.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/solutions/complexation/plot_01_bjerrum_metal_ammine_formation.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_solutions_complexation_plot_01_bjerrum_metal_ammine_formation.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_solutions_complexation_plot_01_bjerrum_metal_ammine_formation.py:


Jannik Bjerrum's stepwise metal-ammine formation
================================================

Copper(II) binds ammonia one ligand at a time,
:math:`Cu(NH_3)_{n-1}^{2+} + NH_3 \rightleftharpoons Cu(NH_3)_n^{2+}`,
with stepwise constants :math:`K_1 > K_2 > K_3 > K_4`. Bjerrum showed
that the whole ladder follows from the free-ligand concentration alone.
The fractions :math:`\alpha_n`
(:func:`~chemistrykit.solutions.complex_fractions`) and the mean number
of bound ligands :math:`\bar n`
(:func:`~chemistrykit.solutions.average_ligand_number`, Bjerrum's
*formation function*) are functions of :math:`p[NH_3]` only. His
"half-:math:`\bar n`" rule runs the logic backwards: where
:math:`\bar n = n - \tfrac12`, :math:`p[NH_3] \approx \log K_n`, so the
constants can be read off a measured formation curve. The stepwise
constants below are of the size Bjerrum measured for Cu(II)-ammonia.

.. GENERATED FROM PYTHON SOURCE LINES 21-55

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.solutions import average_ligand_number, complex_fractions, cumulative_formation_constants, solve_complexation

    log_K = np.array([4.31, 3.67, 3.04, 2.30])
    beta = cumulative_formation_constants(10.0**log_K)

    pL = np.linspace(0.0, 6.0, 600)
    L = 10.0**-pL
    alpha = complex_fractions(L, beta)
    nbar = average_ligand_number(L, beta)

    fig, axes = plt.subplots(1, 2, figsize=(12, 4.8))
    for n, a in enumerate(alpha):
        axes[0].plot(pL, a, label=rf"Cu(NH$_3$)$_{n}^{{2+}}$")
    axes[0].invert_xaxis()
    axes[0].set_xlabel(r"p[NH$_3$] = $-\log$[NH$_3$]")
    axes[0].set_ylabel(r"fraction of copper, $\alpha_n$")
    axes[0].set_title("Species distribution")
    axes[0].legend()

    axes[1].plot(pL, nbar, color="black")
    estimates = []
    for n in range(1, 5):
        p_half = float(np.interp(-(n - 0.5), -nbar, pL))  # nbar decreases with pL
        estimates.append(p_half)
        axes[1].plot(p_half, n - 0.5, "o", color="darkorange")
    axes[1].invert_xaxis()
    axes[1].set_xlabel(r"p[NH$_3$]")
    axes[1].set_ylabel(r"formation function $\bar n$")
    axes[1].set_title(r"Half-$\bar n$ estimates of $\log K_n$ (dots)")
    fig.tight_layout()




.. image-sg:: /api/gallery/solutions/complexation/images/sphx_glr_plot_01_bjerrum_metal_ammine_formation_001.png
   :alt: Species distribution, Half-$\bar n$ estimates of $\log K_n$ (dots)
   :srcset: /api/gallery/solutions/complexation/images/sphx_glr_plot_01_bjerrum_metal_ammine_formation_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 56-58

The half-:math:`\bar n` estimates land close to the true constants; the
small offsets come from overlap between neighboring steps.

.. GENERATED FROM PYTHON SOURCE LINES 58-62

.. code-block:: Python


    for n, (true, est) in enumerate(zip(log_K, estimates, strict=True), start=1):
        print(f"log K{n}: true {true:.2f}, half-nbar estimate {est:.2f}")





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

 .. code-block:: none

    log K1: true 4.31, half-nbar estimate 4.48
    log K2: true 3.67, half-nbar estimate 3.68
    log K3: true 3.04, half-nbar estimate 3.01
    log K4: true 2.30, half-nbar estimate 2.15




.. GENERATED FROM PYTHON SOURCE LINES 63-65

Given only the totals, :func:`~chemistrykit.solutions.solve_complexation`
solves the ligand mass balance for the free ammonia concentration:

.. GENERATED FROM PYTHON SOURCE LINES 65-71

.. code-block:: Python


    eq = solve_complexation(M_total=0.01, L_total=0.05, beta=beta)
    print(f"free NH3 = {eq.free_ligand:.2e} M, nbar = {eq.average_ligand_number:.2f}")
    print("species (M):", np.array2string(eq.species, precision=2))

    plt.show()




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

 .. code-block:: none

    free NH3 = 1.31e-02 M, nbar = 3.69
    species (M): [1.15e-08 3.08e-06 1.89e-04 2.71e-03 7.09e-03]





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

   **Total running time of the script:** (0 minutes 0.083 seconds)


.. _sphx_glr_download_api_gallery_solutions_complexation_plot_01_bjerrum_metal_ammine_formation.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_bjerrum_metal_ammine_formation.ipynb <plot_01_bjerrum_metal_ammine_formation.ipynb>`

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

      :download:`Download Python source code: plot_01_bjerrum_metal_ammine_formation.py <plot_01_bjerrum_metal_ammine_formation.py>`

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

      :download:`Download zipped: plot_01_bjerrum_metal_ammine_formation.zip <plot_01_bjerrum_metal_ammine_formation.zip>`


.. only:: html

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

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