polykin.thermo.eos¤

Virial ¤

Bases: GasEoS

Virial equation of state truncated to the second coefficient.

This EoS is based on the following \(Z(T,P)\) relationship:

\[ Z = 1 + \frac{B_m P}{R T} \]

where \(P\) is the pressure, \(T\) is the temperature, and \(B_m\) is the mixture second virial coefficient. The parameter \(B_m=B_m(T,y)\) is estimated based on the critical properties of the pure components and the mixture composition \(y\).

Important

This equation is an improvement over the ideal gas model, but it should only be used up to moderate pressures such that \(v/v_c > 2\).

References

RC Reid, JM Prausniz, and BE Poling. The properties of gases & liquids 4th edition, 1986, p. 37, 40, 80, 82, 145.

PARAMETER	DESCRIPTION
`Tc`	Critical temperatures of all components [K]. TYPE: `float \| FloatVectorLike(N)`
`Pc`	Critical pressures of all components [Pa]. TYPE: `float \| FloatVectorLike(N)`
`Zc`	Critical compressibility factors of all components. TYPE: `float \| FloatVectorLike(N)`
`w`	Acentric factors of all components. TYPE: `float \| FloatVectorLike(N)`
`name`	Name. TYPE: `str` DEFAULT: `''`

Source code in src/polykin/thermo/eos/virial.py

class Virial(GasEoS):
    r"""Virial equation of state truncated to the second coefficient.

    This EoS is based on the following $Z(T,P)$ relationship:

    $$ Z = 1 + \frac{B_m P}{R T} $$

    where $P$ is the pressure, $T$ is the temperature, and $B_m$ is the mixture
    second virial coefficient. The parameter $B_m=B_m(T,y)$ is estimated based
    on the critical properties of the pure components and the mixture composition
    $y$.

    !!! important

        This equation is an improvement over the ideal gas model, but it
        should only be used up to moderate pressures such that $v/v_c > 2$.

    **References**

    *   RC Reid, JM Prausniz, and BE Poling. The properties of gases &
        liquids 4th edition, 1986, p. 37, 40, 80, 82, 145.

    Parameters
    ----------
    Tc : float | FloatVectorLike (N)
        Critical temperatures of all components [K].
    Pc : float | FloatVectorLike (N)
        Critical pressures of all components [Pa].
    Zc : float | FloatVectorLike (N)
        Critical compressibility factors of all components.
    w : float | FloatVectorLike (N)
        Acentric factors of all components.
    name : str
        Name.
    """

    Tc: FloatVector
    Pc: FloatVector
    Zc: FloatVector
    w: FloatVector

    def __init__(
        self,
        Tc: float | FloatVectorLike,
        Pc: float | FloatVectorLike,
        Zc: float | FloatVectorLike,
        w: float | FloatVectorLike,
        name: str = "",
    ) -> None:

        Tc, Pc, Zc, w = convert_FloatOrVectorLike_to_FloatVector([Tc, Pc, Zc, w])

        N = len(Tc)
        super().__init__(N, name)
        self.Tc = Tc
        self.Pc = Pc
        self.Zc = Zc
        self.w = w

    @functools.cache
    def Bij(self, T: float) -> FloatSquareMatrix:
        r"""Calculate the matrix of interaction virial coefficients.

        The calculation is handled by [`B_mixture`](../../properties/pvt/B_mixture.md).

        Parameters
        ----------
        T : float
            Temperature [K].

        Returns
        -------
        FloatSquareMatrix (N,N)
            Matrix of interaction virial coefficients, $B_{ij}$ [m³/mol].
        """
        return B_mixture(T, self.Tc, self.Pc, self.Zc, self.w)

    def Bm(self, T: float, y: FloatVector) -> float:
        r"""Calculate the second virial coefficient of the mixture.

        $$ B_m = \sum_i \sum_j y_i y_j B_{ij} $$

        **References**

        *   RC Reid, JM Prausniz, and BE Poling. The properties of gases &
            liquids 4th edition, 1986, p. 79.

        Parameters
        ----------
        T : float
            Temperature [K].
        y : FloatVector (N)
            Mole fractions of all components [mol/mol].

        Returns
        -------
        float
            Mixture second virial coefficient, $B_m$ [m³/mol].
        """
        return quadratic_mixing(y, self.Bij(T))

    def Z(self, T: float, P: float, y: FloatVector) -> float:
        r"""Calculate the compressibility factor of the fluid.

        $$ Z = 1 + \frac{B_m P}{R T} $$

        Parameters
        ----------
        T : float
            Temperature [K].
        P : float
            Pressure [Pa].
        y : FloatVector (N)
            Mole fractions of all components [mol/mol].

        Returns
        -------
        float
            Compressibility factor.
        """
        Bm = self.Bm(T, y)
        return 1.0 + Bm * P / (R * T)

    @override
    def gR(self, T: float, P: float, y: FloatVector) -> float:
        """Calculate the molar residual Gibbs energy of the fluid.

        $$ g^R = B_m P $$

        Parameters
        ----------
        T : float
            Temperature [K].
        P : float
            Pressure [Pa].
        y : FloatVector
            Mole fractions of all components [mol/mol].

        Returns
        -------
        float
            Molar residual Gibbs energy [J/mol].
        """
        return self.Bm(T, y) * P

    @override
    def P(self, T: float, v: float, y: FloatVector) -> float:
        r"""Calculate the pressure of the fluid.

        $$ P = \frac{R T}{v - B_m} $$

        Parameters
        ----------
        T : float
            Temperature [K].
        v : float
            Molar volume [m³/mol].
        y : FloatVector (N)
            Mole fractions of all components [mol/mol].

        Returns
        -------
        float
            Pressure [Pa].
        """
        Bm = self.Bm(T, y)
        return R * T / (v - Bm)

    @override
    def phi(self, T: float, P: float, y: FloatVector) -> FloatVector:
        r"""Calculate the fugacity coefficients of all components.

        $$ \ln \hat{\phi}_i
           = \left(2\sum_j {y_jB_{ij}} -B_m \right)\frac{P}{RT} $$

        **References**

        *   RC Reid, JM Prausniz, and BE Poling. The properties of gases &
            liquids 4th edition, 1986, p. 145.

        Parameters
        ----------
        T : float
            Temperature [K].
        P : float
            Pressure [Pa].
        y : FloatVector (N)
            Mole fractions of all components [mol/mol].

        Returns
        -------
        FloatVector (N)
            Fugacity coefficients of all components.
        """
        B = self.Bij(T)
        Bm = self.Bm(T, y)
        return exp((2 * dot(B, y) - Bm) * P / (R * T))

    @override
    def kappa(self, T: float, P: float, y: FloatVector) -> float:
        r"""Calculate the isothermal compressibility coefficient.

        $$ \kappa \equiv
           - \frac{1}{v} \left( \frac{\partial v}{\partial P} \right)_T
           = \frac{1}{P Z} $$

        Parameters
        ----------
        T : float
            Temperature [K].
        P : float
            Pressure [Pa].
        y : FloatVector (N)
            Mole fractions of all components [mol/mol].

        Returns
        -------
        float
            Isothermal compressibility coefficient, $\kappa$ [Pa⁻¹].
        """
        Z = self.Z(T, P, y)
        return 1 / (P * Z)

Bij `cached` ¤

Bij(T: float) -> FloatSquareMatrix

Calculate the matrix of interaction virial coefficients.

The calculation is handled by B_mixture.

PARAMETER	DESCRIPTION
`T`	Temperature [K]. TYPE: `float`

RETURNS	DESCRIPTION
`FloatSquareMatrix(N, N)`	Matrix of interaction virial coefficients, \(B_{ij}\) [m³/mol].

Source code in src/polykin/thermo/eos/virial.py

@functools.cache
def Bij(self, T: float) -> FloatSquareMatrix:
    r"""Calculate the matrix of interaction virial coefficients.

    The calculation is handled by [`B_mixture`](../../properties/pvt/B_mixture.md).

    Parameters
    ----------
    T : float
        Temperature [K].

    Returns
    -------
    FloatSquareMatrix (N,N)
        Matrix of interaction virial coefficients, $B_{ij}$ [m³/mol].
    """
    return B_mixture(T, self.Tc, self.Pc, self.Zc, self.w)

Bm ¤

Bm(T: float, y: FloatVector) -> float

Calculate the second virial coefficient of the mixture.

\[ B_m = \sum_i \sum_j y_i y_j B_{ij} \]

References

RC Reid, JM Prausniz, and BE Poling. The properties of gases & liquids 4th edition, 1986, p. 79.

PARAMETER	DESCRIPTION
`T`	Temperature [K]. TYPE: `float`
`y`	Mole fractions of all components [mol/mol]. TYPE: `FloatVector(N)`

RETURNS	DESCRIPTION
`float`	Mixture second virial coefficient, \(B_m\) [m³/mol].

Source code in src/polykin/thermo/eos/virial.py

def Bm(self, T: float, y: FloatVector) -> float:
    r"""Calculate the second virial coefficient of the mixture.

    $$ B_m = \sum_i \sum_j y_i y_j B_{ij} $$

    **References**

    *   RC Reid, JM Prausniz, and BE Poling. The properties of gases &
        liquids 4th edition, 1986, p. 79.

    Parameters
    ----------
    T : float
        Temperature [K].
    y : FloatVector (N)
        Mole fractions of all components [mol/mol].

    Returns
    -------
    float
        Mixture second virial coefficient, $B_m$ [m³/mol].
    """
    return quadratic_mixing(y, self.Bij(T))

DA ¤

DA(T: float, V: float, n: FloatVector, v0: float) -> float

Calculate the departure of Helmholtz energy.

\[ A(T,V,n) - A^{\circ}(T,V,n)\]

PARAMETER	DESCRIPTION
`T`	Temperature [K]. TYPE: `float`
`V`	Volume [m³]. TYPE: `float`
`n`	Mole amounts of all components [mol]. TYPE: `FloatVector(N)`
`v0`	Molar volume in reference state [m³/mol]. TYPE: `float`

RETURNS	DESCRIPTION
`float`	Helmholtz energy departure, \(A - A^{\circ}\) [J].

Source code in src/polykin/thermo/eos/base.py

def DA(
    self,
    T: float,
    V: float,
    n: FloatVector,
    v0: float,
) -> float:
    r"""Calculate the departure of Helmholtz energy.

    $$ A(T,V,n) - A^{\circ}(T,V,n)$$

    Parameters
    ----------
    T : float
        Temperature [K].
    V : float
        Volume [m³].
    n : FloatVector (N)
        Mole amounts of all components [mol].
    v0 : float
        Molar volume in reference state [m³/mol].

    Returns
    -------
    float
        Helmholtz energy departure, $A - A^{\circ}$ [J].
    """

N `property` ¤

N: int

Number of components.

P ¤

P(T: float, v: float, y: FloatVector) -> float

Calculate the pressure of the fluid.

\[ P = \frac{R T}{v - B_m} \]

PARAMETER	DESCRIPTION
`T`	Temperature [K]. TYPE: `float`
`v`	Molar volume [m³/mol]. TYPE: `float`
`y`	Mole fractions of all components [mol/mol]. TYPE: `FloatVector(N)`

RETURNS	DESCRIPTION
`float`	Pressure [Pa].

Source code in src/polykin/thermo/eos/virial.py

@override
def P(self, T: float, v: float, y: FloatVector) -> float:
    r"""Calculate the pressure of the fluid.

    $$ P = \frac{R T}{v - B_m} $$

    Parameters
    ----------
    T : float
        Temperature [K].
    v : float
        Molar volume [m³/mol].
    y : FloatVector (N)
        Mole fractions of all components [mol/mol].

    Returns
    -------
    float
        Pressure [Pa].
    """
    Bm = self.Bm(T, y)
    return R * T / (v - Bm)

Z ¤

Z(T: float, P: float, y: FloatVector) -> float

Calculate the compressibility factor of the fluid.

\[ Z = 1 + \frac{B_m P}{R T} \]

PARAMETER	DESCRIPTION
`T`	Temperature [K]. TYPE: `float`
`P`	Pressure [Pa]. TYPE: `float`
`y`	Mole fractions of all components [mol/mol]. TYPE: `FloatVector(N)`

RETURNS	DESCRIPTION
`float`	Compressibility factor.

Source code in src/polykin/thermo/eos/virial.py

def Z(self, T: float, P: float, y: FloatVector) -> float:
    r"""Calculate the compressibility factor of the fluid.

    $$ Z = 1 + \frac{B_m P}{R T} $$

    Parameters
    ----------
    T : float
        Temperature [K].
    P : float
        Pressure [Pa].
    y : FloatVector (N)
        Mole fractions of all components [mol/mol].

    Returns
    -------
    float
        Compressibility factor.
    """
    Bm = self.Bm(T, y)
    return 1.0 + Bm * P / (R * T)

aR ¤

aR(T: float, P: float, y: FloatVector) -> float

Calculate the molar residual Helmholtz energy of the fluid.

\[ a^R = g^R - R T (Z - 1) \]

PARAMETER	DESCRIPTION
`T`	Temperature [K]. TYPE: `float`
`P`	Pressure [Pa]. TYPE: `float`
`y`	Mole fractions of all components [mol/mol]. TYPE: `FloatVector`

RETURNS	DESCRIPTION
`float`	Molar residual Helmholtz energy [J/mol].

Source code in src/polykin/thermo/eos/base.py

def aR(self, T: float, P: float, y: FloatVector) -> float:
    r"""Calculate the molar residual Helmholtz energy of the fluid.

    $$ a^R = g^R - R T (Z - 1) $$

    Parameters
    ----------
    T : float
        Temperature [K].
    P : float
        Pressure [Pa].
    y : FloatVector
        Mole fractions of all components [mol/mol].

    Returns
    -------
    float
        Molar residual Helmholtz energy [J/mol].
    """
    return self.gR(T, P, y) - R * T * (self.Z(T, P, y) - 1.0)

beta ¤

beta(T: float, P: float, y: FloatVector) -> float

Calculate the thermal expansion coefficient.

\[ \beta \equiv \frac{1}{v} \left( \frac{\partial v}{\partial T} \right)_{P,y_i} = \frac{1}{T} + \frac{1}{Z} \left( \frac{\partial Z}{\partial T} \right)_{P,y_i} \]

PARAMETER	DESCRIPTION
`T`	Temperature [K]. TYPE: `float`
`P`	Pressure [Pa]. TYPE: `float`
`y`	Mole fractions of all components [mol/mol]. TYPE: `FloatVector(N)`

RETURNS	DESCRIPTION
`float`	Thermal expansion coefficient, \(\beta\) [K⁻¹].

Source code in src/polykin/thermo/eos/base.py

def beta(self, T: float, P: float, y: FloatVector) -> float:
    r"""Calculate the thermal expansion coefficient.

    $$ \beta \equiv
     \frac{1}{v} \left( \frac{\partial v}{\partial T} \right)_{P,y_i}
     = \frac{1}{T}
     + \frac{1}{Z} \left( \frac{\partial Z}{\partial T} \right)_{P,y_i} $$

    Parameters
    ----------
    T : float
        Temperature [K].
    P : float
        Pressure [Pa].
    y : FloatVector (N)
        Mole fractions of all components [mol/mol].

    Returns
    -------
    float
        Thermal expansion coefficient, $\beta$ [K⁻¹].
    """
    dZdT, Z = derivative_centered(lambda T_: self.Z(T_, P, y), T)

    return 1 / T + dZdT / Z

f ¤

f(T: float, P: float, y: FloatVector) -> FloatVector

Calculate the fugacity of all components.

\[ \hat{f}_i = \hat{\phi}_i y_i P \]

PARAMETER	DESCRIPTION
`T`	Temperature [K]. TYPE: `float`
`P`	Pressure [Pa]. TYPE: `float`
`y`	Mole fractions of all components [mol/mol]. TYPE: `FloatVector(N)`

RETURNS	DESCRIPTION
`FloatVector(N)`	Fugacities of all components [Pa].

Source code in src/polykin/thermo/eos/base.py

def f(self, T: float, P: float, y: FloatVector) -> FloatVector:
    r"""Calculate the fugacity of all components.

    $$ \hat{f}_i = \hat{\phi}_i y_i P $$

    Parameters
    ----------
    T : float
        Temperature [K].
    P : float
        Pressure [Pa].
    y : FloatVector (N)
        Mole fractions of all components [mol/mol].

    Returns
    -------
    FloatVector (N)
        Fugacities of all components [Pa].
    """
    return self.phi(T, P, y) * y * P

gR ¤

gR(T: float, P: float, y: FloatVector) -> float

Calculate the molar residual Gibbs energy of the fluid.

\[ g^R = B_m P \]

PARAMETER	DESCRIPTION
`T`	Temperature [K]. TYPE: `float`
`P`	Pressure [Pa]. TYPE: `float`
`y`	Mole fractions of all components [mol/mol]. TYPE: `FloatVector`

RETURNS	DESCRIPTION
`float`	Molar residual Gibbs energy [J/mol].

Source code in src/polykin/thermo/eos/virial.py

@override
def gR(self, T: float, P: float, y: FloatVector) -> float:
    """Calculate the molar residual Gibbs energy of the fluid.

    $$ g^R = B_m P $$

    Parameters
    ----------
    T : float
        Temperature [K].
    P : float
        Pressure [Pa].
    y : FloatVector
        Mole fractions of all components [mol/mol].

    Returns
    -------
    float
        Molar residual Gibbs energy [J/mol].
    """
    return self.Bm(T, y) * P

hR ¤

hR(T: float, P: float, y: FloatVector) -> float

Calculate the molar residual enthalpy of the fluid.

\[ h^R = g^R + T s^R \]

PARAMETER	DESCRIPTION
`T`	Temperature [K]. TYPE: `float`
`P`	Pressure [Pa]. TYPE: `float`
`y`	Mole fractions of all components [mol/mol]. TYPE: `FloatVector`

RETURNS	DESCRIPTION
`float`	Molar residual enthalpy [J/mol].

Source code in src/polykin/thermo/eos/base.py

def hR(self, T: float, P: float, y: FloatVector) -> float:
    r"""Calculate the molar residual enthalpy of the fluid.

    $$ h^R = g^R + T s^R $$

    Parameters
    ----------
    T : float
        Temperature [K].
    P : float
        Pressure [Pa].
    y : FloatVector
        Mole fractions of all components [mol/mol].

    Returns
    -------
    float
        Molar residual enthalpy [J/mol].
    """
    return self.gR(T, P, y) + T * self.sR(T, P, y)

kappa ¤

kappa(T: float, P: float, y: FloatVector) -> float

Calculate the isothermal compressibility coefficient.

\[ \kappa \equiv - \frac{1}{v} \left( \frac{\partial v}{\partial P} \right)_T = \frac{1}{P Z} \]

PARAMETER	DESCRIPTION
`T`	Temperature [K]. TYPE: `float`
`P`	Pressure [Pa]. TYPE: `float`
`y`	Mole fractions of all components [mol/mol]. TYPE: `FloatVector(N)`

RETURNS	DESCRIPTION
`float`	Isothermal compressibility coefficient, \(\kappa\) [Pa⁻¹].

Source code in src/polykin/thermo/eos/virial.py

@override
def kappa(self, T: float, P: float, y: FloatVector) -> float:
    r"""Calculate the isothermal compressibility coefficient.

    $$ \kappa \equiv
       - \frac{1}{v} \left( \frac{\partial v}{\partial P} \right)_T
       = \frac{1}{P Z} $$

    Parameters
    ----------
    T : float
        Temperature [K].
    P : float
        Pressure [Pa].
    y : FloatVector (N)
        Mole fractions of all components [mol/mol].

    Returns
    -------
    float
        Isothermal compressibility coefficient, $\kappa$ [Pa⁻¹].
    """
    Z = self.Z(T, P, y)
    return 1 / (P * Z)

phi ¤

phi(T: float, P: float, y: FloatVector) -> FloatVector

Calculate the fugacity coefficients of all components.

\[ \ln \hat{\phi}_i = \left(2\sum_j {y_jB_{ij}} -B_m \right)\frac{P}{RT} \]

References

RC Reid, JM Prausniz, and BE Poling. The properties of gases & liquids 4th edition, 1986, p. 145.

PARAMETER	DESCRIPTION
`T`	Temperature [K]. TYPE: `float`
`P`	Pressure [Pa]. TYPE: `float`
`y`	Mole fractions of all components [mol/mol]. TYPE: `FloatVector(N)`

RETURNS	DESCRIPTION
`FloatVector(N)`	Fugacity coefficients of all components.

Source code in src/polykin/thermo/eos/virial.py

@override
def phi(self, T: float, P: float, y: FloatVector) -> FloatVector:
    r"""Calculate the fugacity coefficients of all components.

    $$ \ln \hat{\phi}_i
       = \left(2\sum_j {y_jB_{ij}} -B_m \right)\frac{P}{RT} $$

    **References**

    *   RC Reid, JM Prausniz, and BE Poling. The properties of gases &
        liquids 4th edition, 1986, p. 145.

    Parameters
    ----------
    T : float
        Temperature [K].
    P : float
        Pressure [Pa].
    y : FloatVector (N)
        Mole fractions of all components [mol/mol].

    Returns
    -------
    FloatVector (N)
        Fugacity coefficients of all components.
    """
    B = self.Bij(T)
    Bm = self.Bm(T, y)
    return exp((2 * dot(B, y) - Bm) * P / (R * T))

sR ¤

sR(T: float, P: float, y: FloatVector) -> float

Calculate the molar residual entropy of the fluid.

\[ s^R = -\left( \frac{\partial g^R}{\partial T} \right)_{P,y_i} \]

PARAMETER	DESCRIPTION
`T`	Temperature [K]. TYPE: `float`
`P`	Pressure [Pa]. TYPE: `float`
`y`	Mole fractions of all components [mol/mol]. TYPE: `FloatVector`

RETURNS	DESCRIPTION
`float`	Molar residual entropy [J/(mol·K)].

Source code in src/polykin/thermo/eos/base.py

def sR(self, T: float, P: float, y: FloatVector) -> float:
    r"""Calculate the molar residual entropy of the fluid.

    $$ s^R = -\left( \frac{\partial g^R}{\partial T} \right)_{P,y_i} $$

    Parameters
    ----------
    T : float
        Temperature [K].
    P : float
        Pressure [Pa].
    y : FloatVector
        Mole fractions of all components [mol/mol].

    Returns
    -------
    float
        Molar residual entropy [J/(mol·K)].
    """
    return -1 * derivative_centered(lambda T_: self.gR(T_, P, y), T)[0]

v ¤

v(T: float, P: float, y: FloatVector) -> float

Calculate the molar volume the fluid.

\[ v = Z \frac{R T}{P} \]

PARAMETER	DESCRIPTION
`T`	Temperature [K]. TYPE: `float`
`P`	Pressure [Pa]. TYPE: `float`
`y`	Mole fractions of all components [mol/mol]. TYPE: `FloatVector(N)`

RETURNS	DESCRIPTION
`float`	Molar volume of the fluid [m³/mol].

Source code in src/polykin/thermo/eos/base.py

def v(self, T: float, P: float, y: FloatVector) -> float:
    r"""Calculate the molar volume the fluid.

    $$ v = Z \frac{R T}{P} $$

    Parameters
    ----------
    T : float
        Temperature [K].
    P : float
        Pressure [Pa].
    y : FloatVector (N)
        Mole fractions of all components [mol/mol].

    Returns
    -------
    float
        Molar volume of the fluid [m³/mol].
    """
    return self.Z(T, P, y) * R * T / P

vR ¤

vR(T: float, P: float, y: FloatVector) -> float

Calculate the molar residual volume of the fluid.

\[ v^R = \left(Z - 1 \right) \frac{R T}{P} \]

PARAMETER	DESCRIPTION
`T`	Temperature [K]. TYPE: `float`
`P`	Pressure [Pa]. TYPE: `float`
`y`	Mole fractions of all components [mol/mol]. TYPE: `FloatVector`

RETURNS	DESCRIPTION
`float`	Molar residual volume [m³/mol].

Source code in src/polykin/thermo/eos/base.py

def vR(self, T: float, P: float, y: FloatVector) -> float:
    r"""Calculate the molar residual volume of the fluid.

    $$ v^R = \left(Z - 1 \right) \frac{R T}{P} $$

    Parameters
    ----------
    T : float
        Temperature [K].
    P : float
        Pressure [Pa].
    y : FloatVector
        Mole fractions of all components [mol/mol].

    Returns
    -------
    float
        Molar residual volume [m³/mol].
    """
    return R * T / P * (self.Z(T, P, y) - 1.0)

Examples¤

Estimate the molar volume of a 50 mol% ethylene/nitrogen at 350 K and 10 bar.

from polykin.thermo.eos import Virial
import numpy as np

Tc = [282.4, 126.2]    # K
Pc = [50.4e5, 33.9e5]  # Pa
Zc = [0.280, 0.290]
w = [0.089, 0.039]

eos = Virial(Tc, Pc, Zc, w)
v = eos.v(T=350., P=10e5, y=np.array([0.5, 0.5]))

print(f"{v:.2e} m³/mol")

2.87e-03 m³/mol

polykin.thermo.eos¤

Virial ¤

Bij cached ¤

Bm ¤

DA ¤

N property ¤

P ¤

Z ¤

aR ¤

beta ¤

f ¤

gR ¤

hR ¤

kappa ¤

phi ¤

sR ¤

v ¤

vR ¤

Examples¤

Bij `cached` ¤

N `property` ¤