General Equilibrium ICE-Table Solver mechanism

Model scope

The core ICE-table and mass-action approach is standard, but the specification is not complete enough for an unqualified PASS. The exact unresolved ambiguity is the meaning and units of K: Kc, Kp, or activity-based K. The solver also needs explicit balancing validation, phase handling, admissible-extent constraints, residual verification, and safeguarded root-finding. It should honestly label results as ideal dilute-solution concentration calculations unless activities are supported.

• ICE-table setup plus numerical solution of the mass-action K expression is the standard method only after the reaction is balanced and stoichiometric coefficients are applied correctly to every species.
• The specification does not define whether K means Kc, Kp, or an activity-based thermodynamic equilibrium constant. Because the inputs are concentrations, the default must be explicitly Kc or activities approximated by concentrations.
• Newton-Raphson alone is not a sufficiently safe general solver: each iteration must enforce the physically admissible extent interval so that every equilibrium amount remains nonnegative and every denominator remains nonzero.
• The implementation must support negative extents when products are present initially or reactants are absent initially.
• The implementation must reject or separately handle pure solids and liquids, which do not belong in the equilibrium-constant expression, and must distinguish them from aqueous or gaseous concentration terms.
• The implementation must validate that the supplied equation is balanced and that all initial concentrations are nonnegative.

Units and conventions

• All listed concentrations are assumed to be in mol/L (M).
• K is assumed to be dimensionless thermodynamic Kc based on activities, or numerically treated as concentration-based Kc under the ideal dilute-solution approximation; the UI must state which convention is used.
• For a concentration expression, each species concentration is raised to its stoichiometric coefficient and pure solids or liquids are omitted.
• The extent x has units of mol/L when the ICE table is expressed in concentrations.
• If Kp is accepted, gas partial pressures and temperature are required; a pressure K cannot be used with concentration inputs without the appropriate conversion.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "balanced_equation": "A ⇌ B",
    "initial_concentrations": {
      "A": "1.0000000 M",
      "B": "0.0000000 M"
    },
    "Kc": 4
  },
  "expected_outputs": {
    "K_expression": "Kc = [B]/[A] = x/(1-x)",
    "extent_x": "0.80000000 M",
    "equilibrium_concentrations": {
      "A": "0.20000000 M",
      "B": "0.80000000 M"
    },
    "ice_table": {
      "A": {
        "initial": "1.00000000 M",
        "change": "-0.80000000 M",
        "equilibrium": "0.20000000 M"
      },
      "B": {
        "initial": "0.00000000 M",
        "change": "+0.80000000 M",
        "equilibrium": "0.80000000 M"
      }
    },
    "mass_action_check": "0.80000000/0.20000000 = 4.0000000"
  },
  "reference_data": {
    "method": "Solve x/(1-x)=4 within 0 <= x <= 1."
  },
  "working": "Initial: A=1, B=0. Change for A ⇌ B: A decreases by x and B increases by x. Therefore [A]eq=1-x and [B]eq=x. Set Kc=4: x/(1-x)=4. Multiply through: x=4-4x. Add 4x: 5x=4. Divide by 5: x=0.80000000 M. Then [A]eq=1-0.8=0.20000000 M and [B]eq=0+0.8=0.80000000 M. Check: 0.80000000/0.20000000=4.0000000."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

A naive Newton-Raphson implementation initialized at x=0 encounters K=[B]/[A]^2=1/0 for the 2A ⇌ B trap, producing an undefined function value before it can iterate. The solver also must explicitly reject algebraic roots that produce negative concentrations and must expose the extent-direction convention.

Other records

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json
• evidence-tranche8/tranche8-results.json
• BOUNDARY_HAZARD_CATALOG.json