.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/solutions/solubility/plot_02_nernst_solubility_product.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_solubility_plot_02_nernst_solubility_product.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_solutions_solubility_plot_02_nernst_solubility_product.py:


Nernst's solubility product: Ksp for salts of any stoichiometry
=================================================================

Nernst showed that for a sparingly soluble salt :math:`M_pX_q` in
equilibrium with its saturated solution, the product of the ion
concentrations, each raised to its stoichiometric coefficient, is a
constant:

.. math::

   K_{sp} = [M]^p[X]^q = p^p q^q s^{p+q}.

:func:`~chemistrykit.solutions.systems.solubility.ksp_from_molar_solubility`
and its inverse
:func:`~chemistrykit.solutions.systems.solubility.molar_solubility_from_ksp`
convert between the measured molar solubility :math:`s` and
:math:`K_{sp}`. Because the exponent :math:`p+q` depends on
stoichiometry, a smaller :math:`K_{sp}` does not always mean a less
soluble salt.

.. GENERATED FROM PYTHON SOURCE LINES 24-48

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.solutions.systems.solubility import ksp_from_molar_solubility, molar_solubility_from_ksp

    salts = {
        "AgCl (1:1)": (1.8e-10, 1, 1),
        "CaF$_2$ (1:2)": (3.9e-11, 1, 2),
        "Ag$_2$CrO$_4$ (2:1)": (1.1e-12, 2, 1),
        "Ca$_3$(PO$_4$)$_2$ (3:2)": (2.1e-33, 3, 2),
    }

    Ksp_axis = np.logspace(-35, -5, 200)
    fig, ax = plt.subplots(figsize=(7, 5))
    for (name, (Ksp, p, q)), color in zip(salts.items(), ["steelblue", "darkorange", "seagreen", "purple"]):
        ax.loglog(Ksp_axis, molar_solubility_from_ksp(Ksp_axis, p, q), color=color, alpha=0.5)
        s = molar_solubility_from_ksp(Ksp, p, q)
        ax.plot(Ksp, s, "o", color=color, label=f"{name}: s = {s:.1e} M")
    ax.set_xlabel("solubility product $K_{sp}$")
    ax.set_ylabel("molar solubility s (mol/L)")
    ax.set_title(r"Solubility product: $s = (K_{sp}/p^pq^q)^{1/(p+q)}$")
    ax.legend(fontsize=8)
    fig.tight_layout()




.. image-sg:: /api/gallery/solutions/solubility/images/sphx_glr_plot_02_nernst_solubility_product_001.png
   :alt: Solubility product: $s = (K_{sp}/p^pq^q)^{1/(p+q)}$
   :srcset: /api/gallery/solutions/solubility/images/sphx_glr_plot_02_nernst_solubility_product_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 49-52

Ag2CrO4 has a *smaller* Ksp than AgCl yet is *more* soluble -- the
comparison only works between salts of the same stoichiometry. The
round trip s -> Ksp -> s is exact:

.. GENERATED FROM PYTHON SOURCE LINES 52-59

.. code-block:: Python


    for name, (Ksp, p, q) in salts.items():
        s = molar_solubility_from_ksp(Ksp, p, q)
        plain = name.replace("$", "").replace("_", "")
        print(f"{plain:22s}: Ksp = {Ksp:.1e}, s = {s:.3e} M, Ksp from s = {ksp_from_molar_solubility(s, p, q):.2e}")

    plt.show()




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

 .. code-block:: none

    AgCl (1:1)            : Ksp = 1.8e-10, s = 1.342e-05 M, Ksp from s = 1.80e-10
    CaF2 (1:2)            : Ksp = 3.9e-11, s = 2.136e-04 M, Ksp from s = 3.90e-11
    Ag2CrO4 (2:1)         : Ksp = 1.1e-12, s = 6.503e-05 M, Ksp from s = 1.10e-12
    Ca3(PO4)2 (3:2)       : Ksp = 2.1e-33, s = 1.142e-07 M, Ksp from s = 2.10e-33





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

   **Total running time of the script:** (0 minutes 0.075 seconds)


.. _sphx_glr_download_api_gallery_solutions_solubility_plot_02_nernst_solubility_product.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_nernst_solubility_product.ipynb <plot_02_nernst_solubility_product.ipynb>`

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

      :download:`Download Python source code: plot_02_nernst_solubility_product.py <plot_02_nernst_solubility_product.py>`

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

      :download:`Download zipped: plot_02_nernst_solubility_product.zip <plot_02_nernst_solubility_product.zip>`


.. only:: html

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

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