.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/surface/temkin/plot_01_temkin_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_temkin_plot_01_temkin_isotherm.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_surface_temkin_plot_01_temkin_isotherm.py:


The Temkin isotherm: a heat of adsorption that falls with coverage
=====================================================================

Temkin and Pyzhev (1940), modelling ammonia synthesis on iron, dropped
Langmuir's single site energy. In their picture the heat of adsorption
falls linearly as the surface fills, which is the same as a uniform
spread of site energies of width :math:`fRT`. Averaging the Langmuir
coverage over that spread
(:func:`~chemistrykit.surface.systems.temkin.uniform_energy_coverage`)
gives an isotherm that is logarithmic over a wide middle range of
coverage, :math:`\theta \approx (1/f)\ln(K_{max}P)`, the Temkin form. A
single Langmuir curve covers only about two decades of pressure, while
the Temkin curve rises steadily over many. Loading data in that range is
a straight line in :math:`\ln P`, which
:func:`~chemistrykit.surface.systems.temkin.fit_temkin` fits.

.. GENERATED FROM PYTHON SOURCE LINES 20-35

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.surface.systems.langmuir import langmuir_coverage
    from chemistrykit.surface.systems.temkin import fit_temkin, temkin_loading, uniform_energy_coverage

    K_max, f = 1.0e6, 20.0
    P = np.logspace(-8, 2, 400)
    theta_temkin = uniform_energy_coverage(K_max, f, P)
    theta_log = np.log(K_max * P) / f
    theta_langmuir = langmuir_coverage(K_max * np.exp(-f / 2), P)

    mid = (theta_temkin > 0.2) & (theta_temkin < 0.8)
    print(f"Max |exact - (1/f) ln(K_max P)| for 0.2 < theta < 0.8: {np.max(np.abs(theta_temkin - theta_log)[mid]):.2e}")





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

 .. code-block:: none

    Max |exact - (1/f) ln(K_max P)| for 0.2 < theta < 0.8: 9.13e-04




.. GENERATED FROM PYTHON SOURCE LINES 36-37

Fit the loading form q = (RT/b_T) ln(A_T P) to noisy middle-coverage data.

.. GENERATED FROM PYTHON SOURCE LINES 37-45

.. code-block:: Python

    T, A_T, b_T = 300.0, 50.0, 1500.0
    rng = np.random.default_rng(4)
    P_data = np.logspace(-1, 2, 10)
    q_data = temkin_loading(A_T, b_T, P_data, T) + rng.normal(scale=0.02, size=P_data.shape)
    fit = fit_temkin(P_data, q_data, T)
    print(f"True   (A_T, b_T) = ({A_T}, {b_T})")
    print(f"Fitted (A_T, b_T) = ({fit.A_T:.2f}, {fit.b_T:.1f}), R^2 = {fit.r_squared:.5f}")





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

 .. code-block:: none

    True   (A_T, b_T) = (50.0, 1500.0)
    Fitted (A_T, b_T) = (50.16, 1501.7), R^2 = 0.99997




.. GENERATED FROM PYTHON SOURCE LINES 46-64

.. code-block:: Python

    fig, axes = plt.subplots(1, 2, figsize=(11, 4))
    axes[0].semilogx(P, theta_temkin, label="uniform energy spread (exact)")
    axes[0].semilogx(P, np.clip(theta_log, 0, 1), "k--", label=r"Temkin: $(1/f)\ln(K_{max}P)$")
    axes[0].semilogx(P, theta_langmuir, ":", label="single-energy Langmuir")
    axes[0].set_xlabel("P")
    axes[0].set_ylabel(r"coverage $\theta$")
    axes[0].set_title("Temkin: coverage grows as ln P")
    axes[0].legend(fontsize=8)

    axes[1].semilogx(P_data, q_data, "o", label="data")
    P_fine = np.logspace(-1, 2, 100)
    axes[1].semilogx(P_fine, temkin_loading(fit.A_T, fit.b_T, P_fine, T), "k--", label="Temkin fit")
    axes[1].set_xlabel("P")
    axes[1].set_ylabel("loading q")
    axes[1].set_title("Straight line in ln P")
    axes[1].legend()
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/surface/temkin/images/sphx_glr_plot_01_temkin_isotherm_001.png
   :alt: Temkin: coverage grows as ln P, Straight line in ln P
   :srcset: /api/gallery/surface/temkin/images/sphx_glr_plot_01_temkin_isotherm_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.089 seconds)


.. _sphx_glr_download_api_gallery_surface_temkin_plot_01_temkin_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_temkin_isotherm.ipynb <plot_01_temkin_isotherm.ipynb>`

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

      :download:`Download Python source code: plot_01_temkin_isotherm.py <plot_01_temkin_isotherm.py>`

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

      :download:`Download zipped: plot_01_temkin_isotherm.zip <plot_01_temkin_isotherm.zip>`


.. only:: html

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

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