{
  "id": "2201.04009",
  "version": "v2",
  "published": "2022-01-07T15:58:50.000Z",
  "updated": "2022-06-11T19:02:30.000Z",
  "title": "Mathematical Analysis of the van der Waals Equation",
  "authors": [
    "Emil M. Prodanov"
  ],
  "comment": "11 pages, 5 figures",
  "journal": "Physica B: Condensed Matter 640, 414077 (2022)",
  "doi": "10.1016/j.physb.2022.414077",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "The parametric cubic van der Waals polynomial $\\,\\, p V^3 - (R T + b p) V^2 + a V - a b \\,\\,$ is analysed mathematically and some new generic features (theoretically, for any substance) are revealed: the temperature range for applicability of the van der Waals equation, $T > a/(4Rb)$, and the isolation intervals, at any given temperature between $a/(4Rb)$ and the critical temperature $8a/(27Rb)$, of the three volumes on the isobar-isotherm: $\\,\\, 3b/2 < V_A \\le 3b$, $ \\,\\, 2b < V_B < 4b/(3 - \\sqrt{5})$, and $\\,\\, 3b < V_C < b + RT/p$. The unstable states of the van der Waals model have also been generically localized: they lie in an interval within the isolation interval of $V_B$. In the case of unique intersection point of an isotherm with an isobar, the isolation interval of this unique volume is also determined. A discussion on finding the volumes $V_{A, B, C}$, on the premise of Maxwell's hypothesis, is also presented.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-06-11T19:02:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "van der waals equation",
      "mathematical analysis",
      "isolation interval",
      "cubic van der waals polynomial",
      "parametric cubic van der waals"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}