.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/electrochem/butler_volmer/plot_01_tafel.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_butler_volmer_plot_01_tafel.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_electrochem_butler_volmer_plot_01_tafel.py:


Tafel's law: overpotential linear in log current
==================================================

Tafel (1905) found that far from equilibrium the overpotential of
hydrogen evolution grows linearly with :math:`\log_{10} i`:
:math:`\eta = b\log_{10}(i/i_0)`. This example generates noisy
"measured" high-overpotential data, draws the Tafel plot, and uses
:func:`~chemistrykit.electrochem.systems.butler_volmer.fit_tafel_plot`
to recover the Tafel slope `b` and exchange current density :math:`i_0`
by linear regression -- the analysis Tafel's successors still use, and
compares the slope with
:func:`~chemistrykit.electrochem.systems.butler_volmer.tafel_slope`'s
textbook ~118 mV/decade for :math:`\alpha = 0.5`, n = 1.

.. GENERATED FROM PYTHON SOURCE LINES 18-26

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.electrochem.systems.butler_volmer import fit_tafel_plot, tafel_overpotential, tafel_slope

    i0_true, alpha, n = 1.0e-6, 0.5, 1
    b_true = tafel_slope(alpha=alpha, n=n)








.. GENERATED FROM PYTHON SOURCE LINES 27-29

Synthetic measurements: current densities over three decades, with
the overpotential read to +/- 3 mV.

.. GENERATED FROM PYTHON SOURCE LINES 29-36

.. code-block:: Python

    rng = np.random.default_rng(1905)
    i_data = np.logspace(-4, -1, 15)
    eta_data = tafel_overpotential(i_data, i0_true, alpha=alpha, n=n) + rng.normal(0.0, 0.003, i_data.size)
    fit = fit_tafel_plot(eta_data, i_data)
    print(f"true b  = {b_true * 1e3:.1f} mV/decade, fitted b  = {fit.tafel_slope * 1e3:.1f} mV/decade")
    print(f"true i0 = {i0_true:.2e}, fitted i0 = {fit.exchange_current_density:.2e}, R^2 = {fit.r_squared:.4f}")





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

 .. code-block:: none

    true b  = 118.3 mV/decade, fitted b  = 118.2 mV/decade
    true i0 = 1.00e-06, fitted i0 = 9.95e-07, R^2 = 0.9995




.. GENERATED FROM PYTHON SOURCE LINES 37-39

The Tafel plot: a straight line whose extrapolation to eta = 0 gives
log10(i0).

.. GENERATED FROM PYTHON SOURCE LINES 39-51

.. code-block:: Python

    log_i = np.linspace(-6.5, -0.5, 100)
    fig, ax = plt.subplots()
    ax.plot(np.log10(i_data), eta_data, "o", label="measured")
    ax.plot(log_i, fit.predict(10.0**log_i), label=f"Tafel fit, b = {fit.tafel_slope * 1e3:.0f} mV/dec")
    ax.axhline(0.0, color="gray", linewidth=0.8)
    ax.axvline(np.log10(fit.exchange_current_density), color="gray", linestyle=":", label=r"$\log_{10} i_0$")
    ax.set_xlabel(r"$\log_{10}\, i$")
    ax.set_ylabel(r"$\eta$ (V)")
    ax.set_title("Tafel plot")
    ax.legend()
    fig.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/electrochem/butler_volmer/images/sphx_glr_plot_01_tafel_001.png
   :alt: Tafel plot
   :srcset: /api/gallery/electrochem/butler_volmer/images/sphx_glr_plot_01_tafel_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.038 seconds)


.. _sphx_glr_download_api_gallery_electrochem_butler_volmer_plot_01_tafel.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_tafel.ipynb <plot_01_tafel.ipynb>`

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

      :download:`Download Python source code: plot_01_tafel.py <plot_01_tafel.py>`

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

      :download:`Download zipped: plot_01_tafel.zip <plot_01_tafel.zip>`


.. only:: html

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

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