.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/surface/freundlich/plot_01_freundlich_isotherm.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_surface_freundlich_plot_01_freundlich_isotherm.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_surface_freundlich_plot_01_freundlich_isotherm.py:


The Freundlich adsorption isotherm
=====================================

:class:`~chemistrykit.surface.systems.freundlich.FreundlichIsotherm`
models the empirical power law :math:`q=K_fP^{1/n}`, useful for a
heterogeneous surface where Langmuir's single-site-energy assumption
breaks down. Unlike Langmuir, it has no saturation limit, so
extrapolating it far beyond the fitted pressure range is unphysical.
:func:`~chemistrykit.surface.systems.freundlich.fit_freundlich` recovers
`(Kf, n)` via the standard log q vs. log P linearization.

.. GENERATED FROM PYTHON SOURCE LINES 15-26

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.surface.systems.freundlich import FreundlichIsotherm, fit_freundlich
    from chemistrykit.surface.visualizers.surface_plots import plot_isotherm, plot_linearization

    Kf_true, n_true = 4.0, 2.5
    iso = FreundlichIsotherm(Kf=Kf_true, n=n_true)

    print(f"Loading at P=1 (should equal Kf exactly): {iso.loading(1.0):.6f}")





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

 .. code-block:: none

    Loading at P=1 (should equal Kf exactly): 4.000000




.. GENERATED FROM PYTHON SOURCE LINES 27-36

.. code-block:: Python

    rng = np.random.default_rng(1)
    P_data = np.array([0.1, 0.25, 0.5, 1.0, 2.0, 4.0, 8.0])
    q_data = iso.loading(P_data) * (1.0 + rng.normal(scale=0.01, size=P_data.shape))

    fit = fit_freundlich(P_data, q_data)
    print(f"\nTrue (Kf, n)   = ({Kf_true}, {n_true})")
    print(f"Fitted (Kf, n) = ({fit.Kf:.4f}, {fit.n:.4f})")
    print(f"R^2 of linearized fit: {fit.r_squared:.6f}")





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

 .. code-block:: none


    True (Kf, n)   = (4.0, 2.5)
    Fitted (Kf, n) = (4.0055, 2.5087)
    R^2 of linearized fit: 0.999847




.. GENERATED FROM PYTHON SOURCE LINES 37-45

.. code-block:: Python

    fig, axes = plt.subplots(1, 2, figsize=(10, 4))
    plot_isotherm(iso, P_max=10.0, P_data=P_data, q_data=q_data, ax=axes[0])

    x_lin = np.log(P_data)
    y_lin = np.log(q_data)
    plot_linearization(x_lin, y_lin, fit=(1.0 / fit.n, np.log(fit.Kf)), ax=axes[1], xlabel="ln P", ylabel="ln q")
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/surface/freundlich/images/sphx_glr_plot_01_freundlich_isotherm_001.png
   :alt: FreundlichIsotherm isotherm, Isotherm linearization
   :srcset: /api/gallery/surface/freundlich/images/sphx_glr_plot_01_freundlich_isotherm_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.061 seconds)


.. _sphx_glr_download_api_gallery_surface_freundlich_plot_01_freundlich_isotherm.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_freundlich_isotherm.ipynb <plot_01_freundlich_isotherm.ipynb>`

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

      :download:`Download Python source code: plot_01_freundlich_isotherm.py <plot_01_freundlich_isotherm.py>`

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

      :download:`Download zipped: plot_01_freundlich_isotherm.zip <plot_01_freundlich_isotherm.zip>`


.. only:: html

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

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