.. 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_04_lindemann_hinshelwood_falloff.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_04_lindemann_hinshelwood_falloff.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_kinetics_rate_theory_plot_04_lindemann_hinshelwood_falloff.py:


Lindemann-Hinshelwood falloff of a unimolecular rate constant
=============================================================

A unimolecular decomposition needs collisions to energize the molecule,
so its first-order rate constant depends on pressure.
:class:`~chemistrykit.kinetics.LindemannHinshelwood` gives the
steady-state result :math:`k_{uni}=k_1k_2[M]/(k_{-1}[M]+k_2)`. It rises
linearly at low pressure, where activation is the bottleneck, saturates
at :math:`k_\infty=k_1k_2/k_{-1}` at high pressure, and falls to half of
:math:`k_\infty` at :math:`[M]_{1/2}=k_2/k_{-1}`. Left: the falloff curve,
checked against a direct integration of the full three-step mechanism
(:meth:`~chemistrykit.kinetics.LindemannHinshelwood.network`). Right:
Lindemann's linearization :math:`1/k_{uni}` against :math:`1/[M]`. Real
data curve away from this straight line, the discrepancy that led to
Hinshelwood's, and later RRK and RRKM, energy-dependent :math:`k_2`.

.. GENERATED FROM PYTHON SOURCE LINES 20-59

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.kinetics import LindemannHinshelwood

    lh = LindemannHinshelwood(k1=0.1, k_minus1=10.0, k2=100.0)
    M = np.logspace(-1, 4, 200)

    # Integrate the full mechanism for several [M]; the step must resolve A*'s
    # lifetime 1/(k_-1[M] + k2), and the run lasts about 8 lifetimes of A.
    M_check = np.array([0.3, 3.0, 30.0, 300.0])
    k_numeric = []
    for m in M_check:
        dt = 0.2 / (lh.k_minus1 * m + lh.k2)
        result = lh.network(m).integrate((0.0, 8.0 / float(lh.rate_constant(m))), dt=dt)
        A, t = result.concentration("A"), result.t
        i = len(t) // 2
        k_numeric.append(-np.log(A[-1] / A[i]) / (t[-1] - t[i]))

    fig, axes = plt.subplots(1, 2, figsize=(12, 4.8))
    axes[0].loglog(M, lh.rate_constant(M), color="darkorange", label=r"steady-state $k_{uni}$")
    axes[0].loglog(M, lh.k0 * M, ":", color="gray", label=r"low-pressure limit $k_1[M]$")
    axes[0].axhline(lh.k_inf, color="gray", linestyle="--", label=r"$k_\infty$")
    axes[0].axvline(lh.M_half, color="steelblue", linewidth=0.8, label=r"$[M]_{1/2}$")
    axes[0].loglog(M_check, k_numeric, "ko", label="full mechanism, integrated")
    axes[0].set_xlabel("bath-gas concentration [M]")
    axes[0].set_ylabel(r"$k_{uni}$ (s$^{-1}$)")
    axes[0].set_title("Pressure falloff")
    axes[0].legend()

    inv_M = 1.0 / M
    axes[1].plot(inv_M, lh.inverse_rate_constant(M), color="darkorange")
    axes[1].set_xlim(0, 1.0)
    axes[1].set_ylim(0, lh.inverse_rate_constant(1.0) * 1.05)
    axes[1].set_xlabel("1/[M]")
    axes[1].set_ylabel(r"$1/k_{uni}$ (s)")
    axes[1].set_title(r"Lindemann plot: intercept $1/k_\infty$, slope $1/k_1$")
    fig.tight_layout()




.. image-sg:: /api/gallery/kinetics/rate_theory/images/sphx_glr_plot_04_lindemann_hinshelwood_falloff_001.png
   :alt: Pressure falloff, Lindemann plot: intercept $1/k_\infty$, slope $1/k_1$
   :srcset: /api/gallery/kinetics/rate_theory/images/sphx_glr_plot_04_lindemann_hinshelwood_falloff_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 60-62

The two agree to within the steady-state approximation's own error,
of order :math:`k_1[M]/(k_{-1}[M]+k_2)`, about 1% here.

.. GENERATED FROM PYTHON SOURCE LINES 62-67

.. code-block:: Python


    for m, k in zip(M_check, k_numeric, strict=True):
        print(f"[M] = {m:7.1f}: steady state {float(lh.rate_constant(m)):.4f}, integrated {k:.4f} s^-1")

    plt.show()




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

 .. code-block:: none

    [M] =     0.3: steady state 0.0291, integrated 0.0291 s^-1
    [M] =     3.0: steady state 0.2308, integrated 0.2306 s^-1
    [M] =    30.0: steady state 0.7500, integrated 0.7458 s^-1
    [M] =   300.0: steady state 0.9677, integrated 0.9588 s^-1





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

   **Total running time of the script:** (0 minutes 2.347 seconds)


.. _sphx_glr_download_api_gallery_kinetics_rate_theory_plot_04_lindemann_hinshelwood_falloff.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_lindemann_hinshelwood_falloff.ipynb <plot_04_lindemann_hinshelwood_falloff.ipynb>`

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

      :download:`Download Python source code: plot_04_lindemann_hinshelwood_falloff.py <plot_04_lindemann_hinshelwood_falloff.py>`

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

      :download:`Download zipped: plot_04_lindemann_hinshelwood_falloff.zip <plot_04_lindemann_hinshelwood_falloff.zip>`


.. only:: html

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

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