The Energy Cohesion Model (ECM): A Unified Inversion Framework for Extracting Cohesion Energies, Cross-Interactions, and Composition in Gas Mixtures

Energy Cohesion Model (ECM): An Energy-Balance Framework for Extracting Molecular Cohesion Energies, Cross–Interaction Energies, and Unknown Species Contributions in Gas Mixtures Jae Un Kim Department of Physics, Ajou University, Suwon, Republic of Korea [email protected] Abstract This work develops the Energy Cohesion Model (ECM), an energy–balance–based framework for analyzing gas mixtures from macroscopic input–output measurements. Rather than resolving individual molecular signatures, ECM interprets the measured residual energy as a superposition of three contributions: (i) a linear sum of intrinsic cohesion energies of the constituent species, (ii) a mixture–only cross–interaction energy that captures non–ideal cohesive behaviour, and (iii) a non–molecular loss term that accounts for equipment and flow dissipation. A key point of the present formulation is the explicit separation between definition–level energy decomposition and target–level quantities. At the definition level, the residual energy is written as Eres =Elin +Ecross +Eloss, where Eloss >0 is a calibrated loss and Ecross is an additive (possibly positive or negative) mixture interaction term. When one wishes to extract a pure cohesion energy, excluding both mixture interactions and non–molecular dissipation, the target quantity is obtained by subtracting these contributions: Ecoh,pure =Eres −Eloss −Ecross. In this sense, the cross and loss terms appear as positive contributions in the structure of the residual energy, but are removed with negative sign when isolating the pure cohesive component. Within this framework, we: (a) define all variables and symbols, (b) derive the core balance equations using a consistent notation {Eres, Eloss, Elin, Ecross}, (c) provide reconstruction formulas for both single and multiple unknown species using linear algebra, and (d) clarify the physical interpretation of Ecross in terms of non–ideal mixture behaviour. The formulation is intended as a minimal but complete framework for measurement–based cohesion analysis in gas mixtures. 1 Introduction Cohesion and interaction energies in gas mixtures are often inferred from molecular–scale probes such as spectroscopic peaks, mass–to–charge distributions, or chromatographic retention times. These methods can provide highly detailed information, but they rely on species–specific signatures, extensive calibration, and frequently on physical separation of components. The Energy Cohesion Model (ECM) takes a complementary point of view. Instead of attempting to resolve individual molecules, ECM measures the macroscopic energetic response of 1 a gas mixture under a controlled energy input. From a single scalar measurement—the residual energy left in the system—ECM decomposes the response into: •a non–molecular loss term Eloss associated with equipment and flow dissipation, •a linear sum of intrinsic cohesion energies Elin =PixiBi, •and a cross–interaction energy Ecross that is present only in mixtures. In the present paper, we revise and formalize ECM so that the roles of these terms are unambiguous. In particular, we distinguish between: 1. the definition layer, where residual energy is written as a sum of calibrated contributions, and 2. the target layer, where quantities such as “pure cohesion” are extracted by subtracting loss and cross terms. This layered view makes it clear why Ecross and Eloss can be treated as additive contributions in the energy balance, yet appear with minus signs when one isolates a pure cohesion energy. The rest of the paper is organized as follows. Section 2 summarizes the notation. Section 3 develops the core energy balances. Section 4 describes reconstruction of unknown cohesion energies. Section 5 compares ECM with conventional analytical methods. Section 6 presents a concise flow of the ECM procedure, and Sections 7–8 provide discussion and conclusions. 2 Notation and Definitions Throughout this work, energies are treated as effective scalar quantities for a fixed operating condition, meaning that temperature, pressure range, and driving protocol are kept fixed while cohesion quantities are extracted. All energies can be normalized per mole, per unit mass, or per unit volume; ECM does not depend on the specific normalization as long as it is used consistently. Table 1 lists the symbols used in the formulation. tabularx 3 Core Structure of the Energy Cohesion Model 3.1 Residual Energy from Input–Output Under a fixed experimental configuration, ECM begins from a simple energy balance. A controlled input energy Ein is supplied, and a portion Eout leaves the system. The residual energy is defined as Eres =Ein −Eout,(1) and is interpreted as the energy that has been absorbed by the mixture–apparatus system during the process. This residual includes both molecular and non–molecular contributions. 3.2 Non–Molecular Loss Term A part of Eres is not associated with molecular cohesion. Instead, it is consumed by the apparatus itself and by hydrodynamic effects in the flow. We write Eloss =Eequipment +Eflow, Eloss >0 by definition.(2) In practice, Eloss is obtained via calibration runs, for example with inert or very weakly interacting gases (such as He or Ne), or by performing dedicated empty–system measurements. Repeated calibration runs can be used to reduce noise in Eloss. 2 Table 1: Symbols and definitions used in the Energy Cohesion Model (ECM). Symbol Definition xiMole fraction (composition) of species iin the mixture. BiIntrinsic cohesion energy of species iunder the chosen operating condition (energy per mole or per unit mass). Ein Controlled input energy supplied to the system. Eout Energy leaving the system without contributing to molecular cohesion (e.g., transmitted, reflected, or unused energy). Emix Net mixture-level cohesion-related energy inferred from input–output: Emix =Ein −Eout. Eequipment Energy loss due to equipment mechanisms (friction, mechanical damping, internal dissipation). Eflow Energy loss due to flow and hydrodynamic effects (turbulence, shear, entrance and exit losses). Esystem Total baseline system loss: Esystem =Eequipment +Eflow. Eexpected Expected linear cohesion of the mixture, assuming no cross–interactions: Eexpected =PixiBi. EICE Cross–interaction energy (ICE), defined as the residual cohesive energy beyond linear superposition and system loss. uIndex of an unknown species in the mixture. BuEffective intrinsic cohesion energy of the unknown species u. NNumber of distinct chemical species considered in the model. xVector of compositions for a given mixture. BVector of cohesion energies of species. XComposition matrix constructed from multiple mixture experiments. Emix Vector of mixture energies across experiments. Esystem Vector of system baseline losses across experiments. EICE Vector of interaction energies across experiments. 3.3 Expected Linear Cohesion If each species contributed cohesion independently and only through a linear superposition weighted by its composition, the mixture cohesion would be given by Elin =X i xiBi.(3) Here Biis the intrinsic cohesion energy of species i. It depends on the species and on the operating condition, but not on the mixture composition itself. 3.4 Definition of the Cross–Interaction Energy The residual energy Eres may be decomposed into Eres =Elin +Ecross +Eloss,(4) which merely states that the absorbed energy is partitioned into linear cohesion, mixture–only interaction, and non–molecular loss. Rearranging (4) gives the definition Ecross =Eres −Eloss −Elin.(5) 3 Using (1), (2), and (3), this can be written explicitly as Ecross = (Ein −Eout)−(Eequipment +Eflow)−X i xiBi.(6) Two limiting cases are especially informative: •If Ecross >0, the mixture exhibits stronger effective cohesion than predicted by the linear superposition. This can be associated with clustering, association, or cooperative interactions between unlike molecules. •If Ecross <0, the mixture is effectively less cohesive than the linear prediction, as if components hinder each other’s cohesive behaviour. Importantly, Ecross is an additive term in (4), but its value may be positive or negative depending on the underlying physics. 3.5 Pure Cohesion Energy For many purposes, one is interested in a cohesion measure that excludes both mixture–induced interactions and non–molecular loss. We define the pure cohesion energy as Ecoh,pure =Eres −Eloss −Ecross.(7) Substituting (4) into (7) yields Ecoh,pure = (Elin +Ecross +Eloss)−Eloss −Ecross =Elin.(8) Thus, Ecoh,pure is mathematically equal to the linear cohesion Elin. Equation (7) makes explicit why both the cross term and the loss term appear with minus signs when one extracts the pure cohesion from the measured residual energy: they are contributions that are present in Eres but are removed in the construction of an idealized, interaction–free quantity. 3.6 Physical Interpretation of Biand Ecross The intrinsic cohesion energy Bicollects, into a single scalar, all intermolecular effects of species iunder the given operating condition. It implicitly includes attractive forces (e.g. dispersion, dipolar interactions), short–range repulsion in the relevant operating range, and the way these forces transform mechanical input into internal, non–recoverable modes. In principle, Biis related to underlying pair potentials and correlation functions, but ECM does not require a detailed microscopic model. It only assumes that the same Biapplies consistently across all mixtures at the same condition. The cross–interaction term Ecross captures non–additive effects that emerge only when different species coexist. In classical mixture thermodynamics, analogous roles are played by excess properties (e.g. excess enthalpy or excess Gibbs energy). In ECM, Ecross is defined directly at the level of measured mechanical energy: •Ecross = 0 corresponds to an energetically ideal mixture, in which cohesion is fully described by the linear combination of intrinsic energies and the calibrated loss. •Ecross = 0 signals that unlike–molecule collisions, clustering patterns, or other cooperative structures produce an additional (or reduced) cohesive contribution that cannot be represented by simply weighting single–species properties. 4 4 Reconstruction Problems The main utility of ECM lies in its reconstruction capability: given Eres,Eloss, and the composition, one can solve for unknown cohesion energies. 4.1 Known Composition, Known Species If all species in the mixture are known and their cohesion energies Bihave already been characterized, the linear cohesion Elin is fixed by (3). In this case, ECM provides a direct evaluation of the mixture–only cross–interaction energy via (5): Ecross =Eres −Eloss −X i xiBi.(9) When measurements for multiple compositions are available, the behaviour of Ecross as a function of {xi}can be used to check internal consistency: one expects smooth composition dependence rather than arbitrary scatter, within experimental uncertainty. 4.2 Single Unknown Species Consider a mixture where all species except one are known. Let the unknown species be indexed by u, with composition xuand cohesion energy Bu. Equation (3) can be written as Elin =xuBu+X k=u xkBk.(10) From (4) and (5), we also have Elin =Eres −Eloss −Ecross.(11) Combining (10) and (11) yields xuBu+X k=u xkBk=Eres −Eloss −Ecross.(12) Solving for Bugives Bu=Eres −Eloss −Ecross −Pk=uxkBk xu .(13) Thus a single measurement of Eres, together with a known Eloss, a determined Ecross, and known Bkfor all k=u, provides the cohesion energy of the unknown species. 4.3 Multiple Unknown Species and Linear Algebra When several species have unknown cohesion energies, multiple mixture experiments are required and the reconstruction becomes a linear algebra problem. Suppose we perform Mexperiments. In experiment m, the mixture has composition {x(m) i}, and we measure E(m) res and determine E(m) loss and E(m) cross. From (11), E(m) lin =E(m) res −E(m) loss −E(m) cross.(14) On the other hand, by linearity, E(m) lin =X i x(m) iBi.(15) 5 Collecting all Mexperiments, we can write Elin =       E(1) lin E(2) lin . . . E(M) lin       ,B=      B1 B2 . . . BN      , X =       x(1) 1x(1) 2· · · x(1) N x(2) 1x(2) 2· · · x(2) N . . .. . .. . . x(M) 1x(M) 2· · · x(M) N       .(16) Equation (15) becomes Elin =XB.(17) From (14), the linear–cohesion vector can also be written as Elin =Eres −Eloss −Ecross.(18) Combining (17) and (18) leads to XB=Eres −Eloss −Ecross.(19) If Xhas full column rank and M≥N, we can solve for Busing standard linear algebra. For a square, invertible X, B=X−1Eres −Eloss −Ecross.(20) For overdetermined systems, a least–squares solution can be used: B= (X⊤X)−1X⊤Eres −Eloss −Ecross.(21) In this way, ECM provides a transparent structure for reconstructing multiple unknown cohesion energies from mixture experiments. 5 Comparison with Conventional Methods Table 2 contrasts ECM with several widely used analytical methods. Conventional approaches measure molecules directly, whereas ECM measures the mixture as a single energetic entity. Table 2: Comparison between conventional analytical methods and ECM. Method Basis Requirements / Limitations Mass Spectrometry Mass–to–charge ratio detection. Requires high vacuum, ionization, and detailed calibration. Species with overlapping masses or complex fragmentation patterns can be difficult to separate. Gas Chromatography Retention–time separation in columns. Requires specialized stationary phases, carrier gases, and time–consuming separations. Often combined with additional detectors. IR / Raman Spectroscopy Vibrational or rotational absorption and scattering. Requires optical access and reference spectra. Overlapping bands make mixture analysis challenging. ECM (this work) Macroscopic energy balance and cohesion extraction. Requires controlled energy input, output measurement, and estimation of Eloss. Does not rely on species–specific signatures and enables direct evaluation of Ecross and reconstruction of unknown cohesion energies via linear algebra. ECM does not replace spectroscopic or chromatographic methods; instead, it offers an alternative route that can be attractive when: 6 •spectral or mass signatures are ambiguous or unavailable, •instrumentation budget or complexity must be minimized, •unknown species lack well–characterized reference data. 6 Illustrative Structure of ECM Although ECM is algebraically defined, its internal logic can be summarized as a simple sequence: 1. Apply a controlled energy input Ein. 2. Measure Eout and compute Eres =Ein −Eout. 3. Subtract the calibrated loss Eloss to isolate cohesion plus interaction: Eres −Eloss =Elin +Ecross. 4. Compute Elin for a given set of {Bi}using (3). 5. Evaluate Ecross from (5). 6. If some Biare unknown, use one or more experiments and solve the corresponding linear system (19) for B. Conceptually, the central relation is the decomposition Eres =Elin +Ecross +Eloss,(22) with Eloss >0 by definition and Ecross encoding non–ideal mixture effects. The pure cohesion is recovered from Eres via (7). 7 Discussion ECM reframes mixture analysis as an energy–decomposition problem. Instead of attempting to identify each species at the measurement stage, ECM focuses on four scalar quantities: Eres, Eloss,Elin, and Ecross. From these, intrinsic and effective cohesion energies are reconstructed. Several aspects deserve emphasis: •Separation of system and molecular physics. By isolating Eloss through calibration, the remaining structure reflects cohesive effects only. This separation clarifies the physical interpretation of Ecross. •Linear–algebraic structure. Reconstruction of unknown species reduces to solving systems like (19), allowing the use of standard tools such as rank analysis, regularization, and uncertainty quantification. •Unknown and “hidden” species. When composition is partially known, or when an additional component is suspected but not spectroscopically resolved, ECM can still infer an effective Buusing (13) or its multi–species generalizations. In this sense, ECM can act as an energy–based detector for non–ideal or unexpected mixture components. •Scalability. The same mathematical structure applies to a single mixture with one unknown species, or to a library of mixtures with multiple unknowns. The extension from one to many unknowns is handled naturally by the linear–algebra formalism. •Practicality. ECM does not rely on spectral resolution or mass filtering. It is compatible with relatively simple mechanical or thermal input–output measurements, although accurate estimation of Eloss and tight control of the operating condition remain crucial experimental challenges. 7 8 Conclusion The Energy Cohesion Model provides a compact yet complete framework for analyzing gas mixtures through macroscopic energy balance. By decomposing the measured residual energy into (i) a non–molecular loss term Eloss, (ii) an expected linear cohesion Elin, and (iii) a residual cross–interaction energy Ecross, ECM enables: •extraction of intrinsic cohesion energies Bi, •quantification of mixture–only cross–interaction energy, •reconstruction of cohesion energies of unknown species via linear algebra, •and definition of a pure cohesion energy Ecoh,pure =Eres −Eloss −Ecross. The formulation given here specifies all variables and equations required to use ECM in practice. Future work may connect ECM to specific experimental implementations, extend it to liquid or multiphase systems, and compare ECM–derived cohesion tables with independent molecular–scale measurements. References [1] J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, The Molecular Theory of Gases and Liquids, Wiley, New York (1954). [2] F. London, “Zur Theorie und Systematik der Molekularkr¨afte,” Zeitschrift f¨ur Physik,63, 245–279 (1930). [3] D. B. 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