.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/solutions/acid_base/plot_01_ostwald_dilution_law.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_acid_base_plot_01_ostwald_dilution_law.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_solutions_acid_base_plot_01_ostwald_dilution_law.py:


Ostwald's dilution law: weak acids dissociate more when diluted
=================================================================

Dissolving less acid in more water increases the *fraction* of it that
dissociates, even though the absolute :math:`[H^+]` goes down -- the
dilution law Wilhelm Ostwald derived in 1888 by applying the mass-action
law to Arrhenius's dissociation equilibrium,

.. math::

   K_a = \frac{\alpha^2 C_a}{1-\alpha}.

Here the degree of dissociation :math:`\alpha` of acetic acid is
computed from the exact cubic charge-balance solution in
:class:`~chemistrykit.solutions.systems.acid_base.WeakAcid` and compared
against Ostwald's own closed-form solution of the quadratic above (which
ignores water autoionization).

.. GENERATED FROM PYTHON SOURCE LINES 22-43

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.solutions.systems.acid_base import WeakAcid

    Ka = 1.8e-5  # acetic acid
    Ca_values = np.logspace(-6, 0, 80)
    alpha_exact = np.array([WeakAcid(Ca=Ca, Ka=Ka).percent_dissociation() / 100.0 for Ca in Ca_values])

    # Ostwald's quadratic alpha^2 Ca + Ka alpha - Ka = 0, positive root:
    alpha_ostwald = (-Ka + np.sqrt(Ka**2 + 4.0 * Ka * Ca_values)) / (2.0 * Ca_values)

    fig, ax = plt.subplots(figsize=(7, 5))
    ax.semilogx(Ca_values, 100.0 * alpha_exact, color="steelblue", label="exact (WeakAcid)")
    ax.semilogx(Ca_values, 100.0 * alpha_ostwald, color="darkorange", linestyle="--", label=r"Ostwald: $K_a = \alpha^2 C_a/(1-\alpha)$")
    ax.set_xlabel("total acid concentration $C_a$ (mol/L)")
    ax.set_ylabel(r"percent dissociated, $100\alpha$")
    ax.set_title("Ostwald dilution law: acetic acid")
    ax.legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/solutions/acid_base/images/sphx_glr_plot_01_ostwald_dilution_law_001.png
   :alt: Ostwald dilution law: acetic acid
   :srcset: /api/gallery/solutions/acid_base/images/sphx_glr_plot_01_ostwald_dilution_law_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 44-48

The "Ostwald constant" :math:`\alpha^2 C_a/(1-\alpha)` computed from the
exact degree of dissociation stays equal to :math:`K_a` across four
decades of dilution -- the experimental test Ostwald made with
conductivity data for dozens of weak acids:

.. GENERATED FROM PYTHON SOURCE LINES 48-53

.. code-block:: Python


    for Ca, alpha in zip(Ca_values[::20], alpha_exact[::20]):
        print(f"Ca = {Ca:.1e} M: alpha = {alpha:.4f}, alpha^2 Ca/(1-alpha) = {alpha**2 * Ca / (1 - alpha):.3e}")

    plt.show()




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

 .. code-block:: none

    Ca = 1.0e-06 M: alpha = 0.9494, alpha^2 Ca/(1-alpha) = 1.780e-05
    Ca = 3.3e-05 M: alpha = 0.5144, alpha^2 Ca/(1-alpha) = 1.800e-05
    Ca = 1.1e-03 M: alpha = 0.1204, alpha^2 Ca/(1-alpha) = 1.800e-05
    Ca = 3.6e-02 M: alpha = 0.0221, alpha^2 Ca/(1-alpha) = 1.800e-05





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

   **Total running time of the script:** (0 minutes 0.068 seconds)


.. _sphx_glr_download_api_gallery_solutions_acid_base_plot_01_ostwald_dilution_law.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_01_ostwald_dilution_law.ipynb <plot_01_ostwald_dilution_law.ipynb>`

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

      :download:`Download Python source code: plot_01_ostwald_dilution_law.py <plot_01_ostwald_dilution_law.py>`

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

      :download:`Download zipped: plot_01_ostwald_dilution_law.zip <plot_01_ostwald_dilution_law.zip>`


.. only:: html

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

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