.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/electrochem/conductivity/plot_01_kohlrausch_laws.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_electrochem_conductivity_plot_01_kohlrausch_laws.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_electrochem_conductivity_plot_01_kohlrausch_laws.py:


Kohlrausch's laws: independent migration and the square-root law
==================================================================

Kohlrausch found that the molar conductivity of a strong electrolyte
falls linearly with :math:`\sqrt{c}`,
:math:`\Lambda_m = \Lambda_m^\circ - K\sqrt{c}`, and that the
infinite-dilution value :math:`\Lambda_m^\circ` is a sum of independent
ionic contributions. This example extrapolates noisy KCl data to infinite
dilution with
:func:`~chemistrykit.electrochem.systems.conductivity.fit_kohlrausch_law`,
and uses
:func:`~chemistrykit.electrochem.systems.conductivity.limiting_molar_conductivity`
for his classic trick: the limiting conductivity of the *weak* acid acetic
acid, which cannot be extrapolated, from three strong electrolytes.

.. GENERATED FROM PYTHON SOURCE LINES 19-28

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.electrochem.systems.conductivity import (
        fit_kohlrausch_law,
        kohlrausch_molar_conductivity,
        limiting_molar_conductivity,
    )








.. GENERATED FROM PYTHON SOURCE LINES 29-31

Square-root law: synthetic KCl data (Lambda0 from the ionic table,
K ~ 9.2e-3 S m^2 mol^-1 (mol/L)^-1/2) with 0.1% noise.

.. GENERATED FROM PYTHON SOURCE LINES 31-47

.. code-block:: Python

    rng = np.random.default_rng(1875)
    Lambda0_KCl = limiting_molar_conductivity({"K+": 1, "Cl-": 1})
    c = np.linspace(1e-4, 1e-2, 12)
    Lambda_meas = kohlrausch_molar_conductivity(c, Lambda0_KCl, 9.2e-3) * (1 + rng.normal(0.0, 1e-3, c.size))
    fit = fit_kohlrausch_law(c, Lambda_meas)
    print(f"KCl: table Lambda0 = {Lambda0_KCl * 1e4:.1f}, extrapolated = {fit.limiting_molar_conductivity * 1e4:.1f} S cm^2/mol")

    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 4))
    x = np.linspace(0.0, 0.1, 50)
    ax1.plot(np.sqrt(c), Lambda_meas * 1e4, "o", label="KCl data")
    ax1.plot(x, (fit.limiting_molar_conductivity - fit.kohlrausch_coefficient * x) * 1e4, label="Kohlrausch fit")
    ax1.set_xlabel(r"$\sqrt{c}$ (mol/L)$^{1/2}$")
    ax1.set_ylabel(r"$\Lambda_m$ (S cm$^2$ mol$^{-1}$)")
    ax1.set_title("Square-root law")
    ax1.legend()




.. image-sg:: /api/gallery/electrochem/conductivity/images/sphx_glr_plot_01_kohlrausch_laws_001.png
   :alt: Square-root law
   :srcset: /api/gallery/electrochem/conductivity/images/sphx_glr_plot_01_kohlrausch_laws_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    KCl: table Lambda0 = 149.8, extrapolated = 149.9 S cm^2/mol

    <matplotlib.legend.Legend object at 0x1196916a0>



.. GENERATED FROM PYTHON SOURCE LINES 48-49

Independent migration: Lambda0(HAc) = Lambda0(HCl) + Lambda0(NaAc) - Lambda0(NaCl).

.. GENERATED FROM PYTHON SOURCE LINES 49-64

.. code-block:: Python

    salts = {
        "HCl": {"H+": 1, "Cl-": 1},
        "NaAc": {"Na+": 1, "CH3COO-": 1},
        "NaCl": {"Na+": 1, "Cl-": 1},
    }
    values = {name: limiting_molar_conductivity(ions) for name, ions in salts.items()}
    values["HAc"] = values["HCl"] + values["NaAc"] - values["NaCl"]
    for name, v in values.items():
        print(f"Lambda0({name}) = {v * 1e4:.1f} S cm^2/mol")

    ax2.bar(list(values), [v * 1e4 for v in values.values()], color=["gray", "gray", "gray", "crimson"])
    ax2.set_ylabel(r"$\Lambda_m^\circ$ (S cm$^2$ mol$^{-1}$)")
    ax2.set_title("HAc = HCl + NaAc - NaCl")
    fig.tight_layout()
    plt.show()




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

 .. code-block:: none

    Lambda0(HCl) = 425.9 S cm^2/mol
    Lambda0(NaAc) = 91.0 S cm^2/mol
    Lambda0(NaCl) = 126.4 S cm^2/mol
    Lambda0(HAc) = 390.5 S cm^2/mol





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

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


.. _sphx_glr_download_api_gallery_electrochem_conductivity_plot_01_kohlrausch_laws.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_kohlrausch_laws.ipynb <plot_01_kohlrausch_laws.ipynb>`

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

      :download:`Download Python source code: plot_01_kohlrausch_laws.py <plot_01_kohlrausch_laws.py>`

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

      :download:`Download zipped: plot_01_kohlrausch_laws.zip <plot_01_kohlrausch_laws.zip>`


.. only:: html

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

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