.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/photochem/chain_reaction/plot_01_bodenstein_nernst_hcl_chain.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_photochem_chain_reaction_plot_01_bodenstein_nernst_hcl_chain.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_photochem_chain_reaction_plot_01_bodenstein_nernst_hcl_chain.py:


Bodenstein and Nernst: the H2 + Cl2 photochemical chain reaction
===================================================================

Bodenstein (1913) measured that one absorbed photon forms up to
:math:`10^5`-:math:`10^6` HCl molecules in an illuminated H2/Cl2 mixture,
apparently violating photochemical equivalence. Nernst (1918) explained
it: the photon only splits one Cl2 into two Cl atoms, which then start a
chain (Cl + H2 -> HCl + H, H + Cl2 -> HCl + Cl) that runs until two atoms
recombine. :func:`~chemistrykit.photochem.hydrogen_chlorine_chain_network`
integrates this mechanism, and
:func:`~chemistrykit.photochem.chain_quantum_yield` gives the
steady-state yield :math:`\Phi=2k_2[\mathrm{H_2}]/\sqrt{k_t I_{abs}}`.
Rate constants are in arbitrary consistent units.

.. GENERATED FROM PYTHON SOURCE LINES 18-44

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.photochem import chain_quantum_yield, hydrogen_chlorine_chain_network

    j, k2, k3, kt = 1.0e-4, 10.0, 100.0, 100.0
    net = hydrogen_chlorine_chain_network(j, k2, k3, kt, H2_0=1.0, Cl2_0=1.0)
    res = net.integrate((0.0, 30.0), dt=1e-3, method="rk4")

    H2, Cl2, HCl = res.concentration("H2"), res.concentration("Cl2"), res.concentration("HCl")
    I_abs = j * Cl2  # photons absorbed per unit volume per unit time
    phi_numeric = np.gradient(HCl, res.t) / I_abs
    phi_steady = chain_quantum_yield(k2, kt, H2, I_abs)

    fig, ax = plt.subplots()
    ax.plot(res.t, phi_numeric, label="integrated mechanism")
    ax.plot(res.t, phi_steady, "k--", label=r"steady state $2k_2[H_2]/\sqrt{k_t I_{abs}}$")
    ax.axhline(1.0, color="r", ls=":", label=r"Stark-Einstein $\Phi=1$")
    ax.set_yscale("log")
    ax.set_xlabel("Time")
    ax.set_ylabel(r"HCl molecules per photon absorbed, $\Phi$")
    ax.set_title("H$_2$ + Cl$_2$ chain: quantum yield far above 1")
    ax.legend()
    fig.tight_layout()
    print(f"Quantum yield after the induction period: {phi_numeric[-1]:.0f} (steady-state formula {phi_steady[-1]:.0f})")




.. image-sg:: /api/gallery/photochem/chain_reaction/images/sphx_glr_plot_01_bodenstein_nernst_hcl_chain_001.png
   :alt: H$_2$ + Cl$_2$ chain: quantum yield far above 1
   :srcset: /api/gallery/photochem/chain_reaction/images/sphx_glr_plot_01_bodenstein_nernst_hcl_chain_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Quantum yield after the induction period: 179 (steady-state formula 177)




.. GENERATED FROM PYTHON SOURCE LINES 45-47

Chain length grows as the light gets weaker: fewer atoms means slower
radical-radical termination, so each chain runs longer.

.. GENERATED FROM PYTHON SOURCE LINES 47-56

.. code-block:: Python

    I = np.logspace(-8, -2, 50)
    fig2, ax2 = plt.subplots()
    ax2.loglog(I, chain_quantum_yield(k2, kt, 1.0, I))
    ax2.set_xlabel(r"Absorbed photon rate $I_{abs}$")
    ax2.set_ylabel(r"$\Phi_{HCl}$")
    ax2.set_title(r"Chain quantum yield scales as $I_{abs}^{-1/2}$")
    fig2.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/photochem/chain_reaction/images/sphx_glr_plot_01_bodenstein_nernst_hcl_chain_002.png
   :alt: Chain quantum yield scales as $I_{abs}^{-1/2}$
   :srcset: /api/gallery/photochem/chain_reaction/images/sphx_glr_plot_01_bodenstein_nernst_hcl_chain_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.480 seconds)


.. _sphx_glr_download_api_gallery_photochem_chain_reaction_plot_01_bodenstein_nernst_hcl_chain.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_bodenstein_nernst_hcl_chain.ipynb <plot_01_bodenstein_nernst_hcl_chain.ipynb>`

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

      :download:`Download Python source code: plot_01_bodenstein_nernst_hcl_chain.py <plot_01_bodenstein_nernst_hcl_chain.py>`

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

      :download:`Download zipped: plot_01_bodenstein_nernst_hcl_chain.zip <plot_01_bodenstein_nernst_hcl_chain.zip>`


.. only:: html

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

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