.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/plot_06_titration.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_auto_examples_plot_06_titration.py>`
        to download the full example code.

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

.. _sphx_glr_auto_examples_plot_06_titration.py:


Titration curve
===============

The pH during the titration of a weak acid with sodium hydroxide rises in a step around the
equivalence point. A logistic step on top of a constant describes the curve well enough to
locate the equivalence point precisely.

.. GENERATED FROM PYTHON SOURCE LINES 11-22

.. code-block:: Python

    import numpy as np

    import fitwise as fw

    rng = np.random.default_rng(7)
    volume = np.linspace(0, 20, 41)  # mL
    ph = 4.0 + 6.5 / (1 + np.exp(-(volume - 12.3) / 0.45)) + rng.normal(0, 0.05, volume.size)

    result = fw.fit(fw.models.sigmoid + fw.models.constant, volume, ph, yerr=0.05)
    print(result)





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

 .. code-block:: none

    Model:      sigmoid(x; amp, cen, width) + constant(x; c)
    Method:     Trust Region Reflective (scipy least_squares) (auto: nonlinear)
    Converged:  yes (25 evals)
    Data:       41 points, yerr given (absolute)
    Start:      amp, cen, width: step heuristic · c: edge baseline

    Parameter     Value   ± Std.Err    Rel.   Bounds
    amp           6.509       0.018   0.3 %   –
    cen         12.3037      0.0098   0.1 %   –
    width        0.4602      0.0085   1.8 %   –
    c             3.978       0.011   0.3 %   –

    χ²/ndf = 0.673 (37 dof, p = 0.94)    R² = 0.99983    AIC = -12.5
    Correlations > 0.5:  amp–c (-0.62)
    Warnings:   none




.. GENERATED FROM PYTHON SOURCE LINES 23-24

The equivalence point is the midpoint of the step:

.. GENERATED FROM PYTHON SOURCE LINES 24-27

.. code-block:: Python

    cen = result.params["cen"]
    print(f"V_eq = {cen.value:.3f} ± {cen.stderr:.3f} mL")





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

 .. code-block:: none

    V_eq = 12.304 ± 0.010 mL




.. GENERATED FROM PYTHON SOURCE LINES 28-31

.. code-block:: Python

    fig, axes = result.plot()
    axes[0].set_ylabel("pH")
    axes[1].set_xlabel("NaOH added (mL)")



.. image-sg:: /auto_examples/images/sphx_glr_plot_06_titration_001.png
   :alt: plot 06 titration
   :srcset: /auto_examples/images/sphx_glr_plot_06_titration_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 18.503588541666666, 'NaOH added (mL)')




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

   **Total running time of the script:** (0 minutes 0.135 seconds)


.. _sphx_glr_download_auto_examples_plot_06_titration.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_06_titration.ipynb <plot_06_titration.ipynb>`

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

      :download:`Download Python source code: plot_06_titration.py <plot_06_titration.py>`

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

      :download:`Download zipped: plot_06_titration.zip <plot_06_titration.zip>`


.. only:: html

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

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