.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/analytical/titration/plot_01_nernst_redox_titration.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_analytical_titration_plot_01_nernst_redox_titration.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_analytical_titration_plot_01_nernst_redox_titration.py:


The Nernst equation in a potentiometric redox titration
==========================================================

:class:`~chemistrykit.analytical.RedoxTitration` computes the electrode
potential during a redox titration from the Nernst equation,
:math:`E=E^\circ-\frac{RT}{nF}\ln Q`: before equivalence the analyte
couple sets `E`, after it the titrant couple does. At half-equivalence
:math:`E=E^\circ_1` exactly, at twice the equivalence volume
:math:`E=E^\circ_2`, and the endpoint is read, as in the laboratory, at
the steepest rise of `E`.

.. GENERATED FROM PYTHON SOURCE LINES 15-20

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.analytical import RedoxTitration








.. GENERATED FROM PYTHON SOURCE LINES 21-22

Fe2+ titrated by Ce4+ (both one-electron couples) -- symmetric curve:

.. GENERATED FROM PYTHON SOURCE LINES 22-32

.. code-block:: Python

    fe_ce = RedoxTitration(E1_standard=0.771, n1=1, E2_standard=1.72, n2=1, C_analyte=0.100, V_analyte=0.050, C_titrant=0.100)
    V = np.linspace(1e-6, 0.099, 4000)
    curve = fe_ce.curve(V)
    V_eq = fe_ce.equivalence_volume()
    E_half, E_eq, E_double = fe_ce.response_at(np.array([V_eq / 2, V_eq, 2 * V_eq]))
    print(f"Equivalence volume: exact {V_eq * 1000:.3f} mL, steepest-rise estimate {fe_ce.find_equivalence_point(V) * 1000:.3f} mL")
    print(f"E at V_eq/2  = {E_half:.4f} V  (E1 = 0.771 V)")
    print(f"E at V_eq    = {E_eq:.4f} V  ((E1 + E2)/2 = {(0.771 + 1.72) / 2:.4f} V)")
    print(f"E at 2 V_eq  = {E_double:.4f} V  (E2 = 1.72 V)")





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

 .. code-block:: none

    Equivalence volume: exact 50.000 mL, steepest-rise estimate 50.008 mL
    E at V_eq/2  = 0.7710 V  (E1 = 0.771 V)
    E at V_eq    = 1.2455 V  ((E1 + E2)/2 = 1.2455 V)
    E at 2 V_eq  = 1.7200 V  (E2 = 1.72 V)




.. GENERATED FROM PYTHON SOURCE LINES 33-35

The Nernst slope RT/(nF)*ln(10) = 59.16 mV per decade at 25 C: in the
buffer region, every tenfold change in [Fe3+]/[Fe2+] moves E by 59 mV.

.. GENERATED FROM PYTHON SOURCE LINES 35-40

.. code-block:: Python

    V_10 = V_eq * 10 / 11  # [Fe3+]/[Fe2+] = 10
    V_1 = V_eq / 2  # ratio = 1
    dE = fe_ce.response_at(np.array([V_10]))[0] - fe_ce.response_at(np.array([V_1]))[0]
    print(f"\nE change for a tenfold ratio change: {dE * 1000:.2f} mV")





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

 .. code-block:: none


    E change for a tenfold ratio change: 59.16 mV




.. GENERATED FROM PYTHON SOURCE LINES 41-43

An unequal-electron pair (a 2-electron analyte, e.g. Sn2+ -> Sn4+,
titrated by 1-electron Ce4+) is asymmetric: E_eq = (2 E1 + E2)/3.

.. GENERATED FROM PYTHON SOURCE LINES 43-47

.. code-block:: Python

    sn_ce = RedoxTitration(E1_standard=0.139, n1=2, E2_standard=1.72, n2=1, C_analyte=0.050, V_analyte=0.050, C_titrant=0.100)
    curve_sn = sn_ce.curve(V)
    print(f"\nSn2+/Ce4+ E_eq = {sn_ce.response_at(np.array([sn_ce.equivalence_volume()]))[0]:.4f} V (weighted mean {(2 * 0.139 + 1.72) / 3:.4f} V)")





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

 .. code-block:: none


    Sn2+/Ce4+ E_eq = 0.6660 V (weighted mean 0.6660 V)




.. GENERATED FROM PYTHON SOURCE LINES 48-59

.. code-block:: Python

    fig, ax = plt.subplots(figsize=(7, 4.5))
    ax.plot(curve.V * 1000, curve.response, label="Fe$^{2+}$ + Ce$^{4+}$ (n$_1$=n$_2$=1)")
    ax.plot(curve_sn.V * 1000, curve_sn.response, label="Sn$^{2+}$ + Ce$^{4+}$ (n$_1$=2, n$_2$=1)")
    ax.axvline(V_eq * 1000, color="gray", linestyle="--", linewidth=0.8)
    ax.plot([V_eq / 2 * 1000, 2 * V_eq * 1000], [E_half, E_double], "ko", label=r"$E=E^\circ_1$ at $V_{eq}/2$, $E=E^\circ_2$ at $2V_{eq}$")
    ax.set_xlabel("titrant volume (mL)")
    ax.set_ylabel("E (V)")
    ax.set_title("Potentiometric redox titrations (Nernst equation)")
    ax.legend(fontsize=8)
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/analytical/titration/images/sphx_glr_plot_01_nernst_redox_titration_001.png
   :alt: Potentiometric redox titrations (Nernst equation)
   :srcset: /api/gallery/analytical/titration/images/sphx_glr_plot_01_nernst_redox_titration_001.png
   :class: sphx-glr-single-img






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   **Total running time of the script:** (0 minutes 0.098 seconds)


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