.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/surface/catalysis/plot_05_balandin_volcano.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_catalysis_plot_05_balandin_volcano.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_surface_catalysis_plot_05_balandin_volcano.py:


Balandin's volcano curve: catalytic activity peaks at intermediate binding
============================================================================

Balandin's multiplet theory (1929) predicted that when catalytic activity
is plotted against binding strength across a *series of catalysts*, it
rises, peaks and falls again, tracing a "volcano". Bligaard, Norskov and
coworkers (2004) later derived that shape from the Bronsted-Evans-Polanyi
relation. This example reproduces the volcano with the standard textbook
toy model (Chorkendorff & Niemantsverdriet, *Concepts of Modern Catalysis
and Kinetics*, 2nd ed., Wiley-VCH, 2007, Ch. 8). The rate needs an
occupied site to hold the intermediate and an empty site for the next
step, so rate :math:`\propto\theta(1-\theta)`, with :math:`\theta` the
Langmuir coverage
(:func:`~chemistrykit.surface.systems.langmuir.langmuir_coverage`). The
adsorption constant :math:`K` stands in for binding strength. A
hypothetical series of catalysts sits on the two legs of the volcano, and
its summit is exactly Langmuir's half-saturation point :math:`K = 1/P`.
This is not a literal reproduction of Balandin's multiplet equations.

.. GENERATED FROM PYTHON SOURCE LINES 23-40

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.surface.systems.langmuir import LangmuirIsotherm, langmuir_coverage

    P = 1.0
    K_values = np.logspace(-3, 3, 4000)
    theta = langmuir_coverage(K_values, P)
    rate = theta * (1.0 - theta)

    peak_index = int(np.argmax(rate))
    K_peak = K_values[peak_index]
    print(f"Rate-maximizing K: {K_peak:.4f} (theory: 1/P = {1.0 / P:.4f})")
    print(f"Coverage there: {theta[peak_index]:.4f} (theory: exactly 0.5)")
    iso_at_peak = LangmuirIsotherm(K=K_peak, qmax=1.0)
    print(f"Half-saturation pressure of the summit catalyst: {iso_at_peak.half_saturation_pressure():.4f} (= P)")





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

 .. code-block:: none

    Rate-maximizing K: 0.9983 (theory: 1/P = 1.0000)
    Coverage there: 0.4996 (theory: exactly 0.5)
    Half-saturation pressure of the summit catalyst: 1.0017 (= P)




.. GENERATED FROM PYTHON SOURCE LINES 41-43

A hypothetical catalyst series, from weak to strong binders. Plotted
against binding strength, they fall on the two legs of the volcano.

.. GENERATED FROM PYTHON SOURCE LINES 43-49

.. code-block:: Python

    series = {"M1": 0.005, "M2": 0.05, "M3": 0.4, "M4": 3.0, "M5": 30.0, "M6": 300.0}
    for name, K in series.items():
        th = langmuir_coverage(K, P)
        leg = "left (too weak)" if K < 1 / P else "right (too strong)"
        print(f"{name}: K = {K:7.3f}, relative activity = {th * (1 - th) / 0.25:.3f}, {leg}")





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

 .. code-block:: none

    M1: K =   0.005, relative activity = 0.020, left (too weak)
    M2: K =   0.050, relative activity = 0.181, left (too weak)
    M3: K =   0.400, relative activity = 0.816, left (too weak)
    M4: K =   3.000, relative activity = 0.750, right (too strong)
    M5: K =  30.000, relative activity = 0.125, right (too strong)
    M6: K = 300.000, relative activity = 0.013, right (too strong)




.. GENERATED FROM PYTHON SOURCE LINES 50-63

.. code-block:: Python

    fig, ax = plt.subplots(figsize=(7, 4.5))
    ax.loglog(K_values, rate / 0.25, label=r"$4\theta(1-\theta)$")
    for name, K in series.items():
        th = langmuir_coverage(K, P)
        ax.plot(K, th * (1 - th) / 0.25, "o", color="crimson")
        ax.annotate(name, (K, th * (1 - th) / 0.25), textcoords="offset points", xytext=(4, 4))
    ax.axvline(K_peak, color="gray", linestyle="--", linewidth=0.8, label=f"summit at K = {K_peak:.2f}")
    ax.set_xlabel("K (binding-strength proxy)")
    ax.set_ylabel("relative activity")
    ax.set_title("Balandin's volcano curve across a catalyst series")
    ax.legend()
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/surface/catalysis/images/sphx_glr_plot_05_balandin_volcano_001.png
   :alt: Balandin's volcano curve across a catalyst series
   :srcset: /api/gallery/surface/catalysis/images/sphx_glr_plot_05_balandin_volcano_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.091 seconds)


.. _sphx_glr_download_api_gallery_surface_catalysis_plot_05_balandin_volcano.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_05_balandin_volcano.ipynb <plot_05_balandin_volcano.ipynb>`

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

      :download:`Download Python source code: plot_05_balandin_volcano.py <plot_05_balandin_volcano.py>`

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

      :download:`Download zipped: plot_05_balandin_volcano.zip <plot_05_balandin_volcano.zip>`


.. only:: html

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

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