.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/kinetics/networks/plot_03_bateman_consecutive.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_networks_plot_03_bateman_consecutive.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_kinetics_networks_plot_03_bateman_consecutive.py:


Bateman's closed-form solution for consecutive reactions
==========================================================

Bateman (1910) solved the chain of first-order steps
:math:`A \xrightarrow{k_1} B \xrightarrow{k_2} C` (written for radioactive
decay series, but identical for consecutive chemical reactions) in
closed form:

.. math::

    [B](t) = \frac{k_1 [A]_0}{k_2 - k_1}\left(e^{-k_1 t} - e^{-k_2 t}\right),

with the intermediate peaking at
:math:`t_{max} = \ln(k_2/k_1)/(k_2 - k_1)`.
:func:`~chemistrykit.kinetics.systems.networks.consecutive_analytic`
evaluates these formulas; here they are checked against a numerical
integration of the same mechanism with
:class:`~chemistrykit.kinetics.systems.networks.StoichiometricNetwork`.

.. GENERATED FROM PYTHON SOURCE LINES 23-44

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.kinetics.systems.networks import StoichiometricNetwork, consecutive_analytic
    from chemistrykit.kinetics.visualizers.kinetics_plots import plot_concentration_vs_time

    k1, k2, A0 = 1.0, 0.3, 1.0
    net = StoichiometricNetwork.consecutive(k1=k1, k2=k2, A0=A0)
    result = net.integrate((0.0, 15.0), dt=1e-3, method="rk4")
    A_exact, B_exact, C_exact = consecutive_analytic(A0, k1, k2, result.t)
    print(f"max |numeric - Bateman| for [B]: {np.max(np.abs(result.concentration('B') - B_exact)):.2e}")

    fig, axes = plt.subplots(1, 2, figsize=(12, 4.5))
    plot_concentration_vs_time(result, ax=axes[0])
    for y in (A_exact, B_exact, C_exact):
        axes[0].plot(result.t, y, "k--", linewidth=0.8)
    t_max = np.log(k2 / k1) / (k2 - k1)
    axes[0].axvline(t_max, color="gray", linestyle=":", label=f"t_max = {t_max:.2f}")
    axes[0].legend()
    axes[0].set_title("A -> B -> C: numerical (solid) vs Bateman (dashed)")




.. image-sg:: /api/gallery/kinetics/networks/images/sphx_glr_plot_03_bateman_consecutive_001.png
   :alt: A -> B -> C: numerical (solid) vs Bateman (dashed)
   :srcset: /api/gallery/kinetics/networks/images/sphx_glr_plot_03_bateman_consecutive_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    max |numeric - Bateman| for [B]: 4.44e-15

    Text(0.5, 1.0, 'A -> B -> C: numerical (solid) vs Bateman (dashed)')



.. GENERATED FROM PYTHON SOURCE LINES 45-47

The height and timing of the intermediate's peak depend only on the
ratio of the two rate constants.

.. GENERATED FROM PYTHON SOURCE LINES 47-63

.. code-block:: Python


    t = np.linspace(0.0, 15.0, 600)
    for ratio in (0.1, 0.3, 1.0, 3.0, 10.0):
        _, B, _ = consecutive_analytic(A0=1.0, k1=1.0, k2=ratio, t=t)
        axes[1].plot(t, B, label=f"k2/k1 = {ratio:g}")
        if ratio != 1.0:
            tm = np.log(ratio) / (ratio - 1.0)
            _, Bm, _ = consecutive_analytic(1.0, 1.0, ratio, tm)
            axes[1].plot(tm, Bm, "ko", markersize=4)
    axes[1].set_xlabel("t")
    axes[1].set_ylabel("[B]")
    axes[1].set_title("Intermediate peak at t_max = ln(k2/k1)/(k2-k1)")
    axes[1].legend()

    fig.tight_layout()
    plt.show()








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

   **Total running time of the script:** (0 minutes 0.372 seconds)


.. _sphx_glr_download_api_gallery_kinetics_networks_plot_03_bateman_consecutive.py:

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      :download:`Download Python source code: plot_03_bateman_consecutive.py <plot_03_bateman_consecutive.py>`

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