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 introduces the Energy Cohesion Model (ECM), a purely energy-balancebased framework for analyzing gas mixtures. ECM extracts three central quantities from controlled input–output measurements: (i) intrinsic cohesion energies of individual gas species, (ii) a mixture-level cross–interaction energy that captures nonlinear cohesive effects, and (iii) effective cohesion energies of unknown species or unknown mixture components. Unlike spectroscopy, mass spectrometry, or chromatographic separation, ECM does not require species-specific signatures. Instead, it decomposes the measured mixture energy into: a system-level loss term, an expected linear cohesion contribution, and a residual cross–interaction contribution. In this formulation, we (a) define all variables explicitly, (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 the cross–interaction term in terms of non-ideal mixture behavior. The model 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 usually inferred from molecular-scale probes such as spectroscopic peaks, mass-to-charge distributions, or retention times in chromatography. Such techniques provide high-resolution information, but they depend on molecular signatures, instrument-specific calibration, and often species separation. The Energy Cohesion Model (ECM) takes an opposite approach. Instead of probing individual molecules, ECM observes only the macroscopic energetic response of a gas mixture under a controlled energy input. From this single scalar response, ECM separates: 1 •equipmentand flow-related baseline energy losses, •a linear sum of intrinsic cohesion energies, •and a residual cross–interaction energy that exists only in mixtures. Once this energetic structure is established, it becomes possible to: 1. extract intrinsic cohesion energies of known gases, 2. quantify mixture-only interaction energy, 3. reconstruct cohesion energies of unknown species, 4. and treat multi-component reconstruction as a linear algebra problem. The purpose of this paper is to describe ECM completely and unambiguously. We define all symbols and operating assumptions, derive the basic energy balances, and provide explicit reconstruction formulas. In addition, we clarify the physical meaning of the cross– interaction term and discuss how ECM complements, rather than replaces, conventional analytical methods. 2 Notation and Definitions Energies in ECM are treated as effective scalar quantities for a fixed operating condition. That is, temperature, pressure range, and driving protocol are assumed fixed while cohesion quantities are extracted. Table 1 lists all symbols and their meanings. In what follows, all energies can be normalized either per mole, per unit mass, or per unit volume. ECM does not depend on a specific normalization as long as it is used consistently. 3 Core Structure of the Energy Cohesion Model 3.1 Residual Energy from Input–Output Under a fixed experimental configuration, ECM starts from a simple energy balance. A controlled input energy Ein is supplied, and a portion Eout leaves the system. The residual energy is Eres =Ein −Eout,(1) which is interpreted as the energy that has been absorbed by the mixture and the apparatus during the process. This residual contains both molecular and non-molecular contributions. 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). Eres Residual energy available to the mixture after input–output: Eres =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 non-molecular loss (always treated as a positive quantity): Eloss =Eequipment +Eflow. Elin Expected linear cohesion of the mixture, assuming no cross–interactions: Elin =PixiBi. Ecross Cross–interaction energy (mixture-only term), defined as the residual cohesive energy beyond Elin and Eloss. uIndex of an unknown species in the mixture. BuEffective intrinsic cohesion energy of the unknown species u. xVector of compositions for a given mixture. BVector of cohesion energies of species. XComposition matrix constructed from multiple mixture experiments. Eres Vector of residual energies across experiments. Eloss Vector of loss energies across experiments. Ecross Vector of interaction energies across experiments. 3.2 Non-Molecular Loss Term Some part of Eres is not molecular in nature. It is consumed by the apparatus itself and the fluid mechanics of the flow. We write Eloss =Eequipment +Eflow,(2) with Eloss >0 by definition. In practice, Eloss is obtained via calibration runs, e.g. with inert or very weakly interacting gases (such as He or Ne), or via dedicated empty-system measurements. Repeated calibration runs reduce noise in Eloss. 3 3.3 Expected Linear Cohesion If all species contributed cohesion independently and only through linear superposition weighted by composition, the mixture cohesion would be Elin =X i xiBi.(3) Here Biis the intrinsic cohesion energy of species i: it is a property of the species and the operating condition, not of the mixture. 3.4 Definition of the Cross-Interaction Energy In real mixtures, the actual energetic contribution of molecular cohesion differs from the ideal linear expectation in (3). ECM defines the cross–interaction energy Ecross as the difference between the observed cohesion (after removing non-molecular loss) and the expected linear cohesion. We first decompose the residual energy as Eres =Elin +Ecross +Eloss.(4) Solving for Ecross gives Ecross =Eres −Eloss −Elin.(5) Using (1), (2), and (3), this can be written as Ecross = (Ein −Eout)−(Eequipment +Eflow)−X i xiBi.(6) Two limiting cases are useful: •If Ecross >0, the mixture exhibits stronger cohesion than predicted by linear superposition, which may correspond to clustering, association, or cooperative interactions. •If Ecross <0, the mixture is effectively less cohesive than the linear combination, as if components hinder each other’s cohesive behavior. 3.5 Physical Interpretation of Biand Ecross The intrinsic cohesion energy Bisummarizes, in a single scalar, all intermolecular effects of species iunder the given operating condition. It implicitly includes: •attractive forces (e.g. dispersion, dipolar interactions), •repulsive components within the operating range, •and the way these forces convert mechanical input energy into internal, non-recoverable modes. 4 In principle, Biis related to underlying pair potentials and correlation functions, but ECM does not require a specific microscopic model. It only assumes that the same Biapplies consistently across mixtures at the same condition. The cross–interaction term Ecross captures non-additive effects that emerge only when different species are combined. In classical mixture thermodynamics, this role is 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 dissipation: •Ecross = 0 corresponds to an energetically ideal mixture, where cohesion is fully described by a linear combination of {Bi}and the calibrated loss Eloss. •Ecross = 0 signals that unlike-molecule collisions and clustering patterns produce an additional (or reduced) cohesive contribution that cannot be represented by simply weighting single-species properties. Thus Biencodes how a single species stores and dissipates energy, while Ecross encodes how species modify each other’s cohesion when they coexist in the same mixture. 4 Reconstruction Problems The primary utility of ECM lies in reconstruction: given Eres,Eloss, and composition information, we solve for unknown cohesion energies. 4.1 Known Composition, Known Species If all species in the mixture are known and their cohesion energies Biare already characterized, then Elin is fixed by (3). In this case, ECM provides a direct evaluation of the cross–interaction energy: Ecross =Eres −Eloss −X i xiBi.(7) This is the “pure” mixture-only interaction term. When several mixtures with different compositions are available but the same set of {Bi}is used, one can check the internal consistency of ECM by verifying that the inferred Ecross values behave smoothly as a function of composition, rather than fluctuating arbitrarily. 4.2 Single Unknown Species Consider a mixture where all species except one are known. Let uindex the unknown species, with composition xuand cohesion energy Bu. Equation (3) becomes Elin =xuBu+X k=u xkBk.(8) From (4) and (5), we have Elin =Eres −Eloss −Ecross.(9) 5 Combining (8) and (9) yields xuBu+X k=u xkBk=Eres −Eloss −Ecross.(10) Solving for Bugives Bu=Eres −Eloss −Ecross −Pk=uxkBk xu .(11) Thus, with one measurement of Eres, a known loss Eloss, a known Ecross, and known Bkfor all k=u, ECM provides the cohesion energy of the unknown species. 4.3 Multiple Unknown Species and Linear Algebra If several species have unknown cohesion energies, we collect multiple mixture experiments and formulate a linear system. 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 (4), E(m) lin =E(m) res −E(m) loss −E(m) cross.(12) At the same time, by linearity, E(m) lin =X i x(m) iBi.(13) Collect all Mexperiments into vector and matrix form: Elin =      E(1) lin E(2) lin . . . E(M) lin      ,B=      B1 B2 . . . BN      ,(14) and let Xbe the M×Ncomposition matrix with entries x(m) i. Then Elin =XB.(15) From (12), we also have Elin =Eres −Eloss −Ecross.(16) Combining (15) and (16), XB=Eres −Eloss −Ecross.(17) If Xhas full column rank and M≥N, we can solve for Busing ordinary linear algebra. For a square, invertible X, B=X−1(Eres −Eloss −Ecross).(18) For overdetermined systems, a least-squares solution can be used: B=X⊤X−1X⊤(Eres −Eloss −Ecross).(19) In this way, ECM provides a clear linear structure for reconstructing multiple unknown cohesion energies from mixture experiments. 6 5 Comparison with Conventional Methods Table 2 contrasts ECM with several widely used analysis methods. 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 complex calibration. Species with overlapping mass or fragmentation patterns can be difficult to separate. Gas Chromatography Retention-time separation in columns. Requires chemical stationary phases, carrier gases, and timeconsuming separations. Often combined with other detectors. IR / Raman Spectroscopy Molecular vibrational or rotational absorption and scattering. Requires optical access, reference spectra, and distinct peaks. Overlapping bands complicate mixture analysis. ECM (this work) Macroscopic energy balance and cohesion extraction. Requires only controlled energy input, output measurement, and loss estimation Eloss. Does not need species-specific signatures. Enables direct evaluation of mixture interaction energy Ecross and reconstruction of unknown cohesion energies via linear algebra. Conventional approaches measure molecules directly, whereas ECM measures the mixture as a single energetic entity. This difference gives ECM unique advantages when: •spectral or mass signatures are ambiguous, •instrumentation budget is limited, •unknown species do not have well-characterized reference data. 6 Illustrative Structure of ECM Although ECM is defined algebraically, its internal logic can be summarized as a simple flow: •Step 1: Apply a controlled energy input Ein. •Step 2: Measure Eout and compute Eres =Ein −Eout. 7 •Step 3: Subtract the calibrated loss Eloss to isolate molecular cohesion: Ecoh =Eres −Eloss =Elin +Ecross. •Step 4: Compute Elin for a given hypothesis of Bi. •Step 5: Evaluate Ecross =Eres −Eloss −Elin. •Step 6: Adjust or solve for unknown Bifrom one or multiple experiments. Conceptually, the core relation is Eres =Elin +Ecross +Eloss, with Eloss >0 by definition and Ecross capturing non-ideal mixture effects. 7 Discussion ECM rephrases mixture analysis as a problem in energy decomposition. Rather than treating each species separately at the measurement stage, it focuses on four scalar quantities: Eres, Eloss, Elin, Ecross. From these, intrinsic and effective cohesion energies are reconstructed. Several aspects deserve emphasis: •Separation of system physics and molecular physics. By explicitly isolating the loss term Eloss, ECM ensures that the remaining structure reflects purely molecular cohesion. This makes the interpretation of Ecross transparent. •Linear algebraic structure. The reconstruction of unknown species reduces to solving linear systems like (17). This allows the use of Gaussian elimination, rank analysis, and regularization methods. •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 Bufor that component using (11) or its multi-species generalizations. In this sense, ECM acts as an energy-based detector for non-ideal or unexpected mixture components. •Scalability. The same structure applies to a single known mixture, a mixture with one unknown component, or a library of mixtures with multiple unknowns. The mathematical core remains the same. •Practicality. ECM does not rely on spectral resolution or mass filtering. It is compatible with simple mechanical or thermal input–output measurements, with the main experimental challenge being accurate estimation of Eloss and careful control of the operating condition. 8 8 Conclusion The Energy Cohesion Model provides a compact yet complete framework for analyzing gas mixtures through energy balance. By decomposing the measured residual energy into a non-molecular loss term, an expected linear cohesion term, and a residual cross–interaction energy, ECM enables: (i) extraction of intrinsic cohesion energies Bi, (ii) quantification of mixture-only cross–interaction energy Ecross, and (iii) reconstruction of cohesion energies of unknown species via linear algebra. 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. Robinson and J. M. Prausnitz, “Equations of State for Gases and Liquids,” AIChE Journal,6, 264–272 (1960). [4] J. G. M. van der Meulen and P. Mazur, “Nonideal Gas Mixtures and Interaction Contributions,” Physica,17, 1–16 (1951). [5] A. Bejan, Advanced Engineering Thermodynamics, Wiley, Hoboken (2016). [6] J. M. Smith, H. C. Van Ness, and M. M. Abbott, Introduction to Chemical Engineering Thermodynamics, McGraw–Hill, New York (2005). [7] F. M. White, Fluid Mechanics, McGraw–Hill, New York (2011). 9