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ScatchardHildebrand_gamma ¤

ScatchardHildebrand_gamma(
    T: float,
    x: FloatVector,
    delta: FloatVector,
    v: FloatVector,
) -> FloatVector

Calculate the activity coefficients of a multicomponent mixture according to the Scatchard-Hildebrand model.

\[ \ln{\gamma_i} = \frac{v_i}{R T} (\delta_i - \bar{\delta})^2 \]

with:

\[\begin{aligned} \phi_i &= \frac{x_i v_i}{\sum_j x_j v_j} \\ \bar{\delta} &= \sum_j \phi_j \delta_j \end{aligned}\]

where \(x_i\) are the mole fractions, \(\delta_{i}\) are the solubility parameters, and \(v_i\) are the molar volumes.

References

  • Prausnitz, J. M.; Lichtenthaler, R. N.; de Azevedo, E. G. Molecular Thermodynamics of Fluid-Phase Equilibria, 3rd ed.; Prentice Hall, 1999, p. 325.
PARAMETER DESCRIPTION
T

Temperature [K].

TYPE: float

x

Mole fractions of all components [mol/mol].

TYPE: FloatVector(N)

delta

Solubility parameters of all components [(J/L³)^(1/2)].

TYPE: FloatVector(N)

v

Molar volumes of all components [L³/mol].

TYPE: FloatVector(N)

RETURNS DESCRIPTION
FloatVector(N)

Activity coefficients of all components.
See Also
Source code in src/polykin/thermo/acm/scatchard.py 
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def ScatchardHildebrand_gamma(
    T: float,
    x: FloatVector,
    delta: FloatVector,
    v: FloatVector,
) -> FloatVector:
    r"""Calculate the activity coefficients of a multicomponent mixture according to the
    Scatchard-Hildebrand model.

    $$ \ln{\gamma_i} = \frac{v_i}{R T} (\delta_i - \bar{\delta})^2 $$

    with:

    \begin{aligned}
    \phi_i       &= \frac{x_i v_i}{\sum_j x_j v_j} \\
    \bar{\delta} &= \sum_j \phi_j \delta_j
    \end{aligned}

    where $x_i$ are the mole fractions, $\delta_{i}$ are the solubility parameters, and
    $v_i$ are the molar volumes.

    **References**

    *   Prausnitz, J. M.; Lichtenthaler, R. N.; de Azevedo, E. G. Molecular Thermodynamics
        of Fluid-Phase Equilibria, 3rd ed.; Prentice Hall, 1999, p. 325.

    Parameters
    ----------
    T : float
        Temperature [K].
    x : FloatVector (N)
        Mole fractions of all components [mol/mol].
    delta : FloatVector (N)
        Solubility parameters of all components [(J/L³)^(1/2)].
    v : FloatVector (N)
        Molar volumes of all components [L³/mol].

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

    See Also
    --------
    * [`ScatchardHildebrand`](ScatchardHildebrand.md): Related class.
    """
    phi = x * v
    phi /= phi.sum()
    δm = dot(phi, delta)
    return exp(v / (R * T) * (delta - δm) ** 2)