.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/kinetics/rate_theory/plot_03_eyring_transition_state.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_kinetics_rate_theory_plot_03_eyring_transition_state.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_kinetics_rate_theory_plot_03_eyring_transition_state.py:


Eyring's transition-state theory: activation enthalpy and entropy
====================================================================

Transition-state theory (Eyring; Evans and Polanyi, 1935) writes a rate
constant as a universal frequency times an equilibrium constant for
forming the activated complex,

.. math::

    k = \frac{k_BT}{h}\,e^{\Delta S^{\ddagger}/R}\,e^{-\Delta H^{\ddagger}/RT}.

Plotting :math:`\ln(k/T)` against :math:`1/T` (an "Eyring plot") gives
:math:`\Delta H^{\ddagger}` from the slope and :math:`\Delta S^{\ddagger}`
from the intercept -- giving the Arrhenius :math:`E_a` and :math:`A` a
thermodynamic meaning. Here synthetic data generated with
:func:`~chemistrykit.kinetics.systems.rate_theory.eyring_rate_constant`
are fitted with
:func:`~chemistrykit.kinetics.systems.rate_theory.fit_eyring` and, for
comparison, :func:`~chemistrykit.kinetics.systems.arrhenius.fit_arrhenius`.

.. GENERATED FROM PYTHON SOURCE LINES 24-52

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.constants import K_B, H, R
    from chemistrykit.kinetics.systems.arrhenius import fit_arrhenius
    from chemistrykit.kinetics.systems.rate_theory import eyring_rate_constant, fit_eyring

    dH_true, dS_true = 85e3, -25.0  # J/mol, J/(mol K)
    T = np.linspace(290.0, 350.0, 10)
    rng = np.random.default_rng(1935)
    k = eyring_rate_constant(T, dH_true, dS_true) * (1.0 + rng.normal(0.0, 0.02, T.shape))

    eyring = fit_eyring(T, k)
    arrhenius = fit_arrhenius(T, k)
    T_mid = 1.0 / np.mean(1.0 / T)
    print(f"true:    dH = {dH_true / 1000:.1f} kJ/mol, dS = {dS_true:.1f} J/(mol K)")
    print(f"fitted:  dH = {eyring.dH / 1000:.1f} kJ/mol, dS = {eyring.dS:.1f} J/(mol K), dG(298) = {eyring.dG(298.15) / 1000:.1f} kJ/mol")
    print(f"Arrhenius Ea = {arrhenius.Ea / 1000:.1f} kJ/mol  vs  dH + R*T = {(eyring.dH + R * T_mid) / 1000:.1f} kJ/mol")
    print(f"Arrhenius A  = {arrhenius.A:.2e} /s vs e*kB*T/h*exp(dS/R) = {np.e * K_B * T_mid / H * np.exp(eyring.dS / R):.2e} /s")

    fig, axes = plt.subplots(1, 2, figsize=(12, 4.5))
    axes[0].plot(1.0 / T, np.log(k / T), "o", color="steelblue", label="data")
    axes[0].plot(1.0 / T, np.log(eyring.predict(T) / T), color="darkorange", label=r"fit: slope $-\Delta H^{\ddagger}/R$")
    axes[0].set_xlabel("1/T (1/K)")
    axes[0].set_ylabel("ln(k/T)")
    axes[0].set_title(f"Eyring plot: dH = {eyring.dH / 1000:.1f} kJ/mol, dS = {eyring.dS:.0f} J/(mol K)")
    axes[0].legend()




.. image-sg:: /api/gallery/kinetics/rate_theory/images/sphx_glr_plot_03_eyring_transition_state_001.png
   :alt: Eyring plot: dH = 85.0 kJ/mol, dS = -25 J/(mol K)
   :srcset: /api/gallery/kinetics/rate_theory/images/sphx_glr_plot_03_eyring_transition_state_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    true:    dH = 85.0 kJ/mol, dS = -25.0 J/(mol K)
    fitted:  dH = 85.0 kJ/mol, dS = -25.0 J/(mol K), dG(298) = 92.5 kJ/mol
    Arrhenius Ea = 87.7 kJ/mol  vs  dH + R*T = 87.7 kJ/mol
    Arrhenius A  = 8.96e+11 /s vs e*kB*T/h*exp(dS/R) = 8.96e+11 /s

    <matplotlib.legend.Legend object at 0x11c732510>



.. GENERATED FROM PYTHON SOURCE LINES 53-55

The universal frequency k_B T/h (~6e12 /s) is the rate a reaction with
no free-energy barrier would have; the barrier Delta G divides it down.

.. GENERATED FROM PYTHON SOURCE LINES 55-66

.. code-block:: Python


    dG = np.linspace(0.0, 120e3, 200)
    axes[1].semilogy(dG / 1000, K_B * 298.15 / H * np.exp(-dG / (R * 298.15)), color="steelblue")
    axes[1].axvline(eyring.dG(298.15) / 1000, color="crimson", linestyle=":", label="fitted barrier at 298 K")
    axes[1].set_xlabel(r"$\Delta G^{\ddagger}$ (kJ/mol)")
    axes[1].set_ylabel("k at 298 K (1/s)")
    axes[1].set_title("Rate constant set by the free-energy barrier")
    axes[1].legend()

    fig.tight_layout()
    plt.show()








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

   **Total running time of the script:** (0 minutes 0.070 seconds)


.. _sphx_glr_download_api_gallery_kinetics_rate_theory_plot_03_eyring_transition_state.py:

.. only:: html

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

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

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