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 presents a simple method to recover the pure cohesion energy of each species in a gas mixture using only macroscopic residual–energy measurements. The Energy Cohesion Model (ECM) decomposes the measured residual energy into loss, linear cohesion, and cross–interaction terms, allowing the intrinsic cohesion signal to be isolated from mixture–dependent effects. By constructing mixture equations under controlled compositions, the linear cohesion term is extracted and used to reconstruct the intrinsic cohesion energies of the individual species. The results show that cohesion can be determined without molecular identification, separation steps, or spectroscopic calibration. 1 Introduction Measuring cohesion and interaction energies in gas mixtures is difficult. Conventional methods rely on molecular–scale information such as spectroscopic signals, mass–to–charge patterns, or physical separation techniques. These approaches can be precise but require species identification, calibration, and often complex procedures. The Energy Cohesion Model (ECM) uses a different strategy. Instead of resolving individual molecules, ECM measures the macroscopic residual energy after a controlled energy input. This residual energy is decomposed into three parts: •Eloss: energy loss from equipment and flow dissipation, •Elin =PixiBi: a linear sum of the pure cohesion energies, •Ecross: a cross–interaction term that appears only in mixtures. In this work, we clarify the roles of these three terms. When defining the energy balance, Eloss and Ecross appear as additive contributions. However, when extracting the pure cohesion energy, both must be subtracted because they do not represent intrinsic cohesion. The goal of this study is to reconstruct the pure cohesion energy of each species from experimental data. Although the measured residual energy contains loss and cross–interaction contributions, the intrinsic cohesion information exists only in the linear component Elin. Therefore, this model focuses on isolating this linear cohesion term and recovering the underlying cohesion energies it represents. For clarity, the key symbols used throughout the paper are listed below. 1 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). Eres 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). Eloss Total baseline system loss: Esystem =Eequipment +Eflow. Elinear Expected linear cohesion of the mixture, assuming no cross–interactions: Eexpected =PixiBi. Ecross Cross–interaction energy, defined as the residual cohesive energy beyond linear superposition. 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. 2 Core Structure of the Energy Cohesion Model 2.1 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. 2.2 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 2.3 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. 2.4 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) 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. 2.5 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. 3 3 Reconstruction The main utility of ECM lies in its reconstruction capability: given Eres,Eloss, and the composition, one can solve for unknown cohesion energies. 3.1 Known Composition, Known Species When both the mixture composition {xi}and all intrinsic cohesion energies Biare known, the linear component Elin is fully fixed by Eq. (3). Together with the independently calibrated loss term Eloss, the ECM energy balance (4) leaves only one unknown quantity in the measurement, namely the mixture–only cross interaction energy. Using Eq. (5), the cross term is obtained directly as Ecross =Eres −Eloss −Elin. In this regime, ECM does not aim to reconstruct unknown cohesion energies; instead, it isolates the genuine interaction that appears only upon mixing. Mapping Ecross over several compositions provides the physical structure of mixture interactions—its magnitude, sign, and nonlinearity—which later serves as reference information for the reconstruction cases considered in Sections 3.2 and 3.3. In practical terms, this fully–known regime also yields the mixture–only cross interaction energy itself. Since all other contributions are fixed, ECM directly isolates , providing a physically interpretable measure of excess or deficit cohesion arising solely from inter–species interactions. 3.2 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.(7) From (4) and (5), we also have Elin =Eres −Eloss −Ecross.(8) Combining (7) and (8) yields xuBu+X k=u xkBk=Eres −Eloss −Ecross.(9) Solving for Bugives Bu=Eres −Eloss −Ecross −Pk=uxkBk xu .(10) 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. 3.3 Example 1: Single Unknown Species To demonstrate how a single unknown cohesion energy can be reconstructed, consider a mixture consisting of species Aand an unknown species u. Assume the compositions are zA= 0.8, zu= 0.2, and the known cohesion of species Ais BA= 5.0. 4 Suppose the measured residual energy and the estimated loss and cross energies are Eres = 10.0, Eloss = 3.0, Ecross = 0.5. From the decomposition relation (11), Elin =Eres −Eloss −Ecross = 10.0−3.0−0.5 = 6.5. The linear cohesion must also satisfy Elin =zABA+zuBu= 0.8·5.0 + 0.2Bu= 4.0+0.2Bu. Equating the two expressions, 6.5=4.0+0.2Bu, which gives the reconstructed cohesion of the unknown species: Bu= 12.5. This example shows that a single mixture measurement is sufficient to identify the cohesion of an unknown species when the cross and loss energies are determined. 3.4 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 {xi}, and we measure Eres and determine Eloss and Ecross. From (8), Elin =Eres −Eloss −Ecross.(11) On the other hand, by linearity, Elin =X i xiBi.(12) 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       .(13) Equation (12) becomes Elin =XB.(14) From (11), the linear–cohesion vector can also be written as Elin =Eres −Eloss −Ecross.(15) Combining (14) and (15) leads to XB=Eres −Eloss −Ecross.(16) 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.(17) For overdetermined systems, a least–squares solution can be used: B= (X⊤X)−1X⊤Eres −Eloss −Ecross.(18) In this way, ECM provides a transparent structure for reconstructing multiple unknown cohesion energies from mixture experiments. 5 3.5 Example 2: Multiple Unknown Species and Linear Algebra To demonstrate the reconstruction procedure, consider a ternary mixture containing three species A,B, and C. Assume that the cohesion energy of species Ais known, BA= 5.0, while BBand BCare unknown. We perform M= 3 mixture experiments with different compositions. For each experiment m, we measure the residual energy Eres and we determine the loss and cross contributions Eloss and Ecross. The linear cohesion for each experiment is therefore Elin =Eres −Eloss −Ecross. The three mixture compositions are chosen as X=  0.6 0.3 0.1 0.6 0.1 0.3 0.6 0.2 0.2  , where each row represents (xA, xB, xC). For illustration, suppose the measured energies (after removing loss and cross) yield the following linear–cohesion values: Elin =  6.2 5.4 5.8  . We now solve the matrix equation Elin =X B, B =  BA BB BC  . Substituting the known value BA= 5.0, this becomes    6.2 5.4 5.8   =  0.6 0.3 0.1 0.6 0.1 0.3 0.6 0.2 0.2    5.0 BB BC  . Subtracting the known contribution 0.6×5.0 = 3.0 from each equation gives    3.2 2.4 2.8   =  0.3 0.1 0.1 0.3 0.2 0.2  BB BC. Solving this 2 ×2 system yields BB= 9.00, BC= 5.00. Thus, with only three mixture experiments and without any direct spectroscopic identification, the unknown cohesion energies BBand BCare uniquely reconstructed using the ECM linear–algebra framework. 6 4 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: •spectral or mass signatures are ambiguous or unavailable, •instrumentation budget or complexity must be minimized, •unknown species lack well–characterized reference data. 5 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 (16) for B. 7 Conceptually, the central relation is the decomposition Eres =Elin +Ecross +Eloss,(19) with Eloss >0 by definition and Ecross encoding non–ideal mixture effects. The pure cohesion is recovered from Eres via (??). 6 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 (16), 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 (10) 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 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. 8 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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