Phase Properties and Their Derivatives¶

Homogeneous state before equilibrium¶

A thermodynamic property at a supplied \((T,P,\boldsymbol{x})\) state is a different question from the equilibrium phase split of an overall \((T,P,\boldsymbol{z})\) feed. phase_properties evaluates the former directly and never calls a flash solver.

For a selected equation-of-state root,

\[ Z=\frac{Pv}{RT},\qquad f_i=x_i\phi_iP, \]

and, with standard pressure \(p^\circ=1\ \mathrm{bar}\),

\[ \ln\left(\frac{f_i}{p^\circ}\right) = \ln x_i+\ln\phi_i+\ln\left(\frac{P}{p^\circ}\right). \]

The default standard-state convention defines

\[ \mu_i = RT\ln\left(\frac{f_i}{p^\circ}\right). \]

It intentionally does not invent ideal-gas formation terms. Supply an explicit standard state when absolute chemical potentials are required.

The reference-independent residual Gibbs departure is

\[ \frac{g^R}{RT}=\sum_i x_i\ln\phi_i, \]

and the corresponding residual Helmholtz departure is

\[ \frac{a^R}{RT} = \frac{g^R}{RT}-Z+1+\ln Z. \]

These relationships follow the partial-molar-property and fugacity treatment in Michelsen and Mollerup, chapter 1.

Derivative coordinates¶

state_derivatives returns first derivatives at one scalar-\(T\), scalar-\(P\) state in three composition coordinate systems:

softmax logits: unconstrained coordinates mapped to normalized composition;
independent mole fractions: \(x_1,\ldots,x_{n-1}\), with \(x_n=1-\sum_{i=1}^{n-1}x_i\);
mole numbers: evaluated at \(n_i=x_i\) mol on a one-mole basis.

Temperature and pressure derivatives hold composition fixed. Mole-number derivatives hold \(T\) and \(P\) fixed. Because the differentiated properties are intensive, a mole-number derivative at another total amount is the one-mole-basis result divided by that amount in mol.

The returned families include derivatives of:

fugacity coefficients and their logarithms;
fugacities and dimensionless log fugacities;
chemical potentials and reduced chemical potentials;
molar volume; and
molar Gibbs energy with respect to \(T\) and \(P\).

Fugacity derivatives have Pa per coordinate units, chemical-potential derivatives have J/mol per coordinate units, and molar-volume derivatives have m3/mol per coordinate units. Composition must be strictly positive because logarithmic derivatives are singular on the simplex boundary.

Runnable example¶

"""Evaluate homogeneous phase properties and their TP derivatives."""

import torch

from torch_flash import (
    ChemicalState,
    component_set,
    configure,
    peng_robinson_1978,
    phase_properties,
    state_derivatives,
)

runtime = configure(device="cpu", dtype=torch.float64)
model = peng_robinson_1978(component_set(("methane", "n_butane")))
state = ChemicalState(
    temperature=runtime.tensor(300.0),  # K
    pressure=runtime.tensor(5.0e6),  # Pa
    composition=runtime.tensor([0.70, 0.30]),  # mol/mol
)

# This evaluates the supplied homogeneous state; it does not run a flash.
properties = phase_properties(model, state, phase="stable")
derivatives = state_derivatives(model, state, phase="stable")

print(f"Z = {float(properties.compressibility_factor):.6f}")
print(f"molar volume = {float(properties.molar_volume):.6e} m^3/mol")
print("fugacities [Pa] =", properties.fugacities)
print("d ln(f_i/p°) / dT [1/K] =", derivatives.dlog_fugacity_dtemperature)
print("dV_m / dP [m^3/(mol Pa)] =", derivatives.dmolar_volume_dpressure)

# For a binary mixture, the single independent coordinate is x_methane;
# x_n-butane = 1 - x_methane.
print(
    "d mu_i / d x_methane [J/mol] =",
    derivatives.dchemical_potential_dindependent_composition[:, 0],
)

The derivative implementation uses torch.func transformations rather than finite differences. Native model operations remain differentiable with respect to state variables and trainable model tensors unless the API explicitly documents a nondifferentiable root-location boundary.

Caloric properties¶

For scalar states, phase_properties(..., caloric=True) evaluates residual enthalpy and entropy from temperature derivatives of the residual Gibbs energy. Absolute enthalpy, entropy, heat capacity, speed of sound, and Joule-Thomson quantities require an ideal-gas reference model and are exposed through thermal_properties.

from torch_flash import poling_ideal_gas, thermal_properties

ideal = poling_ideal_gas(["methane", "n_butane"])
thermal = thermal_properties(model, state, ideal, phase="stable")
print(thermal.molar_enthalpy)
print(thermal.isobaric_heat_capacity)
print(thermal.speed_of_sound)

The source and validity of the ideal-gas polynomial matter just as much as the residual equation of state. See the parameter guide and reference index.

Numerical cautions¶

A derivative of a selected root is local to that branch; it is not a derivative through a phase switch.
Root coalescence and critical conditions can make response functions ill-conditioned.
Float32 is a separate accuracy study, not an interchangeable equilibrium reference precision.
Do not detach tensors or convert them to NumPy inside a differentiable objective.
Validate derivative shape, units, and coordinates before coupling the result to an optimizer or simulator.