.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/kinetics/rate_laws/plot_02_wilhelmy_sucrose_inversion.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_kinetics_rate_laws_plot_02_wilhelmy_sucrose_inversion.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_kinetics_rate_laws_plot_02_wilhelmy_sucrose_inversion.py:


Wilhelmy's sucrose inversion: the first measured first-order rate law
========================================================================

Wilhelmy (1850) followed the acid-catalyzed inversion of cane sugar by
polarimetry: the optical rotation :math:`\alpha(t)` drifts from the
dextrorotatory sucrose value towards the (levorotatory) value of the
glucose + fructose mixture. Because :math:`\alpha` is linear in the
concentrations, :math:`\alpha(t) - \alpha_\infty` is proportional to the
sucrose still present, and a straight line of
:math:`\ln[(\alpha - \alpha_\infty)/(\alpha_0 - \alpha_\infty)]` against
:math:`t` identifies :math:`-d[\text{sugar}]/dt = k[\text{sugar}]`.

The polarimeter readings below are synthetic (an illustrative rate
constant and rotation angles, plus measurement noise), generated from
:class:`~chemistrykit.kinetics.systems.rate_laws.FirstOrder`.

.. GENERATED FROM PYTHON SOURCE LINES 20-38

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.kinetics.systems.rate_laws import FirstOrder

    k_true = 0.012  # min^-1 (illustrative)
    C0 = 1.0  # relative sucrose concentration
    sucrose = FirstOrder(k=k_true, C0=C0)

    # Rotation of the solution: +alpha_0 for pure sucrose, alpha_inf once fully
    # inverted (fructose's strong levorotation wins over glucose).
    alpha_0, alpha_inf = 13.0, -4.0  # degrees (illustrative)

    rng = np.random.default_rng(1850)
    t = np.linspace(0.0, 240.0, 17)  # minutes
    fraction_left = sucrose.concentration(t) / C0
    alpha = alpha_inf + (alpha_0 - alpha_inf) * fraction_left + rng.normal(0.0, 0.08, t.shape)








.. GENERATED FROM PYTHON SOURCE LINES 39-40

Wilhelmy's test: the log of the remaining rotation is linear in time.

.. GENERATED FROM PYTHON SOURCE LINES 40-66

.. code-block:: Python


    y = np.log((alpha - alpha_inf) / (alpha_0 - alpha_inf))
    slope, intercept = np.polyfit(t, y, 1)
    k_fit = -slope
    print(f"true k = {k_true:.4f} /min, fitted k = {k_fit:.4f} /min")
    print(f"half-life ln2/k = {np.log(2) / k_fit:.1f} min")

    fig, axes = plt.subplots(1, 2, figsize=(11, 4.5))
    t_fine = np.linspace(0.0, 240.0, 300)
    axes[0].plot(t, alpha, "o", color="steelblue", label="polarimeter reading")
    axes[0].plot(t_fine, alpha_inf + (alpha_0 - alpha_inf) * np.exp(-k_fit * t_fine), color="darkorange", label="first-order fit")
    axes[0].axhline(0.0, color="gray", linewidth=0.8)
    axes[0].axhline(alpha_inf, color="gray", linestyle="--", linewidth=0.8, label=r"$\alpha_\infty$ (invert sugar)")
    axes[0].set_xlabel("t (min)")
    axes[0].set_ylabel(r"optical rotation $\alpha$ (deg)")
    axes[0].set_title("Rotation falls through zero as sucrose inverts")
    axes[0].legend()

    axes[1].plot(t, y, "o", color="steelblue")
    axes[1].plot(t_fine, intercept + slope * t_fine, color="darkorange", label=f"slope = -k = {slope:.4f} /min")
    axes[1].set_xlabel("t (min)")
    axes[1].set_ylabel(r"$\ln[(\alpha-\alpha_\infty)/(\alpha_0-\alpha_\infty)]$")
    axes[1].set_title("A straight line: first-order kinetics")
    axes[1].legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/kinetics/rate_laws/images/sphx_glr_plot_02_wilhelmy_sucrose_inversion_001.png
   :alt: Rotation falls through zero as sucrose inverts, A straight line: first-order kinetics
   :srcset: /api/gallery/kinetics/rate_laws/images/sphx_glr_plot_02_wilhelmy_sucrose_inversion_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    true k = 0.0120 /min, fitted k = 0.0120 /min
    half-life ln2/k = 58.0 min




.. GENERATED FROM PYTHON SOURCE LINES 67-69

The half-life does not depend on how much sugar is left: every
:math:`t_{1/2}` interval halves the remaining sucrose again.

.. GENERATED FROM PYTHON SOURCE LINES 69-75

.. code-block:: Python


    for n in range(4):
        tn = n * sucrose.half_life()
        print(f"after {n} half-lives ({tn:6.1f} min): [sugar]/[sugar]_0 = {sucrose.concentration(tn) / C0:.4f}")

    plt.show()




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

 .. code-block:: none

    after 0 half-lives (   0.0 min): [sugar]/[sugar]_0 = 1.0000
    after 1 half-lives (  57.8 min): [sugar]/[sugar]_0 = 0.5000
    after 2 half-lives ( 115.5 min): [sugar]/[sugar]_0 = 0.2500
    after 3 half-lives ( 173.3 min): [sugar]/[sugar]_0 = 0.1250





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

   **Total running time of the script:** (0 minutes 0.073 seconds)


.. _sphx_glr_download_api_gallery_kinetics_rate_laws_plot_02_wilhelmy_sucrose_inversion.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_wilhelmy_sucrose_inversion.ipynb <plot_02_wilhelmy_sucrose_inversion.ipynb>`

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

      :download:`Download Python source code: plot_02_wilhelmy_sucrose_inversion.py <plot_02_wilhelmy_sucrose_inversion.py>`

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

      :download:`Download zipped: plot_02_wilhelmy_sucrose_inversion.zip <plot_02_wilhelmy_sucrose_inversion.zip>`


.. only:: html

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

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