.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/surface/catalysis/plot_02_arrhenius_ostwald_rate_enhancement.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_02_arrhenius_ostwald_rate_enhancement.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_surface_catalysis_plot_02_arrhenius_ostwald_rate_enhancement.py:


Arrhenius and Ostwald: a catalyst speeds a reaction but leaves its equilibrium alone
======================================================================================

Arrhenius's :math:`k = Ae^{-E_a/RT}` explains why a catalyst's lower
activation energy multiplies the rate by :math:`e^{\Delta E_a/RT}`
(:func:`~chemistrykit.surface.systems.catalysis.compare_catalyzed_rate`).
Ostwald's definition adds the key condition that a catalyst cannot move
the equilibrium. It lowers the barrier for the forward and the reverse
reaction by the same amount, so both rate constants grow by the same
factor and :math:`K = k_f/k_r` does not change. Below, the reversible
reaction :math:`A \rightleftharpoons B` reaches equilibrium far sooner
with the catalyst, and it reaches exactly the same equilibrium.

.. GENERATED FROM PYTHON SOURCE LINES 17-25

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.surface.systems.catalysis import compare_catalyzed_rate

    T = 298.15
    Ea_forward, delta_H = 100.0e3, -20.0e3  # J/mol; reverse barrier is Ea_forward - delta_H








.. GENERATED FROM PYTHON SOURCE LINES 26-27

Rate enhancement grows exponentially with the activation-energy drop.

.. GENERATED FROM PYTHON SOURCE LINES 27-31

.. code-block:: Python

    delta_Ea_values = np.linspace(0.0, 50e3, 26)
    enhancements = [compare_catalyzed_rate(Ea_forward, Ea_forward - d, T, A_uncatalyzed=1e13).rate_enhancement for d in delta_Ea_values]
    print(f"Lowering Ea by 30 kJ/mol at {T} K speeds the reaction up {enhancements[15]:.2e}-fold")





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

 .. code-block:: none

    Lowering Ea by 30 kJ/mol at 298.15 K speeds the reaction up 1.80e+05-fold




.. GENERATED FROM PYTHON SOURCE LINES 32-33

Ostwald: the same barrier drop for forward and reverse steps.

.. GENERATED FROM PYTHON SOURCE LINES 33-50

.. code-block:: Python

    delta_Ea = 30e3
    fwd = compare_catalyzed_rate(Ea_forward, Ea_forward - delta_Ea, T, A_uncatalyzed=1e13)
    rev = compare_catalyzed_rate(Ea_forward - delta_H, Ea_forward - delta_H - delta_Ea, T, A_uncatalyzed=1e13)
    K_uncat = fwd.k_uncatalyzed / rev.k_uncatalyzed
    K_cat = fwd.k_catalyzed / rev.k_catalyzed
    print(f"Forward enhancement {fwd.rate_enhancement:.3e}, reverse enhancement {rev.rate_enhancement:.3e}")
    print(f"K without catalyst = {K_uncat:.4f}, K with catalyst = {K_cat:.4f}")


    def approach(kf, kr, t, A0=1.0):
        A_eq = A0 * kr / (kf + kr)
        return A_eq + (A0 - A_eq) * np.exp(-(kf + kr) * t)


    t_cat = np.linspace(0.0, 5.0 / (fwd.k_catalyzed + rev.k_catalyzed), 300)
    t_uncat = t_cat * fwd.rate_enhancement





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

 .. code-block:: none

    Forward enhancement 1.802e+05, reverse enhancement 1.802e+05
    K without catalyst = 3190.4240, K with catalyst = 3190.4240




.. GENERATED FROM PYTHON SOURCE LINES 51-73

.. code-block:: Python

    fig, axes = plt.subplots(1, 3, figsize=(15, 4))
    axes[0].semilogy(delta_Ea_values / 1e3, enhancements, "o-")
    axes[0].set_xlabel(r"$\Delta E_a$ (kJ/mol)")
    axes[0].set_ylabel(r"$k_{cat}/k_{uncat}$")
    axes[0].set_title("Arrhenius: exponential rate enhancement")

    axes[1].plot(t_cat, approach(fwd.k_catalyzed, rev.k_catalyzed, t_cat), label="catalyzed")
    axes[1].plot(t_cat, approach(fwd.k_uncatalyzed, rev.k_uncatalyzed, t_cat), label="uncatalyzed")
    axes[1].set_xlabel("time (s)")
    axes[1].set_ylabel("[A] / [A]$_0$")
    axes[1].set_title("Same time axis: only the catalyzed run moves")
    axes[1].legend()

    axes[2].plot(t_cat / t_cat[-1], approach(fwd.k_catalyzed, rev.k_catalyzed, t_cat), label="catalyzed")
    axes[2].plot(t_uncat / t_uncat[-1], approach(fwd.k_uncatalyzed, rev.k_uncatalyzed, t_uncat), "--", label="uncatalyzed (time rescaled)")
    axes[2].axhline(1.0 / (1.0 + K_cat), color="gray", linewidth=0.8)
    axes[2].set_xlabel("time / run length")
    axes[2].set_ylabel("[A] / [A]$_0$")
    axes[2].set_title("Ostwald: identical equilibrium")
    axes[2].legend()
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/surface/catalysis/images/sphx_glr_plot_02_arrhenius_ostwald_rate_enhancement_001.png
   :alt: Arrhenius: exponential rate enhancement, Same time axis: only the catalyzed run moves, Ostwald: identical equilibrium
   :srcset: /api/gallery/surface/catalysis/images/sphx_glr_plot_02_arrhenius_ostwald_rate_enhancement_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.099 seconds)


.. _sphx_glr_download_api_gallery_surface_catalysis_plot_02_arrhenius_ostwald_rate_enhancement.py:

.. only:: html

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

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

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

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

      :download:`Download zipped: plot_02_arrhenius_ostwald_rate_enhancement.zip <plot_02_arrhenius_ostwald_rate_enhancement.zip>`


.. only:: html

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