.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/plot_07_arrhenius.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_07_arrhenius.py>`
        to download the full example code.

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

.. _sphx_glr_auto_examples_plot_07_arrhenius.py:


Arrhenius law with your own function
====================================

Any Python function can be a model. Here the rate constant of a reaction is measured at
several temperatures and described by the Arrhenius law k = A · exp(−Eₐ / RT).

Your own functions have no built-in heuristic for starting values. Default values in the
signature are used instead; they only need to be of the right order of magnitude.

.. GENERATED FROM PYTHON SOURCE LINES 13-34

.. code-block:: Python

    import numpy as np

    import fitwise as fw

    R = 8.314  # J/(mol K)


    def arrhenius(T, A=1e10, Ea=50e3):
        """Rate constant in 1/s; T in K, Ea in J/mol."""
        return A * np.exp(-Ea / (R * T))


    rng = np.random.default_rng(7)
    T = np.linspace(290, 360, 8)
    k_true = arrhenius(T, A=4.2e12, Ea=78e3)
    k = k_true * (1 + rng.normal(0, 0.03, T.size))  # 3 % relative uncertainty
    dk = 0.03 * k

    result = fw.fit(arrhenius, T, k, yerr=dk)
    print(result)





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

 .. code-block:: none

    Model:      arrhenius(x; A, Ea)
    Method:     Trust Region Reflective (scipy least_squares) (auto: nonlinear)
    Converged:  yes (60 evals)
    Data:       8 points, yerr given (absolute)
    Start:      A, Ea: signature default

    Parameter      Value   ± Std.Err     Rel.   Bounds
    A           4.41e+12    0.67e+12   15.2 %   –
    Ea             78140         410    0.5 %   –

    χ²/ndf = 0.607 (6 dof, p = 0.73)    R² = 0.99944    AIC = -2.3
    Correlations > 0.5:  A–Ea (+1.00)
    Warnings:   none




.. GENERATED FROM PYTHON SOURCE LINES 35-42

The parameters differ by 8 orders of magnitude. fitwise normalizes them internally, so the
solver is not affected. A and Eₐ are strongly correlated (see the ``Correlations`` line of
the report), which is typical for the Arrhenius law: a larger Eₐ can be compensated by a
larger A. Use the covariance when you compute something from both.

The classic Arrhenius plot shows k on a log scale over 1/T, where the law is a straight
line. ``result.eval`` evaluates the fitted model at any temperature:

.. GENERATED FROM PYTHON SOURCE LINES 42-53

.. code-block:: Python

    import matplotlib.pyplot as plt

    T_fine = np.linspace(285, 365, 200)
    fig, ax = plt.subplots()
    ax.errorbar(1000 / T, k, yerr=dk, fmt="o", label="data")
    ax.plot(1000 / T_fine, result.eval(T_fine), label="fit")
    ax.set_yscale("log")
    ax.set_xlabel("1000 / T (1/K)")
    ax.set_ylabel("k (1/s)")
    ax.legend()




.. image-sg:: /auto_examples/images/sphx_glr_plot_07_arrhenius_001.png
   :alt: plot 07 arrhenius
   :srcset: /auto_examples/images/sphx_glr_plot_07_arrhenius_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x7fb59ef71100>



.. GENERATED FROM PYTHON SOURCE LINES 54-56

.. code-block:: Python

    Ea = result.params["Ea"]
    print(f"Ea = {Ea.value / 1e3:.1f} ± {Ea.stderr / 1e3:.1f} kJ/mol")




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

 .. code-block:: none

    Ea = 78.1 ± 0.4 kJ/mol





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

   **Total running time of the script:** (0 minutes 0.092 seconds)


.. _sphx_glr_download_auto_examples_plot_07_arrhenius.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_07_arrhenius.ipynb <plot_07_arrhenius.ipynb>`

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

      :download:`Download Python source code: plot_07_arrhenius.py <plot_07_arrhenius.py>`

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

      :download:`Download zipped: plot_07_arrhenius.zip <plot_07_arrhenius.zip>`


.. only:: html

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

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