.. 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_02_smoluchowski_diffusion_limit.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_02_smoluchowski_diffusion_limit.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_kinetics_rate_theory_plot_02_smoluchowski_diffusion_limit.py:


Smoluchowski's diffusion-limited reaction rate
=================================================

Smoluchowski (1917) asked how fast two species meet if they react on
first contact, so that diffusion alone limits the rate. Solving the
diffusion equation around one reactant gives

.. math::

    k_D = 4\pi D R^* N_A, \qquad
    k(t) = k_D\left(1 + \frac{R^*}{\sqrt{\pi D t}}\right)

(:func:`~chemistrykit.kinetics.systems.rate_theory.smoluchowski_rate_constant`,
:func:`~chemistrykit.kinetics.systems.rate_theory.smoluchowski_transient_rate_constant`).
With the Stokes-Einstein diffusion coefficient the molecular size cancels
and :math:`k_D = 8RT/3\eta`
(:func:`~chemistrykit.kinetics.systems.rate_theory.diffusion_limited_rate_constant`):
the ceiling on any bimolecular rate constant in solution depends only on
the solvent's viscosity.

.. GENERATED FROM PYTHON SOURCE LINES 24-40

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.kinetics.systems.rate_theory import (
        diffusion_limited_rate_constant,
        smoluchowski_rate_constant,
        smoluchowski_transient_rate_constant,
    )

    D = 4e-9  # m^2/s, two small solutes in water
    R_contact = 5e-10  # m
    kD = smoluchowski_rate_constant(D, R_contact)
    print(f"steady-state k_D = {kD * 1000:.3e} L/(mol s)")

    fig, axes = plt.subplots(1, 2, figsize=(12, 4.5))




.. image-sg:: /api/gallery/kinetics/rate_theory/images/sphx_glr_plot_02_smoluchowski_diffusion_limit_001.png
   :alt: plot 02 smoluchowski diffusion limit
   :srcset: /api/gallery/kinetics/rate_theory/images/sphx_glr_plot_02_smoluchowski_diffusion_limit_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    steady-state k_D = 1.514e+10 L/(mol s)




.. GENERATED FROM PYTHON SOURCE LINES 41-43

The transient: at first the reactants' neighbourhoods have not yet been
depleted, so encounters are more frequent than at steady state.

.. GENERATED FROM PYTHON SOURCE LINES 43-53

.. code-block:: Python


    t = np.logspace(-13, -7, 200)
    axes[0].semilogx(t, smoluchowski_transient_rate_constant(D, R_contact, t) * 1000, color="steelblue", label="k(t)")
    axes[0].axhline(kD * 1000, color="gray", linestyle="--", label="steady state $k_D = 4\\pi D R^* N_A$")
    axes[0].axvline(R_contact**2 / (np.pi * D), color="crimson", linestyle=":", label=r"$t = R^{*2}/\pi D$ (k = 2 k_D)")
    axes[0].set_xlabel("t after mixing (s)")
    axes[0].set_ylabel("k (L mol$^{-1}$ s$^{-1}$)")
    axes[0].set_title("Smoluchowski transient decays to the steady rate")
    axes[0].legend(fontsize=8)





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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x11b484050>



.. GENERATED FROM PYTHON SOURCE LINES 54-56

Viscosity sets the ceiling: 8RT/(3 eta) for several solvents at 298 K
(approximate literature viscosities).

.. GENERATED FROM PYTHON SOURCE LINES 56-69

.. code-block:: Python


    solvents = {"diethyl ether": 2.2e-4, "acetone": 3.1e-4, "water": 8.9e-4, "ethanol": 1.07e-3, "ethylene glycol": 1.6e-2, "glycerol": 0.95}
    names = list(solvents)
    k_solvent = np.array([diffusion_limited_rate_constant(298.15, eta) * 1000 for eta in solvents.values()])
    axes[1].barh(names, k_solvent, color="steelblue")
    axes[1].set_xscale("log")
    axes[1].set_xlabel("diffusion-limited k (L mol$^{-1}$ s$^{-1}$)")
    axes[1].set_title("Diffusion limit 8RT/(3 eta) at 298 K")
    for name, kv in zip(names, k_solvent):
        print(f"{name:16s}: {kv:.2e} L/(mol s)")

    fig.tight_layout()
    plt.show()




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

 .. code-block:: none

    diethyl ether   : 3.00e+10 L/(mol s)
    acetone         : 2.13e+10 L/(mol s)
    water           : 7.43e+09 L/(mol s)
    ethanol         : 6.18e+09 L/(mol s)
    ethylene glycol : 4.13e+08 L/(mol s)
    glycerol        : 6.96e+06 L/(mol s)





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

   **Total running time of the script:** (0 minutes 0.081 seconds)


.. _sphx_glr_download_api_gallery_kinetics_rate_theory_plot_02_smoluchowski_diffusion_limit.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_02_smoluchowski_diffusion_limit.ipynb <plot_02_smoluchowski_diffusion_limit.ipynb>`

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

      :download:`Download Python source code: plot_02_smoluchowski_diffusion_limit.py <plot_02_smoluchowski_diffusion_limit.py>`

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

      :download:`Download zipped: plot_02_smoluchowski_diffusion_limit.zip <plot_02_smoluchowski_diffusion_limit.zip>`


.. only:: html

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

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