{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,7]],"date-time":"2026-08-07T15:32:33Z","timestamp":1786116753751,"version":"build-2736575974"},"reference-count":61,"publisher":"American Chemical Society (ACS)","issue":"31","license":[{"start":{"date-parts":[[2026,7,25]],"date-time":"2026-07-25T00:00:00Z","timestamp":1784937600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,8,6]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>The quantum-volume (qv) and Bragg-Williams equations of state, as derived from a lattice gas model, are exploited. The qv equation of state includes a temperature-dependent excluded volume related to the thermal wavelength included in the translational contribution to the molecular partition function, but apart from this contribution intermolecular interactions are not included. The qv equation of state has the correct limiting behavior both in the limit of an ideal gas and in the packing-density limit. It also includes a nonclassical contribution to the internal energy and thereby a many-particle contribution to the kinetic energy, and it supports the view that it is both the position as well as the momentum of a system of particles that are distributed over the volume to maximize the multiplicity of the system. The qv equation of state is extended with intermolecular interactions through the Bragg-Williams mean-field approach, including also a hard-core contribution. For a one-component lattice gas, analytical expressions are presented for the pressure and the critical point as well as for the internal energy, the isochoric heat capacity, and the chemical potential, and results are presented for some simple systems. Given its relative simplicity, essentially on a textbook level, the model shows promising results, in particular for the temperature dependence of the isochoric heat capacity that shows a maximum close to the critical temperature. The presented model is supposed to serve as a foundation for further model extensions as well as for the development of interpretable analytical models in molecular thermodynamics for more complex systems and processes.<\/jats:p>","DOI":"10.1021\/acs.jpcb.6c02888","type":"journal-article","created":{"date-parts":[[2026,7,27]],"date-time":"2026-07-27T05:12:31Z","timestamp":1785129151000},"page":"8026-8036","source":"Crossref","is-referenced-by-count":0,"title":["The Quantum-Volume and Bragg-Williams Equations of\nState"],"prefix":"10.1021","volume":"130","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3384-7585","authenticated-orcid":true,"given":"Per-Olof","family":"\u00c5strand","sequence":"first","affiliation":[{"name":"NTNU - Norwegian University of Science and Technology , , ,","place":["Trondheim, Norway, NO-7481"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"316","published-online":{"date-parts":[[2026,7,25]]},"reference":[{"key":"2026080608132269900_cit1","volume-title":"Statistical Thermodynamics","author":"Fowler","year":"1939"},{"key":"2026080608132269900_cit2","volume-title":"Statistical Mechanics:\nPrinciples and Selected Applications","author":"Hill","year":"1987"},{"key":"2026080608132269900_cit3","volume-title":"Statistical Mechanics\nof Chain Molecules","author":"Flory","year":"1989","edition":"2"},{"key":"2026080608132269900_cit4","volume-title":"Molecular Driving Forces:\nStatistical Thermodynamics in Biology, Chemistry, Physics and Nanoscience","author":"Dill","year":"2011","edition":"2"},{"key":"2026080608132269900_cit5","doi-asserted-by":"publisher","first-page":"464","DOI":"10.3390\/e27050464","article-title":"Phase\nstability and transitions in high-entropy alloys: Insights from lattice\ngas models, computational simulations, and experimental validation","volume":"27","author":"Lach","year":"2025","journal-title":"Entropy"},{"key":"2026080608132269900_cit6","doi-asserted-by":"publisher","first-page":"1009","DOI":"10.3390\/membranes12101009","article-title":"The rigid\nadsorbent lattice fluid model: Thermodynamic consistency and relationship\nto the real adsorbed solution theory","volume":"12","author":"Brandani","year":"2022","journal-title":"Membranes"},{"key":"2026080608132269900_cit7","doi-asserted-by":"publisher","first-page":"2541","DOI":"10.3390\/pr11092541","article-title":"Molecular\nModeling of Supercritical Processes and the Lattice-Gas Model","volume":"11","author":"Tovbin","year":"2023","journal-title":"Processes"},{"key":"2026080608132269900_cit8","doi-asserted-by":"publisher","first-page":"117636","DOI":"10.1016\/j.seppur.2020.117636","article-title":"A critical assessment of the Flory-Huggins\n(FH) theory to predict aqueous two-phase behaviour","volume":"255","author":"Bussamra","year":"2021","journal-title":"Separ. Purif. Technol."},{"key":"2026080608132269900_cit9","doi-asserted-by":"publisher","first-page":"493","DOI":"10.1021\/ed084p493","article-title":"Consistent\napplication of the Boltzmann distribution to residual entropy in crystals","volume":"84","author":"Kozliak","year":"2007","journal-title":"J. Chem. Educ."},{"key":"2026080608132269900_cit10","doi-asserted-by":"publisher","first-page":"1548","DOI":"10.1021\/ed084p1548","article-title":"Configurational\nentropy revisited","volume":"84","author":"Lambert","year":"2007","journal-title":"J. Chem. Educ."},{"key":"2026080608132269900_cit11","doi-asserted-by":"publisher","first-page":"1063","DOI":"10.1021\/ed086p1063","article-title":"Overcoming\nmisconceptions about configurational entropy in condensed phases","volume":"86","author":"Kozliak","year":"2009","journal-title":"J. Chem. Educ."},{"key":"2026080608132269900_cit12","volume-title":"Molecular Thermodynamics","author":"McQuarrie","year":"1999"},{"key":"2026080608132269900_cit13","doi-asserted-by":"publisher","first-page":"12752","DOI":"10.1021\/acs.jpcb.4c06473","article-title":"Lattice models in molecular thermodynamics: Merging\nthe configurational and translational entropies","volume":"128","author":"\u00c5strand","year":"2024","journal-title":"J. Phys. Chem. B"},{"key":"2026080608132269900_cit14","doi-asserted-by":"publisher","first-page":"736","DOI":"10.1021\/acs.jchemed.9b00742","article-title":"Teaching\nthermodynamics with the quantum volume","volume":"97","author":"Yu","year":"2020","journal-title":"J. Chem.\nEduc."},{"key":"2026080608132269900_cit15","doi-asserted-by":"publisher","first-page":"48","DOI":"10.3390\/e19020048","article-title":"Entropy, Shannon\u2019s\nMeasure of Information and Boltzmann\u2019s H-Theorem","volume":"19","author":"Ben-Naim","year":"2017","journal-title":"Entropy"},{"key":"2026080608132269900_cit16","doi-asserted-by":"publisher","first-page":"699","DOI":"10.1098\/rspa.1934.0132","article-title":"The effect of thermal\nagitation on atomic arrangement\nin alloys","volume":"145","author":"Bragg","year":"1934","journal-title":"Proc. R. Soc. A"},{"key":"2026080608132269900_cit17","doi-asserted-by":"publisher","first-page":"540","DOI":"10.1098\/rspa.1935.0165","article-title":"The effect of thermal\nagitation on atomic arrangement\nin alloys - II","volume":"151","author":"Bragg","year":"1935","journal-title":"Proc. R. Soc. A"},{"key":"2026080608132269900_cit18","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1098\/rspa.1935.0188","article-title":"The effect\nof thermal agitation on atomic arrangement in alloys - III","volume":"152","author":"Williams","year":"1935","journal-title":"Proc. R. Soc. A"},{"key":"2026080608132269900_cit19","doi-asserted-by":"publisher","first-page":"1126","DOI":"10.1119\/1.2338545","article-title":"The entropy\nof mixing and assimilation: An information-theoretical perspective","volume":"74","author":"Ben-Naim","year":"2006","journal-title":"Am. J. Phys."},{"key":"2026080608132269900_cit20","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1016\/0378-3812(95)02837-4","article-title":"A new volume dependence for the equation\nof state of hard spheres","volume":"115","author":"Wang","year":"1996","journal-title":"Fluid Phase Equilib."},{"key":"2026080608132269900_cit21","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1016\/S0378-3812(96)03189-5","article-title":"A simplified hard-sphere\nequation of state meeting\nthe high and low density limits","volume":"130","author":"Khoshkbarchi","year":"1997","journal-title":"Fluid Phase\nEquilib."},{"key":"2026080608132269900_cit22","doi-asserted-by":"publisher","first-page":"4643","DOI":"10.1021\/ie504919b","article-title":"Discrete modelling: Thermodynamics\nbased on Shannon entropy and discrete states of molecules","volume":"54","author":"Pfleger","year":"2015","journal-title":"Ind. Eng. Chem. Res."},{"key":"2026080608132269900_cit23","doi-asserted-by":"publisher","first-page":"955","DOI":"10.1103\/PhysRev.50.955","article-title":"The complete\nequation of state of one, two and three-dimensional gases of hard\nelastic spheres","volume":"50","author":"Tonks","year":"1936","journal-title":"Phys. Rev."},{"key":"2026080608132269900_cit24","doi-asserted-by":"publisher","first-page":"377","DOI":"10.1016\/0375-9601(68)91065-7","article-title":"The hard\ncore quantum gas at high temperature","volume":"27","author":"Hemmer","year":"1968","journal-title":"Phys.\nLett. A"},{"key":"2026080608132269900_cit25","doi-asserted-by":"publisher","first-page":"295","DOI":"10.1103\/PhysRev.178.295","article-title":"Quantum-mechanical\nequation of state of hard-sphere gas at high temperature","volume":"178","author":"Jancovici","year":"1969","journal-title":"Phys. Rev."},{"key":"2026080608132269900_cit26","doi-asserted-by":"publisher","first-page":"3923","DOI":"10.1063\/1.453841","article-title":"Monte Carlo simulation of the hard-sphere fluid with\nquantum correction and estimate of its free energy","volume":"88","author":"Yoon","year":"1988","journal-title":"J. Chem. Phys."},{"key":"2026080608132269900_cit27","doi-asserted-by":"publisher","first-page":"160","DOI":"10.1016\/j.molliq.2017.04.134","article-title":"A hard sphere fluid with quantum correction in nanospherical\npores: A DFT study","volume":"238","author":"Helmi","year":"2017","journal-title":"J. Mol. Liq."},{"key":"2026080608132269900_cit28","doi-asserted-by":"publisher","first-page":"305","DOI":"10.1088\/0034-4885\/12\/1\/314","article-title":"Molecular distribution and equation of state of gases","volume":"12","author":"de Boer","year":"1949","journal-title":"Rep. Prog. Phys."},{"key":"2026080608132269900_cit29","doi-asserted-by":"publisher","first-page":"729","DOI":"10.1016\/S0031-8914(36)80346-2","article-title":"The quantum theory\nof the non-ideal gas. I. Deviations from classical\ntheory","volume":"3","author":"Uhlenbeck","year":"1936","journal-title":"Physica"},{"key":"2026080608132269900_cit30","doi-asserted-by":"publisher","first-page":"915","DOI":"10.1016\/S0031-8914(37)80189-5","article-title":"The quantum\ntheory of the non-ideal gas. II. Behaviour\nat low temperatures","volume":"4","author":"Beth","year":"1937","journal-title":"Physica"},{"key":"2026080608132269900_cit31","volume-title":"In Molecular Theory of\nLiquids and Gases","author":"de Boer","year":"1964"},{"key":"2026080608132269900_cit32","doi-asserted-by":"publisher","first-page":"054503","DOI":"10.1063\/1.4891806","article-title":"The critical\ncompressibility factor value: Associative fluids and liquid alkali\nmetals","volume":"141","author":"Kulinskii","year":"2014","journal-title":"J. Chem. Phys."},{"key":"2026080608132269900_cit33","volume-title":"Intermolecular Forces","author":"Maitland","year":"1981"},{"key":"2026080608132269900_cit34","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-63896-2","volume-title":"Molecular\nPhysical Chemistry for Engineering Applications","author":"Danes\u0327","year":"2021"},{"key":"2026080608132269900_cit35","doi-asserted-by":"publisher","first-page":"133","DOI":"10.1080\/00268970701843147","article-title":"Ab initio potential\nenergy curve for the neon atom pair and thermophysical properties\nof the dilute neon gas. I. Neon-neon interatomic potential and rovibrational\nspectra","volume":"106","author":"Hellmann","year":"2008","journal-title":"Mol. Phys."},{"key":"2026080608132269900_cit36","doi-asserted-by":"publisher","first-page":"10255","DOI":"10.1063\/1.477721","article-title":"Accurate ab initio rovibronic spectrum of the X1states in Ar2","volume":"109","author":"Ferna\u0301ndez","year":"1998","journal-title":"J. Chem. Phys."},{"key":"2026080608132269900_cit37","doi-asserted-by":"publisher","first-page":"204307","DOI":"10.1063\/1.4921623","article-title":"Accurate ab initio potential for the krypton dimer and transport properties\nof the low-density krypton gas","volume":"142","author":"Waldrop","year":"2015","journal-title":"J. Chem. Phys."},{"key":"2026080608132269900_cit38","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1080\/00268976.2017.1359347","article-title":"A relativistic\ncoupled-cluster interaction potential and rovibrational constants\nfor the xenon dimer","volume":"116","author":"Jerabek","year":"2018","journal-title":"Mol. Phys."},{"key":"2026080608132269900_cit39","doi-asserted-by":"publisher","first-page":"10146","DOI":"10.1021\/jp9034375","article-title":"An assessment\nof theoretical methods\nfor nonbonded interactions: Comparison to complete basis set limit\ncoupled-cluster potential energy curves for the benzene dimer, the\nmethane dimer, benzene-methane, and benzene H2S","volume":"113","author":"Sherrill","year":"2009","journal-title":"J. Phys. Chem. A"},{"key":"2026080608132269900_cit40","doi-asserted-by":"publisher","first-page":"2029","DOI":"10.1071\/CH9802029","article-title":"Generalized van der Waals theory. II. Quantum effects\non the equation of state","volume":"33","author":"Hooper","year":"1980","journal-title":"Aust. J. Chem."},{"key":"2026080608132269900_cit41","doi-asserted-by":"publisher","first-page":"615","DOI":"10.1088\/0034-4885\/30\/2\/306","article-title":"The theory\nof equilibrium critical phenomena","volume":"30","author":"Fisher","year":"1967","journal-title":"Rep. Prog.\nPhys."},{"key":"2026080608132269900_cit42","volume-title":"Introduction to Phase\nTransitions and Critical Phenomena","author":"Stanley","year":"1971"},{"key":"2026080608132269900_cit43","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1146\/annurev.pc.37.100186.001201","article-title":"Thermodynamic\nbehavior of fluids near the critical\npoint","volume":"37","author":"Sengers","year":"1986","journal-title":"Annu. Rev. Phys. Chem."},{"key":"2026080608132269900_cit44","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-20879-0","volume-title":"Thermodynamics in Earth\nand Planetary Sciences","author":"Ganguly","year":"2020","edition":"2"},{"key":"2026080608132269900_cit45","volume-title":"Introduction to Modern\nStatistical Mechanics","author":"Chandler","year":"1987"},{"key":"2026080608132269900_cit46","volume-title":"Statistical Mechanics:\ntheory and Molecular Simulation","author":"Tuckerman","year":"2010"},{"key":"2026080608132269900_cit47","volume-title":"An Introduction to\nStatistical Thermodynamics","author":"Hill","year":"1986"},{"key":"2026080608132269900_cit48","doi-asserted-by":"publisher","first-page":"541","DOI":"10.2475\/ajs.291.6.541","article-title":"Critical phenomena in hydrothermal\nsystems: State,\nthermodynamic, electrostatic, and transport properties of H2O in the critical region","volume":"291","author":"Johnson","year":"1991","journal-title":"Am. J. Sci."},{"key":"2026080608132269900_cit49","doi-asserted-by":"publisher","first-page":"A1599","DOI":"10.1103\/PhysRev.136.A1599","article-title":"Specific\nheat of a gas near the critical point","volume":"136","author":"Fisher","year":"1964","journal-title":"Phys.\nRev."},{"key":"2026080608132269900_cit50","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1016\/0375-9601(74)90687-2","article-title":"The critical point of neon","volume":"49","author":"Peck","year":"1974","journal-title":"Phys.\nLett. A"},{"key":"2026080608132269900_cit51","doi-asserted-by":"publisher","first-page":"779","DOI":"10.1063\/1.556037","article-title":"A new equation of state for argon\ncovering the fluid region for temperatures from the melting line to\n700 K at pressures up to 1000 MPa","volume":"28","author":"Tegeler","year":"1999","journal-title":"J. Phys.\nChem. Ref. Data"},{"key":"2026080608132269900_cit52","doi-asserted-by":"publisher","first-page":"496","DOI":"10.1103\/PhysRevLett.20.496","article-title":"Specific heat of xenon near the critical\npoint","volume":"20","author":"Edwards","year":"1968","journal-title":"Phys. Rev. Lett."},{"key":"2026080608132269900_cit53","doi-asserted-by":"publisher","first-page":"266","DOI":"10.1016\/0031-8914(74)90124-4","article-title":"Density fluctuations\nand the specific heat near the\ncritical point","volume":"74","author":"Rice","year":"1974","journal-title":"Physica"},{"key":"2026080608132269900_cit54","doi-asserted-by":"publisher","first-page":"490","DOI":"10.1016\/0031-8914(74)90376-0","article-title":"Density fluctuations\nand the specific heat near the\ncritical point. II","volume":"78","author":"Rice","year":"1974","journal-title":"Physica"},{"key":"2026080608132269900_cit55","doi-asserted-by":"publisher","first-page":"194502","DOI":"10.1063\/1.4803855","article-title":"Intermolecular interactions\nand the thermodynamic properties\nof supercritical fluids","volume":"138","author":"Yigzawe","year":"2013","journal-title":"J. Chem. Phys."},{"key":"2026080608132269900_cit56","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1016\/j.molliq.2017.03.120","article-title":"Accurate calculation of near-critical heat capacities CP and CV of argon using\nmolecular dynamics","volume":"237","author":"Nichele","year":"2017","journal-title":"J. Mol. Liq."},{"key":"2026080608132269900_cit57","doi-asserted-by":"publisher","first-page":"121","DOI":"10.1016\/j.fluid.2008.11.017","article-title":"Some observations regarding\nthe prediction\nof isochoric heat capacities by engineering EOS models","volume":"277","author":"Polishuk","year":"2009","journal-title":"Fluid Phase Equilib."},{"key":"2026080608132269900_cit58","doi-asserted-by":"publisher","first-page":"5654","DOI":"10.1021\/acs.jced.0c00649","article-title":"Heat capacities of\nfluids: The performance of various equations of state","volume":"65","author":"Zhu","year":"2020","journal-title":"J. Chem. Eng. Data"},{"key":"2026080608132269900_cit59","doi-asserted-by":"publisher","first-page":"3073","DOI":"10.1002\/aic.14911","article-title":"Incorporating\ncritical\ndivergence of isochoric heat capacity into the soft-SAFT equation\nof state","volume":"61","author":"Llovell","year":"2015","journal-title":"AIChE J."},{"key":"2026080608132269900_cit60","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1119\/1.2142789","article-title":"Teaching the principles of statistical\ndynamics","volume":"74","author":"Ghosh","year":"2006","journal-title":"Am. J. Phys."},{"key":"2026080608132269900_cit61","doi-asserted-by":"publisher","first-page":"010901","DOI":"10.1063\/1.5012990","article-title":"Maximum caliber\nis a general variational\nprinciple for dynamical systems","volume":"148","author":"Dixit","year":"2018","journal-title":"J. Chem. Phys."}],"container-title":["The Journal of Physical Chemistry B"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pubs.acs.org\/doi\/pdf\/10.1021\/acs.jpcb.6c02888","content-type":"application\/pdf","content-version":"vor","intended-application":"unspecified"},{"URL":"https:\/\/pubs.acs.org\/jpcbfk\/article-pdf\/130\/31\/8026\/66254852\/acs.jpcb.6c02888.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/pubs.acs.org\/jpcbfk\/article-pdf\/130\/31\/8026\/66254852\/acs.jpcb.6c02888.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,6]],"date-time":"2026-08-06T12:19:21Z","timestamp":1786018761000},"score":1,"resource":{"primary":{"URL":"https:\/\/pubs.acs.org\/jpcbfk\/article\/130\/31\/8026\/5232320\/The-Quantum-Volume-and-Bragg-Williams-Equations-of"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,7,25]]},"references-count":61,"journal-issue":{"issue":"31","published-online":{"date-parts":[[2026,7,25]]},"published-print":{"date-parts":[[2026,8,6]]}},"URL":"https:\/\/doi.org\/10.1021\/acs.jpcb.6c02888","relation":{"has-preprint":[{"id-type":"doi","id":"10.26434\/chemrxiv.15002536\/v2","asserted-by":"object"}]},"ISSN":["1520-6106","1520-5207"],"issn-type":[{"value":"1520-6106","type":"print"},{"value":"1520-5207","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,7,25]]}}}