Complete Density-Root Enumeration of Pressure-Density Isotherms via the Find-and-Hide Strategy

• Complete density-root enumeration for any pressure-explicit equation of state • Deterministic 1D Find engine: extrema → monotone brackets → no missed roots • Bracketed solves yield all roots with predictable, low EOS-evaluation counts • Pure fluids (cyclohexane, n-alkanes, benzene) show 1/3/5-root regimes • Asymmetric ternary PC-SAFT mixture confirms re-entrant 1/3/5-root regions An algorithm is presented for complete density-root enumeration of pressure-density isotherms generated by equations of state that provide an evaluable mapping p = p ( T , ρ , x ) as a function of density (including Helmholtz-energy mixture models in which pressure is obtained from derivatives of a Helmholtz potential). The strategy is constructed within the Find-and-Hide framework and remains solver-agnostic, requiring only black-box evaluations of the isotherm and, when available, optional derivative information to accelerate the localization of extrema. The key idea is to exploit the one-dimensional structure of isotherm inversion: isotherm extrema are determined at fixed temperature, the admissible density domain is decomposed into monotone intervals, and each interval is searched with a bracket-preserving procedure to guarantee that no real density root is missed. This yields an enumeration workflow with predictable evaluation counts per root and eliminates the implicit “at most three roots” assumption common in classical equation-of-state inversion routines. The method is demonstrated using PC-SAFT as a demanding case study (up to five real density roots) and is further validated using a GERG-2008-type equimolar nitrogen+argon mixture example that exhibits seven real density roots at fixed p = p ( T , ρ , x ) . Numerical experiments for cyclohexane at 120 K, complemented by additional isotherm families for n-alkanes and benzene and by an asymmetric ternary-mixture case ( n − C 4 / n − C 8 / n − C 12 ; z = 0.05 / 0.05 / 0.90 ), illustrate compound-, mixture-, and temperature-dependent root multiplicity, including re-entrant three-root windows at elevated pressures. These results support treating equation-of-state inversion as an enumeration problem whenever phase-equilibrium or stability calculations may be sensitive to missing density solutions.