Quaternion-Based Kinetic Operator for Modeling Competitive and Incompatible Chemical Reactions Incorporating Angular Error Probabilities

Quaternion-Based Kinetic Operator for Modeling Competitive and Incompatible Chemical Reactions Incorporating Angular Error Probabilities | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Quaternion-Based Kinetic Operator for Modeling Competitive and Incompatible Chemical Reactions Incorporating Angular Error Probabilities Carlos Riveros This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7341735/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract We propose a novel kinetic operator based on quaternion probability formalism combined with angular error metrics to model chemical reaction systems involving multiple competing and mutually incompatible pathways. The method introduces a quaternion representation of reaction probabilities, where each component corresponds to an independent yet non-coexisting reaction pathway. The angular error is derived from experimentally measurable quantities by treating the theoretical and experimental results as orthogonal components of a right triangle, allowing a geometric interpretation of the deviation. The integration of this angular error into the quaternionic probability framework enables the identification of the dominant reaction under competition, even in the absence of equilibrium. The approach is extended to account for fractional-order kinetics and a modified Arrhenius operator, allowing for complex temperature-dependent behaviors. Two case studies are presented: (1) competition between three reactants producing two mutually exclusive products, and (2) competition between amino acids for a catalyst. Numerical simulations demonstrate the feasibility and potential of the model in predicting reaction dominance under noisy experimental conditions. This framework opens new avenues for integrating algebraic operator theory, fractional calculus, and statistical geometry into reaction kinetics. Quaternion probability angular error competitive reactions fractional kinetics Arrhenius operator chemical dynamics Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7341735","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":502063267,"identity":"f331217e-581a-4422-9f6c-11cf534d954f","order_by":0,"name":"Carlos Riveros","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA+ElEQVRIiWNgGAWjYHACxgMgkg3COSAHIpmBWAafHhQtxjAtPAS1wNiJDYS08LeffXDgwx+GfD6J5GMfPvy5k77h/OGHnwsY7HBqkTiTbnBwZhuDZZtEWvLMmW3PcjfcSDOWnsGQjFOLAUMaw2HeBgYDNp4zxsy8DYeBWhjMmHkYDuDWwv+M4TDPH5CW85+Zef4cTjc4f/wbfi0SQFt42IBa2HuYmXnYDicYHMjBb4vEjWcMQL9IALW0GTMC/WI480ZOsTSPAW6/8PenMT748MfGQL6Z+TEDMMTk+c4f3/iZp8JODpcWmGUYDiagYRSMglEwCkYBXgAAM+lQQ0s9DOYAAAAASUVORK5CYII=","orcid":"","institution":"University of Santiago Chile","correspondingAuthor":true,"prefix":"","firstName":"Carlos","middleName":"","lastName":"Riveros","suffix":""}],"badges":[],"createdAt":"2025-08-11 02:53:13","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7341735/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7341735/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":89421339,"identity":"a1a83923-4100-43ad-90e4-e7e9dd877010","added_by":"auto","created_at":"2025-08-19 18:50:45","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":415936,"visible":true,"origin":"","legend":"","description":"","filename":"QuaternionBasedKineticOperatorforModelingCompetitiveandIncompatibleChemicalReactionsIncorporatingAngularErrorProbabilities.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7341735/v1_covered_48b1fc89-1646-49e2-a271-0c1035b3327b.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Quaternion-Based Kinetic Operator for Modeling Competitive and Incompatible Chemical Reactions Incorporating Angular Error Probabilities","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Quaternion probability, angular error, competitive reactions, fractional kinetics, Arrhenius operator, chemical dynamics","lastPublishedDoi":"10.21203/rs.3.rs-7341735/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7341735/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe propose a novel kinetic operator based on quaternion probability formalism combined with angular error metrics to model chemical reaction systems involving multiple competing and mutually incompatible pathways. The method introduces a quaternion representation of reaction probabilities, where each component corresponds to an independent yet non-coexisting reaction pathway. The angular error is derived from experimentally measurable quantities by treating the theoretical and experimental results as orthogonal components of a right triangle, allowing a geometric interpretation of the deviation. The integration of this angular error into the quaternionic probability framework enables the identification of the dominant reaction under competition, even in the absence of equilibrium. The approach is extended to account for fractional-order kinetics and a modified Arrhenius operator, allowing for complex temperature-dependent behaviors. Two case studies are presented: (1) competition between three reactants producing two mutually exclusive products, and (2) competition between amino acids for a catalyst. 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