{
  "version": 1,
  "topic": "PV11",
  "title": "Why a small pressure datum mistake can distort a gas calculation",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Gas density error from a missing atmosphere",
    "premise": "Patm = 14.7 psia, T = 600 °R, Z = 0.85, molecular weight 18.82855 lb/lbmol, R = 10.732 field units.",
    "formula": "ρcorrect = Pabs M/(ZRT); ρwrong = (Pabs − 14.7)M/(ZRT)",
    "variables": "The fixed-Z control isolates pressure reference. The wrong column is deliberately invalid as a state input.",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Absolute pressure (psia) = 100",
        "explanation": "Patm = 14.7 psia, T = 600 °R, Z = 0.85, molecular weight 18.82855 lb/lbmol, R = 10.732 field units."
      },
      {
        "label": "Select the relation",
        "calculation": "ρcorrect = Pabs M/(ZRT); ρwrong = (Pabs − 14.7)M/(ZRT)",
        "explanation": "The fixed-Z control isolates pressure reference. The wrong column is deliberately invalid as a state input."
      },
      {
        "label": "Substitute the values",
        "calculation": "At 100 psia the wrong pressure is 85.3; density ratio = 85.3/100 = 0.853, a 14.7% underestimate",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "Correct density (lb/ft³) = 0.344006015; Wrong density (lb/ft³) = 0.29343713; Underestimate (%) = 14.7",
        "explanation": "The relative datum error grows at low absolute pressure under these fixed-property controls."
      }
    ],
    "result": "Correct density (lb/ft³) = 0.344006015; Wrong density (lb/ft³) = 0.29343713; Underestimate (%) = 14.7",
    "interpretation": "The relative datum error grows at low absolute pressure under these fixed-property controls.",
    "control": "Absolute pressure (psia)",
    "columns": [
      "Absolute pressure (psia)",
      "Correct density (lb/ft³)",
      "Wrong density (lb/ft³)",
      "Underestimate (%)"
    ],
    "rows": [
      [
        25,
        0.08600150365774338,
        0.035432619506990275,
        58.8
      ],
      [
        50,
        0.17200300731548676,
        0.12143412316473365,
        29.4
      ],
      [
        100,
        0.3440060146309735,
        0.29343713048022046,
        14.7
      ],
      [
        500,
        1.7200300731548677,
        1.6694611890041147,
        2.94
      ],
      [
        3000,
        10.320180438929206,
        10.269611554778454,
        0.49
      ]
    ],
    "baseline": 2,
    "rowNotes": [
      "Absolute pressure (psia) = 25. Correct density (lb/ft³): 0.0860015037; Wrong density (lb/ft³): 0.0354326195; Underestimate (%): 58.8.",
      "Absolute pressure (psia) = 50. Correct density (lb/ft³): 0.172003007; Wrong density (lb/ft³): 0.121434123; Underestimate (%): 29.4.",
      "Absolute pressure (psia) = 100. Correct density (lb/ft³): 0.344006015; Wrong density (lb/ft³): 0.29343713; Underestimate (%): 14.7.",
      "Absolute pressure (psia) = 500. Correct density (lb/ft³): 1.72003007; Wrong density (lb/ft³): 1.66946119; Underestimate (%): 2.94.",
      "Absolute pressure (psia) = 3000. Correct density (lb/ft³): 10.3201804; Wrong density (lb/ft³): 10.2696116; Underestimate (%): 0.49."
    ],
    "provenance": "Original synthetic teaching calculation. Equations and fixed controls are stated here; these are not field measurements.",
    "questions": [
      {
        "question": "Reproduce the baseline Correct density (lb/ft³) without rounding intermediate values.",
        "answer": "At 100 psia the wrong pressure is 85.3; density ratio = 85.3/100 = 0.853, a 14.7% underestimate. Correct density (lb/ft³) = 0.344006015. The relative datum error grows at low absolute pressure under these fixed-property controls."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Absolute pressure (psia) changes from 25 to 3000. Other controls remain as stated in the premise. Correct density (lb/ft³) changes from 0.0860015037 to 10.3201804."
      },
      {
        "question": "What would be needed to apply this example to a different physical system?",
        "answer": "Replace the synthetic controls with compatible documented inputs and check the relation's assumptions. The fixed-Z control isolates pressure reference. The wrong column is deliberately invalid as a state input. The relative datum error grows at low absolute pressure under these fixed-property controls."
      }
    ]
  },
  "recorded": null
}