.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/thermo/equilibrium/plot_02_gibbs_energy_minimization.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_thermo_equilibrium_plot_02_gibbs_energy_minimization.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_thermo_equilibrium_plot_02_gibbs_energy_minimization.py:


Gibbs's equilibrium criterion: minimum total Gibbs energy
=========================================================

Gibbs showed that at fixed temperature and pressure a system reaches
equilibrium at the minimum of its total Gibbs energy
:math:`G = \sum_i n_i\mu_i`. For :math:`N_2O_4 \rightleftharpoons 2NO_2`
this example plots :math:`G(\xi)` from
:func:`~chemistrykit.thermo.systems.equilibrium.gibbs_energy_of_mixture`
at several temperatures and marks the composition returned by
:func:`~chemistrykit.thermo.systems.equilibrium.solve_equilibrium_composition`,
which sits at each minimum. The slope :math:`dG/d\xi = \sum_i\nu_i\mu_i`
(the reaction Gibbs energy) crosses zero there.

.. GENERATED FROM PYTHON SOURCE LINES 17-47



.. image-sg:: /api/gallery/thermo/equilibrium/images/sphx_glr_plot_02_gibbs_energy_minimization_001.png
   :alt: Total Gibbs energy; dots = solver's equilibrium, Reaction Gibbs energy vanishes at the minimum
   :srcset: /api/gallery/thermo/equilibrium/images/sphx_glr_plot_02_gibbs_energy_minimization_001.png
   :class: sphx-glr-single-img





.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.thermo.systems.equilibrium import gibbs_energy_of_mixture, solve_equilibrium_composition

    dH, dS = 57_200.0, 175.8  # N2O4 -> 2 NO2, J/mol and J/(mol K)
    xi = np.linspace(0.002, 0.998, 400)

    fig, axes = plt.subplots(1, 2, figsize=(12, 4.8))
    for T, color in [(280.0, "steelblue"), (320.0, "seagreen"), (360.0, "crimson")]:
        gf = [0.0, (dH - T * dS) / 2.0]
        G = np.array([gibbs_energy_of_mixture([1 - x, 2 * x], gf, T) for x in xi])
        res = solve_equilibrium_composition(("N2O4", "NO2"), [[-1.0], [2.0]], [1.0, 0.0], gf, T)
        G_eq = gibbs_energy_of_mixture(res.n, gf, T)
        axes[0].plot(xi, (G - G[0]) / 1000.0, color=color, label=f"T = {T:.0f} K")
        axes[0].plot(res.extents[0], (G_eq - G[0]) / 1000.0, "o", color=color)
        axes[1].plot(xi, np.gradient(G, xi) / 1000.0, color=color)
        axes[1].axvline(res.extents[0], color=color, linestyle=":")
    axes[0].set_xlabel(r"extent of reaction $\xi$ (mol)")
    axes[0].set_ylabel(r"$G(\xi) - G(0)$ (kJ)")
    axes[0].set_title("Total Gibbs energy; dots = solver's equilibrium")
    axes[0].legend()
    axes[1].axhline(0.0, color="gray", linewidth=0.8)
    axes[1].set_ylim(-30, 30)
    axes[1].set_xlabel(r"extent of reaction $\xi$ (mol)")
    axes[1].set_ylabel(r"$dG/d\xi = \sum_i \nu_i \mu_i$ (kJ/mol)")
    axes[1].set_title("Reaction Gibbs energy vanishes at the minimum")
    fig.tight_layout()

    plt.show()


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

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


.. _sphx_glr_download_api_gallery_thermo_equilibrium_plot_02_gibbs_energy_minimization.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_gibbs_energy_minimization.ipynb <plot_02_gibbs_energy_minimization.ipynb>`

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

      :download:`Download Python source code: plot_02_gibbs_energy_minimization.py <plot_02_gibbs_energy_minimization.py>`

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

      :download:`Download zipped: plot_02_gibbs_energy_minimization.zip <plot_02_gibbs_energy_minimization.zip>`


.. only:: html

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

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