.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/solutions/activity/plot_03_davies_equation.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_solutions_activity_plot_03_davies_equation.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_solutions_activity_plot_03_davies_equation.py:


The Davies equation: activity coefficients that turn back up
==============================================================

Measured activity coefficients fall with ionic strength, pass through a
minimum, and rise again in concentrated solutions -- behaviour no pure
screening law reproduces. Davies added an empirical term linear in
:math:`I` to Guntelberg's form,

.. math::

   \log_{10}\gamma = -A z^2\left(\frac{\sqrt{I}}{1+\sqrt{I}} - 0.3\,I\right),

implemented by
:func:`~chemistrykit.solutions.systems.activity.activity_coefficient_davies`.
The comparison below shows the minimum near :math:`I \approx 0.4` mol/L
and the upturn that the Debye-Huckel family misses.

.. GENERATED FROM PYTHON SOURCE LINES 21-43

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.solutions.systems.activity import (
        activity_coefficient_davies,
        activity_coefficient_debye_huckel_extended,
        activity_coefficient_debye_huckel_limiting,
    )

    I_range = np.linspace(1e-4, 2.0, 400)

    fig, ax = plt.subplots(figsize=(7, 5))
    ax.plot(I_range, [activity_coefficient_debye_huckel_limiting(1, I) for I in I_range], color="gray", linestyle=":", label="Debye-Huckel limiting")
    ax.plot(I_range, [activity_coefficient_debye_huckel_extended(1, I) for I in I_range], color="darkorange", linestyle="--", label="Guntelberg (Ba = 1)")
    ax.plot(I_range, [activity_coefficient_davies(1, I, b=0.2) for I in I_range], color="seagreen", alpha=0.7, label="Davies, b = 0.2 (1938)")
    ax.plot(I_range, [activity_coefficient_davies(1, I) for I in I_range], color="steelblue", label="Davies, b = 0.3 (1962)")
    ax.set_xlabel("ionic strength I (mol/L)")
    ax.set_ylabel(r"activity coefficient $\gamma$, z = 1")
    ax.set_title("Davies equation vs. the Debye-Huckel family")
    ax.legend(fontsize=8)
    fig.tight_layout()




.. image-sg:: /api/gallery/solutions/activity/images/sphx_glr_plot_03_davies_equation_001.png
   :alt: Davies equation vs. the Debye-Huckel family
   :srcset: /api/gallery/solutions/activity/images/sphx_glr_plot_03_davies_equation_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 44-45

Location of the Davies minimum for b = 0.3, found on a fine grid:

.. GENERATED FROM PYTHON SOURCE LINES 45-52

.. code-block:: Python


    I_fine = np.linspace(0.01, 2.0, 20001)
    gamma_fine = np.array([activity_coefficient_davies(1, I) for I in I_fine])
    i_min = int(np.argmin(gamma_fine))
    print(f"Davies minimum: gamma = {gamma_fine[i_min]:.4f} at I = {I_fine[i_min]:.3f} mol/L")

    plt.show()




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

 .. code-block:: none

    Davies minimum: gamma = 0.7309 at I = 0.395 mol/L





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

   **Total running time of the script:** (0 minutes 0.055 seconds)


.. _sphx_glr_download_api_gallery_solutions_activity_plot_03_davies_equation.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_03_davies_equation.ipynb <plot_03_davies_equation.ipynb>`

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

      :download:`Download Python source code: plot_03_davies_equation.py <plot_03_davies_equation.py>`

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

      :download:`Download zipped: plot_03_davies_equation.zip <plot_03_davies_equation.zip>`


.. only:: html

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

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_