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 (ICE) 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 system baseline losses, an expected linear cohesion contribution, and a residual interaction term. In this strengthened formulation, we (a) define all variables explicitly, (b) derive the core balance equations, (c) provide reconstruction formulas for both single and multiple unknown species using linear algebra, and (d) discuss physical interpretation of the cross–interaction energy 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 (ICE) 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 (ICE), 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 ICE 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 Basic Energy Balance 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 remainder, Emix =Ein −Eout,(1) is interpreted as the sum of system losses and molecular cohesion effects within the gas mixture. 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. 3.2 System Baseline Loss Some part of Emix is not molecular in nature. It is consumed by the apparatus itself and the fluid mechanics of the flow. We write Esystem =Eequipment +Eflow.(2) In practice, Esystem 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 Esystem. 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 Eexpected =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 Cross-Interaction Energy (ICE) In real mixtures, the actual energetic contribution of molecular cohesion differs from the ideal linear expectation in (3). ECM defines the cross–interaction energy (ICE) as the difference between the observed cohesion (after removing system loss) and the expected linear cohesion. We decompose Emix as Emix =Esystem +Eexpected +EICE.(4) Solving for EICE gives EICE =Emix −Esystem −Eexpected.(5) Using (1), (2), and (3), this can be written as EICE = (Ein −Eout)−(Eequipment +Eflow)−X i xiBi.(6) Two limiting cases are useful: •If EICE >0, the mixture exhibits stronger cohesion than predicted by linear superposition, which may correspond to clustering, association, or cooperative interactions. •If EICE <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 EICE 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, dipole 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 ICE term EICE 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, EICE is defined directly at the level of measured mechanical energy dissipation: •EICE = 0 corresponds to an energetically ideal mixture, where cohesion is fully described by a linear combination of {Bi}. •EICE = 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 EICE 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 Emix,Esystem, 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 Eexpected is fixed by (3). In this case, ECM provides a direct evaluation of the cross–interaction energy: EICE =Emix −Esystem −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 EICE 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 Eexpected =xuBu+X k=u xkBk.(8) From (4) and (5), we have Eexpected =Emix −Esystem −EICE.(9) 5 Combining (8) and (9) yields xuBu+X k=u xkBk=Emix −Esystem −EICE.(10) Solving for Bugives Bu=Emix −Esystem −EICE −Pk=uxkBk xu .(11) Thus, with one measurement of Emix, a known system baseline Esystem, a known EICE, 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) mix and determine E(m) system and E(m) ICE. From (4), E(m) expected =E(m) mix −E(m) system −E(m) ICE.(12) At the same time, by linearity, E(m) expected =X i x(m) iBi.(13) Collect all Mexperiments into vector and matrix form: Eexpected =       E(1) expected E(2) expected . . . E(M) expected       ,B=      B1 B2 . . . BN      ,(14) and let Xbe the M×Ncomposition matrix with entries x(m) i. Then Eexpected =XB.(15) From (12), we also have Eexpected =Emix −Esystem −EICE.(16) Combining (15) and (16), XB=Emix −Esystem −EICE.(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(Emix −Esystem −EICE).(18) For overdetermined systems, a least-squares solution can be used: B=X⊤X−1X⊤(Emix −Esystem −EICE).(19) In this way, ECM provides a clear linear structure for reconstructing multiple unknown cohesion energies from mixture experiments. 6 5 Illustrative Numerical Example To illustrate the algebra of ECM, we present a simple hypothetical example. Numerical values are chosen for clarity and do not correspond to any particular physical gas. 5.1 Single Unknown Species Consider a binary mixture of a known species Aand an unknown species u. The composition is xA= 0.8, xu= 0.2. Suppose that under a given protocol we have: •measured input and output energies such that Emix =Ein −Eout = 10.0 J, •calibrated baseline loss Esystem = 3.0 J, •interaction energy from independent analysis (or assumed negligible) EICE = 0.5 J, •known intrinsic cohesion of species Aas BA= 5.0 J. From (4), Eexpected =Emix −Esystem −EICE = 10.0−3.0−0.5=6.5 J.(20) On the other hand, Eexpected =xABA+xuBu= 0.8·5.0+0.2·Bu.(21) Equating the two expressions for Eexpected, 6.5 = 4.0 + 0.2Bu⇒Bu= 12.5 J.(22) This example shows how a single experiment can determine the effective cohesion energy of one unknown species when composition and EICE are known. 5.2 Two Unknown Cohesion Energies via Linear Algebra Now consider a ternary system with three species A,B, and C, where BAis known but BB and BCare unknown. We perform three experiments with different compositions {x(m) i}and obtain E(m) expected from (12). For simplicity, we assume E(m) ICE = 0 in this example. Let X=  0.6 0.3 0.1 0.6 0.1 0.3 0.6 0.2 0.2  ,B=  BA BB BC  ,Eexpected =   E(1) expected E(2) expected E(3) expected   . 7 Assume BA= 4.0 J and the measured values are Eexpected =  3.4 J 3.0 J 3.2 J  . We then solve the linear system XB=Eexpected for B. Writing BAseparately and subtracting its contribution, one can form a reduced system for (BB, BC) or solve the full 3 ×3 system directly. In either case, this numerical example shows how ECM equations are used in practice: measured mixture energies, known compositions, and known baselines are combined to recover unknown cohesion energies. 6 Assumptions and Limitations The Energy Cohesion Model is intentionally minimalist. Its power comes from compressing complex microscopic physics into a small number of effective energies. This compression requires several assumptions and has clear limitations. 6.1 Fixed Operating Condition All ECM quantities (Bi,EICE,Esystem) are defined for a fixed operating condition: •temperature and pressure range, •driving protocol (pulse shape, duration, frequency), •geometry and flow configuration. If these conditions change significantly, the values of Biand EICE must be re-calibrated. ECM does not assume any specific functional dependence on temperature or pressure. 6.2 Separable System Loss ECM assumes that Esystem can be separated from molecular cohesion and estimated independently (e.g. via inert-gas calibration). If geometry or flow regime changes between calibration and measurement, the extracted cohesion energies may be biased. 6.3 Effective Linearity of Single-Species Contributions The decomposition Eexpected =X i xiBi assumes that single-species cohesion contributions add linearly when considered at the level of effective energies. Strongly non-linear composition effects are then folded into EICE. In systems where such non-linearity is extreme or composition-dependent phase changes occur, ECM must be interpreted with care. 8 6.4 Measurement Noise and Ill-Conditioning The reconstruction step, especially for multiple unknown species, involves solving linear systems like (17). If the composition matrix Xis nearly singular or poorly conditioned, small measurement errors in Emix or Esystem can lead to large uncertainties in B. In practice, this requires: •designing mixtures to make Xwell-conditioned, •using repeated experiments and averaging, •and applying regularization or uncertainty analysis when needed. 6.5 Lumped Interaction Energy By design, EICE is a lumped quantity. It does not resolve pairwise cross-terms Eij individually unless additional structure or assumptions are added. ECM therefore provides a compact total interaction measure rather than a full interaction matrix. 7 Future Work Several natural extensions of ECM can be explored in subsequent studies: •Temperatureand pressure-dependent cohesion tables. By repeating ECM measurements across a grid of thermodynamic states, one can construct Bi(T, p) and EICE(T, p) tables, analogous to equation-of-state parameter maps. •Connection to thermodynamic excess functions. The relation between EICE and excess enthalpy or excess Gibbs energy can be investigated, clarifying how ECMderived quantities map onto classical mixture theory. •Extension to liquid and multiphase systems. While ECM is formulated here for gases, the same energy-balance logic can be generalized to liquids, aerosols, or gas–liquid mixtures with appropriate modifications. •Experimental implementation and uncertainty quantification. Detailed designs for ECM apparatus, including actuator choice, sensor placement, and data acquisition, can be developed and combined with uncertainty propagation analysis. •Library-based identification of unknown species. Once a database of {Bi}is available, ECM could be used as a screening tool: measured effective Buvalues for unknown species can be compared against the library to narrow down candidate identities, especially in combination with weak spectroscopic or mass information. These directions would move ECM from a purely theoretical framework toward a practical tool for laboratory and field analysis of complex gas mixtures. 9