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.. "api/gallery/kinetics/rate_theory/plot_01_collision_theory.py"
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.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_kinetics_rate_theory_plot_01_collision_theory.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_kinetics_rate_theory_plot_01_collision_theory.py:


Trautz-Lewis collision theory: rate constants from molecular collisions
==========================================================================

Trautz (1916) and Lewis (1918) computed a gas-phase rate constant as the
number of collisions per unit time times the Boltzmann fraction with
enough energy,

.. math::

    k = P\,\sigma \sqrt{\frac{8k_BT}{\pi\mu}}\,N_A\,e^{-E_a/RT},

evaluated by
:func:`~chemistrykit.kinetics.systems.rate_theory.collision_theory_rate_constant`.
The left panel shows its two ingredients -- the collision rate grows only
as :math:`\sqrt{T}`, while the energetic fraction grows exponentially.
The right panel shows that an Arrhenius fit to collision-theory rate
constants returns :math:`E_a + RT/2`, because the prefactor itself
depends weakly on temperature. Lewis's test case was
:math:`2\,\text{HI} \to \text{H}_2 + \text{I}_2`; the molecular
parameters below are illustrative values of that order.

.. GENERATED FROM PYTHON SOURCE LINES 25-65



.. image-sg:: /api/gallery/kinetics/rate_theory/images/sphx_glr_plot_01_collision_theory_001.png
   :alt: Where collision theory's temperature dependence comes from, Apparent activation energy = Ea + RT/2
   :srcset: /api/gallery/kinetics/rate_theory/images/sphx_glr_plot_01_collision_theory_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    collision rate constant at 700 K: 2.078e-03 L/(mol s)
    Arrhenius fit: Ea_app = 186.75 kJ/mol; Ea + RT/2 = 186.77 kJ/mol






|

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.constants import ATOMIC_MASS_UNIT, R
    from chemistrykit.kinetics.systems.arrhenius import fit_arrhenius
    from chemistrykit.kinetics.systems.rate_theory import collision_theory_rate_constant

    m_HI = 127.91 * ATOMIC_MASS_UNIT
    mu = m_HI / 2.0  # identical collision partners
    d = 3.5e-10  # collision diameter, m (illustrative)
    sigma = np.pi * d**2
    Ea = 184e3  # J/mol, of the order of the HI decomposition barrier

    T = np.linspace(550.0, 800.0, 60)
    collision_part = collision_theory_rate_constant(T, sigma, mu)  # every collision reacts
    k = collision_theory_rate_constant(T, sigma, mu, Ea=Ea)

    fig, axes = plt.subplots(1, 2, figsize=(12, 4.5))
    axes[0].semilogy(T, collision_part / collision_part[0], label=r"collision rate $\propto\sqrt{T}$")
    axes[0].semilogy(T, np.exp(-Ea / (R * T)) / np.exp(-Ea / (R * T[0])), label=r"energetic fraction $e^{-E_a/RT}$")
    axes[0].semilogy(T, k / k[0], "k--", label="k (product)")
    axes[0].set_xlabel("T (K)")
    axes[0].set_ylabel("relative to value at 550 K")
    axes[0].set_title("Where collision theory's temperature dependence comes from")
    axes[0].legend()

    fit = fit_arrhenius(T, k)
    T_mid = 1.0 / np.mean(1.0 / T)
    print(f"collision rate constant at 700 K: {collision_theory_rate_constant(700.0, sigma, mu, Ea=Ea) * 1000:.3e} L/(mol s)")
    print(f"Arrhenius fit: Ea_app = {fit.Ea / 1000:.2f} kJ/mol; Ea + RT/2 = {(Ea + 0.5 * R * T_mid) / 1000:.2f} kJ/mol")

    axes[1].plot(1.0 / T, np.log(k), color="steelblue", label="collision theory")
    axes[1].plot(1.0 / T, np.log(fit.predict(T)), "k--", linewidth=0.8, label=f"Arrhenius fit, Ea = {fit.Ea / 1000:.1f} kJ/mol")
    axes[1].set_xlabel("1/T (1/K)")
    axes[1].set_ylabel("ln k  (k in m^3/(mol s))")
    axes[1].set_title("Apparent activation energy = Ea + RT/2")
    axes[1].legend()

    fig.tight_layout()
    plt.show()


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

   **Total running time of the script:** (0 minutes 0.116 seconds)


.. _sphx_glr_download_api_gallery_kinetics_rate_theory_plot_01_collision_theory.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_collision_theory.ipynb <plot_01_collision_theory.ipynb>`

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      :download:`Download Python source code: plot_01_collision_theory.py <plot_01_collision_theory.py>`

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