.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/thermo/mixtures/plot_06_nrtl_liquid_liquid_split.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_thermo_mixtures_plot_06_nrtl_liquid_liquid_split.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_thermo_mixtures_plot_06_nrtl_liquid_liquid_split.py:


NRTL: a non-ideal liquid that splits into two phases
====================================================

Wilson's model cannot predict liquid-liquid immiscibility: its Gibbs
energy of mixing stays convex whatever the parameters. Renon and
Prausnitz's non-random two-liquid model
(:class:`~chemistrykit.thermo.NRTLSolution`) adds a non-randomness
parameter :math:`\alpha` and can. When

.. math::

    \frac{\Delta G_{mix}}{RT} = x_1\ln x_1 + x_2\ln x_2 + \frac{G^E}{RT}

develops two minima, a single liquid lowers its Gibbs energy by splitting
into two phases whose compositions share a common tangent. Below, NRTL
parameters typical of a partially miscible organic-water pair give such a
split, located by equating each component's activity
:math:`x_i\gamma_i` in the two phases.

.. GENERATED FROM PYTHON SOURCE LINES 23-67

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy.optimize import fsolve

    from chemistrykit.constants import R
    from chemistrykit.thermo import NRTLSolution, WilsonSolution

    T = 298.15
    nrtl = NRTLSolution(tau12=2.2, tau21=1.8, P1_star=1.0, P2_star=1.0, alpha=0.2)
    wilson = WilsonSolution(Lambda12=0.05, Lambda21=0.05, P1_star=1.0, P2_star=1.0)

    x = np.linspace(1e-4, 1 - 1e-4, 2000)


    def g_mix(model, x1):
        return x1 * np.log(x1) + (1 - x1) * np.log(1 - x1) + model.excess_gibbs(x1, T) / (R * T)


    def equal_activities(z):
        xa, xb = z
        g1a, g2a = nrtl.activity_coefficients(xa)
        g1b, g2b = nrtl.activity_coefficients(xb)
        return [np.log(xa * g1a) - np.log(xb * g1b), np.log((1 - xa) * g2a) - np.log((1 - xb) * g2b)]


    xa, xb = fsolve(equal_activities, [0.02, 0.95])
    assert xb - xa > 0.5, "expected two distinct liquid phases"
    ga, gb = g_mix(nrtl, xa), g_mix(nrtl, xb)

    fig, axes = plt.subplots(1, 2, figsize=(12, 4.8))
    axes[0].plot(x, g_mix(nrtl, x), color="darkorange", label="NRTL")
    axes[0].plot([xa, xb], [ga, gb], "k--", label="common tangent")
    axes[0].plot([xa, xb], [ga, gb], "ko")
    axes[0].set_xlabel("mole fraction of component 1")
    axes[0].set_ylabel(r"$\Delta G_{mix}/RT$")
    axes[0].set_title(f"NRTL: the liquid splits into x = {xa:.3f} and {xb:.3f}")
    axes[0].legend()

    axes[1].plot(x, g_mix(wilson, x), color="steelblue")
    axes[1].set_xlabel("mole fraction of component 1")
    axes[1].set_ylabel(r"$\Delta G_{mix}/RT$")
    axes[1].set_title("Wilson, even strongly non-ideal: always one phase")
    fig.tight_layout()




.. image-sg:: /api/gallery/thermo/mixtures/images/sphx_glr_plot_06_nrtl_liquid_liquid_split_001.png
   :alt: NRTL: the liquid splits into x = 0.056 and 0.957, Wilson, even strongly non-ideal: always one phase
   :srcset: /api/gallery/thermo/mixtures/images/sphx_glr_plot_06_nrtl_liquid_liquid_split_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 68-70

A single phase is stable only where :math:`\Delta G_{mix}` is convex;
NRTL's curvature changes sign, Wilson's never does.

.. GENERATED FROM PYTHON SOURCE LINES 70-75

.. code-block:: Python


    print("NRTL curvature changes sign:", bool((np.diff(g_mix(nrtl, x), 2) < 0).any()))
    print("Wilson curvature changes sign:", bool((np.diff(g_mix(wilson, x), 2) < 0).any()))

    plt.show()




.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    NRTL curvature changes sign: True
    Wilson curvature changes sign: False





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.110 seconds)


.. _sphx_glr_download_api_gallery_thermo_mixtures_plot_06_nrtl_liquid_liquid_split.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_06_nrtl_liquid_liquid_split.ipynb <plot_06_nrtl_liquid_liquid_split.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_06_nrtl_liquid_liquid_split.py <plot_06_nrtl_liquid_liquid_split.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_06_nrtl_liquid_liquid_split.zip <plot_06_nrtl_liquid_liquid_split.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

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