.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/surface/catalysis/plot_04_constable_compensation.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_04_constable_compensation.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_surface_catalysis_plot_04_constable_compensation.py:


Constable's compensation effect across a catalyst series
===========================================================

Constable (1925) found that across a series of similar catalysts, a
higher activation energy :math:`E_a` came with a higher pre-exponential
factor :math:`A`, so :math:`\ln A` rose linearly with :math:`E_a`:

.. math::

   \ln A = \alpha + \frac{E_a}{RT_{iso}}.

When that holds, every catalyst in the series has the same rate constant
at the *isokinetic temperature* :math:`T_{iso}`, and their Arrhenius lines
all cross there. This example builds such a series and uses
:func:`~chemistrykit.surface.systems.catalysis.compare_catalyzed_rate`,
which takes separate :math:`E_a` and :math:`A` values for the two pathways.
It then recovers :math:`T_{iso}` from the slope of the Constable plot.
The rate enhancement over a reference catalyst is exactly 1 at
:math:`T_{iso}`.

.. GENERATED FROM PYTHON SOURCE LINES 24-41

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.constants import R
    from chemistrykit.surface.systems.catalysis import compare_catalyzed_rate

    T_iso = 550.0  # K
    alpha = 5.0
    Ea_series = np.array([60e3, 75e3, 90e3, 105e3, 120e3])
    lnA_series = alpha + Ea_series / (R * T_iso)

    rng = np.random.default_rng(7)
    lnA_measured = lnA_series + rng.normal(scale=0.2, size=Ea_series.shape)
    slope, intercept = np.polyfit(Ea_series, lnA_measured, 1)
    print(f"True isokinetic temperature:           {T_iso:.1f} K")
    print(f"Recovered from the Constable plot:     {1.0 / (R * slope):.1f} K")





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

 .. code-block:: none

    True isokinetic temperature:           550.0 K
    Recovered from the Constable plot:     557.1 K




.. GENERATED FROM PYTHON SOURCE LINES 42-44

Rate enhancement of each catalyst over the lowest-Ea member, above,
at, and below T_iso: the ordering flips at T_iso.

.. GENERATED FROM PYTHON SOURCE LINES 44-49

.. code-block:: Python

    ref_Ea, ref_A = Ea_series[0], np.exp(lnA_series[0])
    for T in [450.0, T_iso, 650.0]:
        ratios = [compare_catalyzed_rate(ref_Ea, Ea, T, A_uncatalyzed=ref_A, A_catalyzed=np.exp(lnA)).rate_enhancement for Ea, lnA in zip(Ea_series, lnA_series)]
        print(f"T = {T:5.1f} K: k/k_ref = " + ", ".join(f"{r:.3g}" for r in ratios))





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

 .. code-block:: none

    T = 450.0 K: k/k_ref = 1, 0.482, 0.233, 0.112, 0.0542
    T = 550.0 K: k/k_ref = 1, 1, 1, 1, 1
    T = 650.0 K: k/k_ref = 1, 1.66, 2.74, 4.54, 7.53




.. GENERATED FROM PYTHON SOURCE LINES 50-68

.. code-block:: Python

    invT = np.linspace(1 / 800.0, 1 / 400.0, 200)
    fig, axes = plt.subplots(1, 2, figsize=(11, 4))
    for Ea, lnA in zip(Ea_series, lnA_series):
        axes[0].plot(1e3 * invT, lnA - Ea / R * invT, label=f"$E_a$={Ea / 1e3:.0f} kJ/mol")
    axes[0].axvline(1e3 / T_iso, color="gray", linestyle="--", linewidth=0.8)
    axes[0].set_xlabel("1000 / T (1/K)")
    axes[0].set_ylabel("ln k")
    axes[0].set_title("Arrhenius lines cross at $T_{iso}$")
    axes[0].legend(fontsize=8)

    axes[1].plot(Ea_series / 1e3, lnA_measured, "o", label="catalyst series")
    axes[1].plot(Ea_series / 1e3, slope * Ea_series + intercept, "k--", label="linear fit")
    axes[1].set_xlabel("$E_a$ (kJ/mol)")
    axes[1].set_ylabel("ln A")
    axes[1].set_title("Constable plot: ln A rises with $E_a$")
    axes[1].legend()
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/surface/catalysis/images/sphx_glr_plot_04_constable_compensation_001.png
   :alt: Arrhenius lines cross at $T_{iso}$, Constable plot: ln A rises with $E_a$
   :srcset: /api/gallery/surface/catalysis/images/sphx_glr_plot_04_constable_compensation_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.076 seconds)


.. _sphx_glr_download_api_gallery_surface_catalysis_plot_04_constable_compensation.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_04_constable_compensation.ipynb <plot_04_constable_compensation.ipynb>`

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

      :download:`Download Python source code: plot_04_constable_compensation.py <plot_04_constable_compensation.py>`

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

      :download:`Download zipped: plot_04_constable_compensation.zip <plot_04_constable_compensation.zip>`


.. only:: html

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

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