.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/electrochem/nernst/plot_02_debye_huckel_activity.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_nernst_plot_02_debye_huckel_activity.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_electrochem_nernst_plot_02_debye_huckel_activity.py:


Debye-Hückel activity corrections to the Nernst equation
==========================================================

Debye and Hückel (1923) showed that each ion's cloud of counter-ions
lowers its activity below its concentration, by an amount growing with
ionic strength and ionic charge. This example uses
:func:`~chemistrykit.electrochem.systems.nernst.nernst_potential_with_activity`
(built on :mod:`chemistrykit.solutions.systems.activity`) to show how the
potential of a Cu2+/Cu electrode drifts away from the ideal Nernst value
as the solution gets more concentrated, and that the correction vanishes
in the dilute limit.

.. GENERATED FROM PYTHON SOURCE LINES 16-21

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.electrochem.systems.nernst import nernst_potential, nernst_potential_with_activity








.. GENERATED FROM PYTHON SOURCE LINES 22-25

Cu2+ + 2e- -> Cu(s) in CuSO4 solution: Q = 1/a(Cu2+). The sulfate
counter-ion contributes to the ionic strength but not to Q
(stoichiometric coefficient 0); the solid copper is at unit activity.

.. GENERATED FROM PYTHON SOURCE LINES 25-33

.. code-block:: Python

    E_standard, n = 0.34, 2
    c_cu = np.logspace(-5, -1, 40)
    E_ideal = np.array([nernst_potential(E_standard, n, 1.0 / c) for c in c_cu])
    E_dh = np.array([nernst_potential_with_activity(E_standard, n, [c, c, 1.0], [2, -2, 0], [-1.0, 0.0, 0.0]) for c in c_cu])

    for c, e_i, e_a in zip(c_cu[::13], E_ideal[::13], E_dh[::13], strict=True):
        print(f"[Cu2+] = {c:8.1e} M: ideal E = {e_i:.4f} V, Debye-Hückel E = {e_a:.4f} V, shift = {1e3 * (e_a - e_i):+.2f} mV")





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

 .. code-block:: none

    [Cu2+] =  1.0e-05 M: ideal E = 0.1921 V, Debye-Hückel E = 0.1917 V, shift = -0.38 mV
    [Cu2+] =  2.2e-04 M: ideal E = 0.2315 V, Debye-Hückel E = 0.2298 V, shift = -1.72 mV
    [Cu2+] =  4.6e-03 M: ideal E = 0.2710 V, Debye-Hückel E = 0.2638 V, shift = -7.22 mV
    [Cu2+] =  1.0e-01 M: ideal E = 0.3104 V, Debye-Hückel E = 0.2871 V, shift = -23.33 mV




.. GENERATED FROM PYTHON SOURCE LINES 34-46

.. code-block:: Python

    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 4))
    ax1.semilogx(c_cu, E_ideal, label="ideal (activity = concentration)")
    ax1.semilogx(c_cu, E_dh, "--", label="Debye-Hückel activities")
    ax1.set_xlabel(r"[Cu$^{2+}$] (mol/L)")
    ax1.set_ylabel("E (V vs. SHE)")
    ax1.legend()
    ax2.semilogx(c_cu, 1e3 * (E_dh - E_ideal))
    ax2.set_xlabel(r"[Cu$^{2+}$] (mol/L)")
    ax2.set_ylabel("Activity correction (mV)")
    ax2.set_title("Correction vanishes at infinite dilution")
    fig.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/electrochem/nernst/images/sphx_glr_plot_02_debye_huckel_activity_001.png
   :alt: Correction vanishes at infinite dilution
   :srcset: /api/gallery/electrochem/nernst/images/sphx_glr_plot_02_debye_huckel_activity_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.094 seconds)


.. _sphx_glr_download_api_gallery_electrochem_nernst_plot_02_debye_huckel_activity.py:

.. only:: html

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

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

      :download:`Download Python source code: plot_02_debye_huckel_activity.py <plot_02_debye_huckel_activity.py>`

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

      :download:`Download zipped: plot_02_debye_huckel_activity.zip <plot_02_debye_huckel_activity.zip>`


.. only:: html

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

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