.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/electrochem/voltammetry/plot_03_randles_sevcik_cyclic_voltammetry.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_voltammetry_plot_03_randles_sevcik_cyclic_voltammetry.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_electrochem_voltammetry_plot_03_randles_sevcik_cyclic_voltammetry.py:


The Randles-Ševčík equation: peak current in cyclic voltammetry
=================================================================

Randles and Ševčík (1948) independently showed that the peak current of
a reversible linear-sweep voltammogram grows with the square root of the
scan rate, :math:`i_p = 0.4463\,nFAC\sqrt{nFvD/RT}`. This example
evaluates
:func:`~chemistrykit.electrochem.systems.voltammetry.randles_sevcik_peak_current`
over a range of scan rates, and recovers the diffusion coefficient from
the slope of the :math:`i_p` vs. :math:`\sqrt{v}` plot -- the standard
diagnostic for a diffusion-controlled electrode reaction.

.. GENERATED FROM PYTHON SOURCE LINES 16-29

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.constants import FARADAY, STANDARD_TEMPERATURE, R
    from chemistrykit.electrochem.systems.voltammetry import randles_sevcik_peak_current

    n, A, C, D_true = 1, np.pi * (1.5e-3) ** 2, 1.0, 2.4e-9  # 3 mm disc, 1 mM ferrocene-like couple
    v = np.array([0.01, 0.025, 0.05, 0.1, 0.25, 0.5, 1.0])
    rng = np.random.default_rng(1948)
    ip = randles_sevcik_peak_current(v, n, A, C, D_true) * (1 + rng.normal(0.0, 0.01, v.size))
    for vi, ipi in zip(v, ip, strict=True):
        print(f"v = {vi * 1e3:6.0f} mV/s: i_p = {ipi * 1e6:6.2f} uA")





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

 .. code-block:: none

    v =     10 mV/s: i_p =   9.28 uA
    v =     25 mV/s: i_p =  14.79 uA
    v =     50 mV/s: i_p =  20.67 uA
    v =    100 mV/s: i_p =  28.77 uA
    v =    250 mV/s: i_p =  47.61 uA
    v =    500 mV/s: i_p =  66.40 uA
    v =   1000 mV/s: i_p =  92.80 uA




.. GENERATED FROM PYTHON SOURCE LINES 30-31

Slope of i_p vs sqrt(v) through the origin gives D.

.. GENERATED FROM PYTHON SOURCE LINES 31-45

.. code-block:: Python

    slope = np.sum(ip * np.sqrt(v)) / np.sum(v)
    D_fit = (slope / (0.4463 * n * FARADAY * A * C)) ** 2 * R * STANDARD_TEMPERATURE / (n * FARADAY)
    print(f"\nD true = {D_true:.2e} m^2/s, D from Randles-Sevcik plot = {D_fit:.2e} m^2/s")

    x = np.linspace(0.0, 1.0, 50)
    fig, ax = plt.subplots()
    ax.plot(np.sqrt(v), ip * 1e6, "o", label="peak currents")
    ax.plot(x, slope * x * 1e6, label=r"linear in $\sqrt{v}$")
    ax.set_xlabel(r"$\sqrt{v}$ (V/s)$^{1/2}$")
    ax.set_ylabel(r"$i_p$ ($\mu$A)")
    ax.set_title("Randles-Ševčík plot")
    ax.legend()
    fig.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/electrochem/voltammetry/images/sphx_glr_plot_03_randles_sevcik_cyclic_voltammetry_001.png
   :alt: Randles-Ševčík plot
   :srcset: /api/gallery/electrochem/voltammetry/images/sphx_glr_plot_03_randles_sevcik_cyclic_voltammetry_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    D true = 2.40e-09 m^2/s, D from Randles-Sevcik plot = 2.41e-09 m^2/s





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

   **Total running time of the script:** (0 minutes 0.034 seconds)


.. _sphx_glr_download_api_gallery_electrochem_voltammetry_plot_03_randles_sevcik_cyclic_voltammetry.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_randles_sevcik_cyclic_voltammetry.ipynb <plot_03_randles_sevcik_cyclic_voltammetry.ipynb>`

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

      :download:`Download Python source code: plot_03_randles_sevcik_cyclic_voltammetry.py <plot_03_randles_sevcik_cyclic_voltammetry.py>`

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

      :download:`Download zipped: plot_03_randles_sevcik_cyclic_voltammetry.zip <plot_03_randles_sevcik_cyclic_voltammetry.zip>`


.. only:: html

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

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