.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/electrochem/voltammetry/plot_01_cottrell_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_electrochem_voltammetry_plot_01_cottrell_equation.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_electrochem_voltammetry_plot_01_cottrell_equation.py:


The Cottrell equation: current decay after a potential step
=============================================================

Cottrell (1903) solved the diffusion problem for a planar electrode whose
potential is suddenly stepped so that the electroactive species is
consumed at the surface: the current decays as :math:`t^{-1/2}`,
:math:`i = nFAC\sqrt{D/(\pi t)}`. This example plots
:func:`~chemistrykit.electrochem.systems.voltammetry.cottrell_current`,
checks the constant :math:`i\sqrt{t}` signature, and recovers the
diffusion coefficient from a noisy "measured" transient via the Cottrell
plot of :math:`i` against :math:`t^{-1/2}`.

.. GENERATED FROM PYTHON SOURCE LINES 16-29

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.constants import FARADAY
    from chemistrykit.electrochem.systems.voltammetry import cottrell_current

    n, A, C, D_true = 1, 7.07e-6, 1.0, 7.0e-10  # 3 mm disc, 1 mM, D in m^2/s
    t = np.linspace(0.05, 5.0, 80)
    i_true = cottrell_current(t, n, A, C, D_true)
    print(
        f"i*sqrt(t) at t = 0.1 s and 4 s: {cottrell_current(0.1, n, A, C, D_true) * np.sqrt(0.1):.4e}, {cottrell_current(4.0, n, A, C, D_true) * 2.0:.4e} A s^1/2"
    )





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

 .. code-block:: none

    i*sqrt(t) at t = 0.1 s and 4 s: 1.0183e-05, 1.0183e-05 A s^1/2




.. GENERATED FROM PYTHON SOURCE LINES 30-32

Cottrell plot: i vs t^-1/2 is a line through the origin with slope
nFAC sqrt(D/pi).

.. GENERATED FROM PYTHON SOURCE LINES 32-52

.. code-block:: Python

    rng = np.random.default_rng(1903)
    i_meas = i_true * (1 + rng.normal(0.0, 0.01, t.size))
    slope = np.sum(i_meas * t**-0.5) / np.sum(t**-1.0)
    D_fit = np.pi * (slope / (n * FARADAY * A * C)) ** 2
    print(f"D true = {D_true:.2e} m^2/s, D from Cottrell plot = {D_fit:.2e} m^2/s")

    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 4))
    ax1.plot(t, i_meas * 1e6, ".", label="measured")
    ax1.plot(t, i_true * 1e6, label="Cottrell")
    ax1.set_xlabel("t (s)")
    ax1.set_ylabel(r"i ($\mu$A)")
    ax1.set_title(r"$t^{-1/2}$ decay after a potential step")
    ax1.legend()
    ax2.plot(t**-0.5, i_meas * 1e6, ".")
    ax2.plot(t**-0.5, slope * t**-0.5 * 1e6)
    ax2.set_xlabel(r"$t^{-1/2}$ (s$^{-1/2}$)")
    ax2.set_ylabel(r"i ($\mu$A)")
    ax2.set_title("Cottrell plot")
    fig.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/electrochem/voltammetry/images/sphx_glr_plot_01_cottrell_equation_001.png
   :alt: $t^{-1/2}$ decay after a potential step, Cottrell plot
   :srcset: /api/gallery/electrochem/voltammetry/images/sphx_glr_plot_01_cottrell_equation_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    D true = 7.00e-10 m^2/s, D from Cottrell plot = 7.00e-10 m^2/s





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

   **Total running time of the script:** (0 minutes 0.053 seconds)


.. _sphx_glr_download_api_gallery_electrochem_voltammetry_plot_01_cottrell_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_01_cottrell_equation.ipynb <plot_01_cottrell_equation.ipynb>`

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

      :download:`Download Python source code: plot_01_cottrell_equation.py <plot_01_cottrell_equation.py>`

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

      :download:`Download zipped: plot_01_cottrell_equation.zip <plot_01_cottrell_equation.zip>`


.. only:: html

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

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