences guide estimation. In physical terms, ECM does not attempt to resolve microscopic mechanisms directly; instead, it asks: *Which internal quantities impose distinguishable footprints on macroscopic energy?* This separates Cases that admit exact reconstruction from those that cannot in principle yield unique solutions. 3 4 Case Analysis for Reconstruction Case 0: Single-Species Mixture With only one species present, Ecross = 0. Residual becomes: Eres =Ebind +Eloss. If equipment loss is calibrated: Ebind =Eres −Eloss. This is the only scenario where binding can be recovered exactly without additional information. No structural ambiguity exists because the system has no hidden degrees of freedom. Repetition experiments are unnecessary except for numerical stabilization. Case 1: Multi-Species, Composition Known Given xi: Ebind =X i xiBi. Residual: Eres =X i xiBi+Ecross +Eloss. Because composition is fixed, only cross and loss contribute ambiguity. A single experiment constrains their combined effect, but multiple conditions reduce uncertainty. Nevertheless, this Case retains partial identifiability and often performs close to exact reconstruction in practice. 4 Case 2: Species Count Known, Composition Unknown Unknowns: xiand Bi. E(k) res = n X i=1 xiB(k) i+E(k) cross +E(k) loss. Because internal parameters exceed the observable dimension, multiple independent conditions are required to form a solvable system. One experiment can never separate xiand Bibecause they appear multiplicatively. Only changes across distinct conditions induce identifiable structural variation. Case 3: Species Count Unknown Here even the dimensionality of internal structure is missing. Distinguishable species leave distinct signatures in Eres: differences in temperature dependence, pressure response, or experimental condition produce recognizable clusters. Inferring nrelies on: •the number of distinct response modes, •curvature of the singular value spectrum, •consistency of patterns across varied conditions. Once ˆnis inferred, estimation of xiand Biproceeds as in Case 2. Case 4: Fully Unknown System No direct information is available: Eest bind =Eres −b Ecross −b Eloss. 5 Cross/loss must be estimated via: •smoothing across conditions, •non-negativity constraints, •structural monotonicity, •physically plausible response patterns. This Case does not admit exact reconstruction, but ECM still provides the closest consistent estimate, making it useful when mixture information is unavailable. 5 Discussion ECM clarifies a long-standing ambiguity in energy-based mixture analysis: internal mechanisms become indistinguishable unless structural differences are induced across conditions. The framework identifies exactly which quantities can be separated and under what assumptions. Identifiability does not come from precision but from structure. Even perfect energy measurement cannot isolate binding if the system has more hidden degrees of freedom than observable constraints. Conversely, even coarse macroscopic data can isolate binding when the system is structurally constrained (Case 0–1). 6 Conclusion This expanded formulation of ECM shows that macroscopic energy differences contain far more information about binding contributions than previously recognized. By classifying all possible information states, ECM provides both the limits and the capabilities of energy-only reconstruction. 6 Future developments may incorporate temperature-dependent binding, nonlinear effects, or dynamic adjustments to extend ECM into broader physical regimes. References 1. Bird, R. B., Stewart, W. E., & Lightfoot, E. N. Transport Phenomena. 2. Atkins, P., & de Paula, J. Physical Chemistry. 3. Strang, G. Linear Algebra and Its Applications. 4. Hansen, P. C. Rank-Deficient and Discrete Ill-posed Problems. 5. MacKay, D. J. C. Information Theory, Inference, and Learning Algorithms. 7