Energy Cohesion Model (ECM): An Energy-Balance Framework for Extracting Molecular Cohesion Energies, Cross–Interaction Energies, and Unknown Species Contributions 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 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: Eres =Ein −Eout. Eequip 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 Operational Principle The ECM device operates by generating a short, controlled pressure pulse inside a sealed measurement chamber containing the gas mixture. A mechanical actuator creates the pulse by a rapid back–and–forth displacement of a driving plate. This displacement injects a well-defined mechanical energy Ein into the chamber. A pressure–sensing module, mounted on the opposite side, records the portion of the pulse that returns to the actuator or leaves the chamber. This outgoing component is denoted Eout. Because the geometry of the chamber and the pulse duration are fixed, the difference between input and output energy represents the total energy absorbed by the system: Eres =Ein −Eout. This absorbed energy includes both molecular contributions (cohesion and interaction effects of the gas mixture) and non–molecular contributions (mechanical damping, turbulence, wall friction). Eres =Elin +Ecross +Eloss,(1) The key feature is that the pressure pulse transmits energy through compression and rarefaction, ensuring that the measurement depends only on bulk mechanical response—not on optical, chemical, or spectroscopic signatures. Repeated pulses with identical drive conditions allow averaging, suppressing sensor noise and ensuring that Eres represents a stable macroscopic response of the mixture. 2 3 Decomposition and Extraction of Energy Terms After the residual energy Eres is measured, ECM resolves it into three independent contributions: baseline losses, linear cohesion, and mixture-specific cross interactions. 3.1 Baseline Loss Term A portion of the absorbed energy does not originate from molecular cohesion but from mechanical and hydrodynamic dissipation. This loss term is defined as Eloss =Eequipment +Eflow, Eloss >0. It is obtained experimentally through calibration runs using inert or weakly interacting gases, or with an empty chamber. Because these effects do not depend on mixture composition, they can be subtracted uniformly across all measurements. 3.2 Linear Cohesion Term If each species contributed independently to cohesion, the mixture cohesion would be the molefraction–weighted sum of the intrinsic cohesion energies: Elin =X i xiBi. Elin =Eres −Eloss −Ecross.(2) Here Biis the intrinsic cohesion energy of species i, which depends on the operating condition but not on mixture composition. When either the composition {xi}or the intrinsic energies {Bi} are known, Elin can be computed directly. 3.3 Cross Interaction Term Any deviation from linear superposition is captured by the cross–interaction energy, Ecross =Eres −Eloss −Elin.(3) Positive values indicate cooperative interactions, while negative values indicate interference. Since Ecross = 0 for single–species samples, it is inherently a multi–species effect. Because Ecross is obtained by subtraction of independently measurable quantities, its stability reflects the internal consistency of the ECM experiment. Even small variations in composition produce detectable shifts in Ecross, making it a sensitive indicator for mixture-driven effects that cannot be accessed through Elin alone. Thus, Ecross serves both as a diagnostic check on measurement accuracy and as a quantitative descriptor of interaction strength within the mixture. 3.4 Practical Extraction Workflow For each mixture: 1. Perform calibration to determine Eloss. 2. Measure Ein and Eout, then compute Eres =Ein −Eout. 3 3. Compute Elin if composition or intrinsic energies are unknown. (If the linear cohesion term is not known in advance, it can be recovered through inverse reconstruction. (Chapter 4)) 4. Obtain the cross contribution: Ecross =Eres −Eloss −Elin. These extracted quantities form the basis for the reconstruction procedures that follow. 4 Reconstruction Reconstruction in the ECM framework refers to determining the unknown coupling strengths Bi and, when necessary, the compositions x(k) iand even the number of species N, from measurable quantities. All reconstruction begins with the fundamental linear relation E(k) lin =E(k) res −E(k) loss −E(k) cross = N X i=1 x(k) iBi. Depending on whether the species count Nand the compositions x(k) iare known, four distinct scenarios arise. We present each case with the required method, solvability condition, and a concrete numerical example. 4.1 Case 1: Nknown, x(k) iknown Explanation This is the only fully determined linear system. With M≥Nlinearly independent experiments, E(k) lin = N X i=1 x(k) iBi has a unique solution. Solvability Condition rank(X)=N. Solution Formula B= (X⊤X)−1X⊤Elin. Numerical Example For N= 2: X=  0.7 0.3 0.2 0.8 0.5 0.5  ,Elin =  2.10 2.75 2.40  , the reconstruction yields B=2.82 2.06. 4 4.2 Case 2: Nknown, x(k) iunknown Explanation Here Nis fixed but the compositions are not. Reconstruction requires estimating the unknown x(k) iunder the constraints x(k) i≥0,X i x(k) i= 1. Estimation Step For each experiment: E(k) lin ≈ N X i=1 x(k) iBi, so (X, B) are obtained via constrained least-squares: (Xest, Best) = arg min X,B ∥XB −Elin∥2 2. Numerical Example Given tentative couplings BA= 3.0, BB= 2.0: 2.0=3x(1) A+ 2(1 −x(1) A),2.4 = 3x(2) A+ 2(1 −x(2) A), yielding x(1) A= 0.0, x(2) A= 0.4, which updates Xand then B. 4.3 Case 3: Nunknown, x(k) iknown Explanation Now compositions are known, but the species count is not. Thus the first task is to determine N. In ECM, the two most appropriate and theoretically consistent methods are: •(i) Residual Minimization •(ii) Rank-Growth Analysis These two complement each other: residual minimization identifies the best-fitting model, while rank-growth provides a lower bound for N. (i) Residual Minimization For a candidate N, reconstruct the couplings: B(N) = (X⊤X)−1X⊤Elin, then evaluate the residual R(N) = ∥XB(N)−Elin∥2. The optimal species count minimizes the residual: Nest = arg min NR(N). 5 (ii) Rank-Growth Analysis Because Xcannot have rank exceeding N, rank(X)≤N, the observed rank growth gives a guaranteed lower bound on N. With sufficiently diverse experiments, the rank typically reaches N. Numerical Example Suppose three experiments provide known compositions, and residual tests give R(1) = 0.88, R(2) = 0.21, R(3) = 0.05. Residual minimization selects Nest = 3. Rank of Xis observed to be 2, implying N≥2. Combined, this yields the consistent estimate N= 3. 4.4 Case 4: Nunknown, x(k) iunknown Explanation This is the most challenging case. Both Nand the compositions must be inferred from the measurements. The reconstruction pipeline integrates the same two estimation tools used in Case 3: •Residual Minimization (selects the best-fitting Nand refines Xand B) •Rank-Growth Analysis (provides a lower bound and stabilizes the search) Reconstruction Pipeline 1. Species-count estimation For each candidate N, estimate Xand Band compute R(N) = ∥Xest(N)Best(N)−Elin∥2. Select Nest = arg min NR(N), Nest ≥rank(Xest). 2. Composition estimation For each experiment k, x(k) i≥0,X i x(k) i= 1, obtained via constrained least-squares. 3. Coupling reconstruction Best = (X⊤ estXest)−1X⊤ estElin. 6 Numerical Example Data: E(1) lin = 1.9, E(2) lin = 2.1, E(3) lin = 2.4. Residual tests: R(1) = 1.12, R(2) = 0.33, R(3) = 0.07. Thus Nest = 3. Rank-growth shows rank(Xest) = 2, hence N≥2. Combined: N= 3. Solving E(k) lin =x(k) ABA+(1−x(k) A)BB under the constraints yields x(1) A= 0.1, x(2) A= 0.3, x(3) A= 0.5, and BA= 3.05, BB= 1.95. Interpretation Because both Nand xiare estimated, this case yields approximate solutions. Accuracy improves with the diversity of experiments and the consistency of the estimated compositions. Final Remark. In this chapter, the reconstruction is performed using only the linear structure E(k) lin = N X i=1 x(k) iBi, because this term alone governs the solvable, rank–determined part of the ECM framework. Once the compositions x(k) iand intrinsic binding strengths Biare obtained, the remaining portion of the measured energy is simply E(k) meas −E(k) lin =E(k) loss +E(k) cross. These nonlinear components do not influence the reconstruction of x(k) iand Bi, and therefore are acknowledged here only as residual contributions that appear after the linear solution is determined. ECM Advantage. Although the reconstruction procedure may appear complex, the ECM framework provides capabilities that conventional approaches cannot achieve. •Simultaneous recovery of compositions and binding strengths. Unlike traditional models, ECM allows xiand Bito be reconstructed at the same time from the same data. •Applicability even when the number of species is unknown. The rank structure of the linear term enables the estimation of the number of unknown species, which is impossible in standard methods. 7 •Only one observable is required. The framework reconstructs the entire mixture behavior using a single scalar measurement Emeas, without additional spectroscopy or composition analyses. •Mathematically strict solvability conditions. The rank condition provides a clear criterion for when an inverse problem has a unique solution, removing ambiguity found in empirical fitting approaches. •Nonlinear terms become clean residuals. After the linear reconstruction, the remaining energy naturally isolates into Eloss and Ecross, making interaction effects transparent and interpretable. 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: •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. 8 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 5. Evaluate Ecross from 6. If some Biare unknown, use one or more experiments and solve the corresponding linear system for B. Conceptually, the central relation is the decomposition Eres =Elin +Ecross +Eloss,(4) with Eloss >0 by definition and Ecross encoding non–ideal mixture effects. The pure cohesion is recovered from Eres via 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, 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 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. 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, 9