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polykin.thermo.eos¤

IdealGas ¤

Bases: GasEoS

Ideal gas equation of state.

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

\[ Z = 1 \]
PARAMETER DESCRIPTION
N

Number of components.

TYPE: int

name

Name.

TYPE: str DEFAULT: ''

Source code in src/polykin/thermo/eos/idealgas.py 
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class IdealGas(GasEoS):
    r"""Ideal gas equation of state.

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

    $$ Z = 1 $$

    Parameters
    ----------
    N : int
        Number of components.
    name : str
        Name.
    """

    def __init__(self, N: int, name: str = "") -> None:

        super().__init__(N, name)

    def Z(self, T=None, P=None, y=None) -> float:
        r"""Calculate the compressibility factor of the fluid.

        $$ Z = 1 $$

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

        Returns
        -------
        float
            Compressibility factor.
        """
        return 1.0

    @override
    def gR(self, T=None, P=None, y=None) -> float:
        """Calculate the molar residual Gibbs energy of the fluid.

        $$ g^R = 0 $$

        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 0.0

    @override
    def P(self, T: float, v: float, y=None) -> float:
        r"""Calculate the pressure of the fluid.

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

        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].
        """
        return R * T / v

    @override
    def phi(self, T=None, P=None, y=None) -> FloatVector:
        r"""Calculate the fugacity coefficients of all components.

        $$ \hat{\phi}_i = 1 $$

        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.
        """
        return np.ones(self.N)

    @override
    def beta(self, T: float, P=None, y=None) -> float:
        r"""Calculate the thermal expansion coefficient.

        $$ \beta \equiv
           \frac{1}{v} \left( \frac{\partial v}{\partial T} \right)_P
           = \frac{1}{T} $$

        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].
        """
        return 1 / T

    @override
    def kappa(self, T: float, P: float, y=None) -> float:
        r"""Calculate the isothermal compressibility coefficient.

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

        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].
        """
        return 1 / P

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 
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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=None) -> float

Calculate the pressure of the fluid.

\[ P = \frac{R T}{v} \]
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) DEFAULT: None

RETURNS DESCRIPTION
float

Pressure [Pa].
Source code in src/polykin/thermo/eos/idealgas.py 
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@override
def P(self, T: float, v: float, y=None) -> float:
    r"""Calculate the pressure of the fluid.

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

    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].
    """
    return R * T / v

Z ¤

Z(T=None, P=None, y=None) -> float

Calculate the compressibility factor of the fluid.

\[ Z = 1 \]
PARAMETER DESCRIPTION
T

Temperature [K].

TYPE: float DEFAULT: None

P

Pressure [Pa].

TYPE: float DEFAULT: None

y

Mole fractions of all components [mol/mol].

TYPE: FloatVector(N) DEFAULT: None

RETURNS DESCRIPTION
float

Compressibility factor.
Source code in src/polykin/thermo/eos/idealgas.py 
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def Z(self, T=None, P=None, y=None) -> float:
    r"""Calculate the compressibility factor of the fluid.

    $$ Z = 1 $$

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

    Returns
    -------
    float
        Compressibility factor.
    """
    return 1.0

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 
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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=None, y=None) -> float

Calculate the thermal expansion coefficient.

\[ \beta \equiv \frac{1}{v} \left( \frac{\partial v}{\partial T} \right)_P = \frac{1}{T} \]
PARAMETER DESCRIPTION
T

Temperature [K].

TYPE: float

P

Pressure [Pa].

TYPE: float DEFAULT: None

y

Mole fractions of all components [mol/mol].

TYPE: FloatVector(N) DEFAULT: None

RETURNS DESCRIPTION
float

Thermal expansion coefficient, \(\beta\) [K⁻¹].
Source code in src/polykin/thermo/eos/idealgas.py 
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@override
def beta(self, T: float, P=None, y=None) -> float:
    r"""Calculate the thermal expansion coefficient.

    $$ \beta \equiv
       \frac{1}{v} \left( \frac{\partial v}{\partial T} \right)_P
       = \frac{1}{T} $$

    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].
    """
    return 1 / T

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 
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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=None, P=None, y=None) -> float

Calculate the molar residual Gibbs energy of the fluid.

\[ g^R = 0 \]
PARAMETER DESCRIPTION
T

Temperature [K].

TYPE: float DEFAULT: None

P

Pressure [Pa].

TYPE: float DEFAULT: None

y

Mole fractions of all components [mol/mol].

TYPE: FloatVector DEFAULT: None

RETURNS DESCRIPTION
float

Molar residual Gibbs energy [J/mol].
Source code in src/polykin/thermo/eos/idealgas.py 
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@override
def gR(self, T=None, P=None, y=None) -> float:
    """Calculate the molar residual Gibbs energy of the fluid.

    $$ g^R = 0 $$

    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 0.0

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 
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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=None) -> float

Calculate the isothermal compressibility coefficient.

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

Temperature [K].

TYPE: float

P

Pressure [Pa].

TYPE: float

y

Mole fractions of all components [mol/mol].

TYPE: FloatVector(N) DEFAULT: None

RETURNS DESCRIPTION
float

Isothermal compressibility coefficient, \(\kappa\) [Pa⁻¹].
Source code in src/polykin/thermo/eos/idealgas.py 
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@override
def kappa(self, T: float, P: float, y=None) -> float:
    r"""Calculate the isothermal compressibility coefficient.

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

    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].
    """
    return 1 / P

phi ¤

phi(T=None, P=None, y=None) -> FloatVector

Calculate the fugacity coefficients of all components.

\[ \hat{\phi}_i = 1 \]
PARAMETER DESCRIPTION
T

Temperature [K].

TYPE: float DEFAULT: None

P

Pressure [Pa].

TYPE: float DEFAULT: None

y

Mole fractions of all components [mol/mol].

TYPE: FloatVector(N) DEFAULT: None

RETURNS DESCRIPTION
FloatVector(N)

Fugacity coefficients of all components.
Source code in src/polykin/thermo/eos/idealgas.py 
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@override
def phi(self, T=None, P=None, y=None) -> FloatVector:
    r"""Calculate the fugacity coefficients of all components.

    $$ \hat{\phi}_i = 1 $$

    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.
    """
    return np.ones(self.N)

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 
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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 
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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 
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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 gas at 0°C and 1 atm.

from polykin.thermo.eos import IdealGas

eos = IdealGas(1)
v = eos.v(T=273.15, P=1.01325e5, y=None)

print(f"{v:.2e} m³/mol")
2.24e-02 m³/mol