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MolecularACM ¤

Abstract base class for activity coefficient models for molecular (i.e., non-polymeric) systems.

A molecular ACM is defined in terms of a molar excess Gibbs energy model,

\[ g^{E} = g^{E}(T, x_i) \]

typically evaluated at constant pressure. All other thermodynamic solution properties are derived from \(g^{E}\).

To implement a specific molecular ACM, subclasses must:

  • Implement the gE method.
  • Preferably override the gamma method for efficiency.
Source code in src/polykin/thermo/acm/base.py 
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class MolecularACM(ACM):
    """Abstract base class for activity coefficient models for molecular
    (i.e., non-polymeric) systems.

    A molecular ACM is defined in terms of a molar excess Gibbs energy model,

    $$ g^{E} = g^{E}(T, x_i) $$

    typically evaluated at constant pressure. All other thermodynamic
    solution properties are derived from $g^{E}$.

    To implement a specific molecular ACM, subclasses must:

    * Implement the `gE` method.
    * Preferably override the `gamma` method for efficiency.
    """

    @abstractmethod
    def gE(self, T: Number, x: FloatVector) -> Number:
        r"""Calculate the molar excess Gibbs energy.

        $$ g^{E} \equiv g - g^{id} $$

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

        Returns
        -------
        float
            Molar excess Gibbs energy [J/mol].
        """

    def gamma(self, T: float, x: FloatVector) -> FloatVector:
        r"""Calculate the activity coefficients based on mole fraction.

        $$ \ln \gamma_i = \frac{1}{RT}
           \left( \frac{\partial (n g^E)}{\partial n_i} \right)_{T,P,n_j} $$

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

        Returns
        -------
        FloatVector (N)
            Activity coefficients of all components.
        """

        def GE(n: np.ndarray):
            """Total excess Gibbs energy."""
            nT = n.sum()
            return nT * self.gE(T, n / nT)

        dGEdn = jacobian_forward(GE, x)

        return exp(dGEdn / (R * T))

    def Dgmix(self, T: float, x: FloatVector) -> float:
        r"""Calculate the molar Gibbs energy of mixing.

        $$ \Delta_{mix} g = g^E + R T \sum_i {x_i \ln{x_i}} $$

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

        Returns
        -------
        float
            Molar Gibbs energy of mixing [J/mol].
        """
        return self.gE(T, x) - T * self._Dsmix_ideal(T, x)

    def Dhmix(self, T: float, x: FloatVector) -> float:
        r"""Calculate the molar enthalpy of mixing.

        $$ \Delta_{mix} h = h^{E} $$

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

        Returns
        -------
        float
            Molar enthalpy of mixing [J/mol].
        """
        return self.hE(T, x)

    def Dsmix(self, T: float, x: FloatVector) -> float:
        r"""Calculate the molar entropy of mixing.

        $$ \Delta_{mix} s = s^{E} - R \sum_i {x_i \ln{x_i}} $$

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

        Returns
        -------
        float
            Molar entropy of mixing [J/(mol·K)].
        """
        return self.sE(T, x) + self._Dsmix_ideal(T, x)

    def hE(self, T: float, x: FloatVector) -> float:
        r"""Calculate the molar excess enthalpy.

        $$ h^{E} = g^{E} + T s^{E} $$

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

        Returns
        -------
        float
            Molar excess enthalpy [J/mol].
        """
        return self.gE(T, x) + T * self.sE(T, x)

    def sE(self, T: float, x: FloatVector) -> float:
        r"""Calculate the molar excess entropy.

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

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

        Returns
        -------
        float
            Molar excess entropy [J/(mol·K)].
        """
        return -1 * derivative_complex(lambda T_: self.gE(T_, x), T)[0]

    def activity(self, T: float, x: FloatVector) -> FloatVector:
        r"""Calculate the activities.

        $$ a_i = x_i \gamma_i $$

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

        Returns
        -------
        FloatVector (N)
            Activities of all components.
        """
        return x * self.gamma(T, x)

Dgmix ¤

Dgmix(T: float, x: FloatVector) -> float

Calculate the molar Gibbs energy of mixing.

\[ \Delta_{mix} g = g^E + R T \sum_i {x_i \ln{x_i}} \]
PARAMETER DESCRIPTION
T

Temperature [K].

TYPE: float

x

Mole fractions of all components [mol/mol].

TYPE: FloatVector(N)

RETURNS DESCRIPTION
float

Molar Gibbs energy of mixing [J/mol].
Source code in src/polykin/thermo/acm/base.py 
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def Dgmix(self, T: float, x: FloatVector) -> float:
    r"""Calculate the molar Gibbs energy of mixing.

    $$ \Delta_{mix} g = g^E + R T \sum_i {x_i \ln{x_i}} $$

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

    Returns
    -------
    float
        Molar Gibbs energy of mixing [J/mol].
    """
    return self.gE(T, x) - T * self._Dsmix_ideal(T, x)

Dhmix ¤

Dhmix(T: float, x: FloatVector) -> float

Calculate the molar enthalpy of mixing.

\[ \Delta_{mix} h = h^{E} \]
PARAMETER DESCRIPTION
T

Temperature [K].

TYPE: float

x

Mole fractions of all components [mol/mol].

TYPE: FloatVector(N)

RETURNS DESCRIPTION
float

Molar enthalpy of mixing [J/mol].
Source code in src/polykin/thermo/acm/base.py 
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def Dhmix(self, T: float, x: FloatVector) -> float:
    r"""Calculate the molar enthalpy of mixing.

    $$ \Delta_{mix} h = h^{E} $$

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

    Returns
    -------
    float
        Molar enthalpy of mixing [J/mol].
    """
    return self.hE(T, x)

Dsmix ¤

Dsmix(T: float, x: FloatVector) -> float

Calculate the molar entropy of mixing.

\[ \Delta_{mix} s = s^{E} - R \sum_i {x_i \ln{x_i}} \]
PARAMETER DESCRIPTION
T

Temperature [K].

TYPE: float

x

Mole fractions of all components [mol/mol].

TYPE: FloatVector(N)

RETURNS DESCRIPTION
float

Molar entropy of mixing [J/(mol·K)].
Source code in src/polykin/thermo/acm/base.py 
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def Dsmix(self, T: float, x: FloatVector) -> float:
    r"""Calculate the molar entropy of mixing.

    $$ \Delta_{mix} s = s^{E} - R \sum_i {x_i \ln{x_i}} $$

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

    Returns
    -------
    float
        Molar entropy of mixing [J/(mol·K)].
    """
    return self.sE(T, x) + self._Dsmix_ideal(T, x)

N property ¤

N: int

Number of components.

activity ¤

activity(T: float, x: FloatVector) -> FloatVector

Calculate the activities.

\[ a_i = x_i \gamma_i \]
PARAMETER DESCRIPTION
T

Temperature [K].

TYPE: float

x

Mole fractions of all components [mol/mol].

TYPE: FloatVector(N)

RETURNS DESCRIPTION
FloatVector(N)

Activities of all components.
Source code in src/polykin/thermo/acm/base.py 
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def activity(self, T: float, x: FloatVector) -> FloatVector:
    r"""Calculate the activities.

    $$ a_i = x_i \gamma_i $$

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

    Returns
    -------
    FloatVector (N)
        Activities of all components.
    """
    return x * self.gamma(T, x)

gE abstractmethod ¤

gE(T: Number, x: FloatVector) -> Number

Calculate the molar excess Gibbs energy.

\[ g^{E} \equiv g - g^{id} \]
PARAMETER DESCRIPTION
T

Temperature [K].

TYPE: float

x

Mole fractions of all components [mol/mol].

TYPE: FloatVector(N)

RETURNS DESCRIPTION
float

Molar excess Gibbs energy [J/mol].
Source code in src/polykin/thermo/acm/base.py 
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@abstractmethod
def gE(self, T: Number, x: FloatVector) -> Number:
    r"""Calculate the molar excess Gibbs energy.

    $$ g^{E} \equiv g - g^{id} $$

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

    Returns
    -------
    float
        Molar excess Gibbs energy [J/mol].
    """

gamma ¤

gamma(T: float, x: FloatVector) -> FloatVector

Calculate the activity coefficients based on mole fraction.

\[ \ln \gamma_i = \frac{1}{RT} \left( \frac{\partial (n g^E)}{\partial n_i} \right)_{T,P,n_j} \]
PARAMETER DESCRIPTION
T

Temperature [K].

TYPE: float

x

Mole fractions of all components [mol/mol].

TYPE: FloatVector(N)

RETURNS DESCRIPTION
FloatVector(N)

Activity coefficients of all components.
Source code in src/polykin/thermo/acm/base.py 
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def gamma(self, T: float, x: FloatVector) -> FloatVector:
    r"""Calculate the activity coefficients based on mole fraction.

    $$ \ln \gamma_i = \frac{1}{RT}
       \left( \frac{\partial (n g^E)}{\partial n_i} \right)_{T,P,n_j} $$

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

    Returns
    -------
    FloatVector (N)
        Activity coefficients of all components.
    """

    def GE(n: np.ndarray):
        """Total excess Gibbs energy."""
        nT = n.sum()
        return nT * self.gE(T, n / nT)

    dGEdn = jacobian_forward(GE, x)

    return exp(dGEdn / (R * T))

hE ¤

hE(T: float, x: FloatVector) -> float

Calculate the molar excess enthalpy.

\[ h^{E} = g^{E} + T s^{E} \]
PARAMETER DESCRIPTION
T

Temperature [K].

TYPE: float

x

Mole fractions of all components [mol/mol].

TYPE: FloatVector(N)

RETURNS DESCRIPTION
float

Molar excess enthalpy [J/mol].
Source code in src/polykin/thermo/acm/base.py 
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def hE(self, T: float, x: FloatVector) -> float:
    r"""Calculate the molar excess enthalpy.

    $$ h^{E} = g^{E} + T s^{E} $$

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

    Returns
    -------
    float
        Molar excess enthalpy [J/mol].
    """
    return self.gE(T, x) + T * self.sE(T, x)

sE ¤

sE(T: float, x: FloatVector) -> float

Calculate the molar excess entropy.

\[ s^{E} = -\left(\frac{\partial g^{E}}{\partial T}\right)_{P,x_i} \]
PARAMETER DESCRIPTION
T

Temperature [K].

TYPE: float

x

Mole fractions of all components [mol/mol].

TYPE: FloatVector(N)

RETURNS DESCRIPTION
float

Molar excess entropy [J/(mol·K)].
Source code in src/polykin/thermo/acm/base.py 
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def sE(self, T: float, x: FloatVector) -> float:
    r"""Calculate the molar excess entropy.

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

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

    Returns
    -------
    float
        Molar excess entropy [J/(mol·K)].
    """
    return -1 * derivative_complex(lambda T_: self.gE(T_, x), T)[0]