{"title":"Einstein-Smoluchowski-type relations for real gases","authors":"Alexander E. Dubinov","doi":"10.1007/s10910-025-01711-2","DOIUrl":null,"url":null,"abstract":"<div><p>Derivation of exact explicit Einstein-Smolukhowski (ES) relations for non-ideal real gases is purpose of this paper. The ES method of the derivation was modified for this purpose. The new method is based on the rule of differentiation of inverse functions known in mathematics. The modified method turned out to be more effective than the traditional one: the fact is that the modified method always works, while the traditional method is effective only for a small number of simple equations of state (EoS). The method has been tested for four popular EoS: the Lorentz EoS, the Van der Waals EoS, the Peng-Robinson EoS, and the Dieterici EoS. As a result, exact explicit ES formulas for gases obeying these EoS were derived. It was found that the ratio of the diffusion coefficient to the particle mobility coefficient depends not only on the gas temperature, but also on its concentration for all examples of gases. The derived exact formulas can be used to debug codes that simulate molecular dynamics.</p></div>","PeriodicalId":648,"journal":{"name":"Journal of Mathematical Chemistry","volume":"63 4","pages":"1116 - 1125"},"PeriodicalIF":2.0000,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Chemistry","FirstCategoryId":"92","ListUrlMain":"https://link.springer.com/article/10.1007/s10910-025-01711-2","RegionNum":3,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
Derivation of exact explicit Einstein-Smolukhowski (ES) relations for non-ideal real gases is purpose of this paper. The ES method of the derivation was modified for this purpose. The new method is based on the rule of differentiation of inverse functions known in mathematics. The modified method turned out to be more effective than the traditional one: the fact is that the modified method always works, while the traditional method is effective only for a small number of simple equations of state (EoS). The method has been tested for four popular EoS: the Lorentz EoS, the Van der Waals EoS, the Peng-Robinson EoS, and the Dieterici EoS. As a result, exact explicit ES formulas for gases obeying these EoS were derived. It was found that the ratio of the diffusion coefficient to the particle mobility coefficient depends not only on the gas temperature, but also on its concentration for all examples of gases. The derived exact formulas can be used to debug codes that simulate molecular dynamics.
期刊介绍：
The Journal of Mathematical Chemistry (JOMC) publishes original, chemically important mathematical results which use non-routine mathematical methodologies often unfamiliar to the usual audience of mainstream experimental and theoretical chemistry journals. Furthermore JOMC publishes papers on novel applications of more familiar mathematical techniques and analyses of chemical problems which indicate the need for new mathematical approaches.
Mathematical chemistry is a truly interdisciplinary subject, a field of rapidly growing importance. As chemistry becomes more and more amenable to mathematically rigorous study, it is likely that chemistry will also become an alert and demanding consumer of new mathematical results. The level of complexity of chemical problems is often very high, and modeling molecular behaviour and chemical reactions does require new mathematical approaches. Chemistry is witnessing an important shift in emphasis: simplistic models are no longer satisfactory, and more detailed mathematical understanding of complex chemical properties and phenomena are required. From theoretical chemistry and quantum chemistry to applied fields such as molecular modeling, drug design, molecular engineering, and the development of supramolecular structures, mathematical chemistry is an important discipline providing both explanations and predictions. JOMC has an important role in advancing chemistry to an era of detailed understanding of molecules and reactions.
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