.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/solutions/titration/plot_02_gran_plot.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_titration_plot_02_gran_plot.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_solutions_titration_plot_02_gran_plot.py:


Gran's plot: finding the equivalence point by linear extrapolation
====================================================================

Near the equivalence point a titration curve is steep and hard to
sample. Gran observed that before the equivalence point of a strong acid
titrated with a strong base,

.. math::

   G = (V_a + V_b)\,10^{-\mathrm{pH}} = C_a V_a - C_b V_b,

a straight line in the titrant volume :math:`V_b` that reaches zero
exactly at the equivalence volume. Fitting it with
:func:`~chemistrykit.solutions.systems.titration.gran_plot` recovers
:math:`V_e` from noisy pre-equivalence readings alone, without knowing
either concentration.

.. GENERATED FROM PYTHON SOURCE LINES 21-56

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.solutions.systems.titration import StrongAcidStrongBaseTitration, gran_plot

    titration = StrongAcidStrongBaseTitration(Ca=0.100, Va=0.050, Cb=0.100)
    Ve = titration.equivalence_volume()

    rng = np.random.default_rng(0)
    Vb_data = np.linspace(0.30 * Ve, 0.95 * Ve, 12)
    pH_data = titration.pH_at(Vb_data) + rng.normal(0.0, 0.01, Vb_data.size)  # +-0.01 pH meter noise

    result = gran_plot(Vb_data, pH_data, Va=titration.Va)

    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 4.5))
    Vb_curve = np.linspace(1e-6, 1.4 * Ve, 500)
    ax1.plot(Vb_curve * 1000, titration.pH_at(Vb_curve), color="gray", label="exact titration curve")
    ax1.plot(Vb_data * 1000, pH_data, "o", color="steelblue", label="measured points used")
    ax1.axvline(Ve * 1000, color="gray", linestyle=":")
    ax1.set_xlabel("volume of NaOH added (mL)")
    ax1.set_ylabel("pH")
    ax1.set_title("Pre-equivalence data only")
    ax1.legend(fontsize=8)

    V_line = np.linspace(0, result.equivalence_volume, 2)
    ax2.plot(result.Vb * 1000, result.G * 1000, "o", color="steelblue", label="Gran function")
    ax2.plot(V_line * 1000, (result.intercept + result.slope * V_line) * 1000, color="darkorange", label="least-squares line")
    ax2.axhline(0, color="gray", linewidth=0.7)
    ax2.axvline(Ve * 1000, color="gray", linestyle=":", label="true $V_e$")
    ax2.set_xlabel("volume of NaOH added (mL)")
    ax2.set_ylabel(r"$G = (V_a+V_b)10^{-\mathrm{pH}}$ (mmol)")
    ax2.set_title("Gran plot: x-intercept = equivalence volume")
    ax2.legend(fontsize=8)
    fig.tight_layout()




.. image-sg:: /api/gallery/solutions/titration/images/sphx_glr_plot_02_gran_plot_001.png
   :alt: Pre-equivalence data only, Gran plot: x-intercept = equivalence volume
   :srcset: /api/gallery/solutions/titration/images/sphx_glr_plot_02_gran_plot_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 57-59

Recovered quantities -- the slope is :math:`-C_b` and the intercept is
:math:`C_aV_a`:

.. GENERATED FROM PYTHON SOURCE LINES 59-64

.. code-block:: Python


    print(f"Gran equivalence volume: {result.equivalence_volume * 1000:.3f} mL (true {Ve * 1000:.3f} mL)")
    print(f"Slope = {result.slope:.4f} mol/L (-Cb = -0.1000); intercept = {result.intercept * 1000:.4f} mmol (CaVa = 5.0000)")

    plt.show()




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

 .. code-block:: none

    Gran equivalence volume: 50.049 mL (true 50.000 mL)
    Slope = -0.0994 mol/L (-Cb = -0.1000); intercept = 4.9768 mmol (CaVa = 5.0000)





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

   **Total running time of the script:** (0 minutes 0.078 seconds)


.. _sphx_glr_download_api_gallery_solutions_titration_plot_02_gran_plot.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_gran_plot.ipynb <plot_02_gran_plot.ipynb>`

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

      :download:`Download Python source code: plot_02_gran_plot.py <plot_02_gran_plot.py>`

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

      :download:`Download zipped: plot_02_gran_plot.zip <plot_02_gran_plot.zip>`


.. only:: html

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

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