.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/surface/bet/plot_01_bet_isotherm.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_surface_bet_plot_01_bet_isotherm.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_surface_bet_plot_01_bet_isotherm.py:


The BET multilayer adsorption isotherm
=========================================

:class:`~chemistrykit.surface.systems.bet.BETIsotherm` generalizes
Langmuir to multilayer adsorption. As the saturation vapor pressure
:math:`P_0 \to \infty` at fixed :math:`K=C/P_0`, multilayer condensation
is suppressed and BET reduces exactly to the Langmuir isotherm -- a
reduction verified here numerically, not just asserted.

.. GENERATED FROM PYTHON SOURCE LINES 13-23

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.surface.systems.bet import BETIsotherm, fit_bet
    from chemistrykit.surface.systems.langmuir import langmuir_coverage
    from chemistrykit.surface.visualizers.surface_plots import plot_isotherm, plot_linearization

    Vm_true, C_true, P0 = 5.0, 80.0, 10.0
    iso = BETIsotherm(Vm=Vm_true, C=C_true, P0=P0)








.. GENERATED FROM PYTHON SOURCE LINES 24-26

Fit (Vm, C) from synthetic data restricted to the usual reliable BET
range, 0.05 < P/P0 < 0.35.

.. GENERATED FROM PYTHON SOURCE LINES 26-35

.. code-block:: Python

    rng = np.random.default_rng(2)
    P_data = np.array([0.5, 1.0, 1.5, 2.0, 2.5, 3.0])
    V_data = iso.loading(P_data) * (1.0 + rng.normal(scale=0.005, size=P_data.shape))

    fit = fit_bet(P_data, V_data, P0=P0)
    print(f"True (Vm, C)   = ({Vm_true}, {C_true})")
    print(f"Fitted (Vm, C) = ({fit.Vm:.4f}, {fit.C:.4f})")
    print(f"R^2 of linearized fit: {fit.r_squared:.6f}")





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

 .. code-block:: none

    True (Vm, C)   = (5.0, 80.0)
    Fitted (Vm, C) = (5.0341, 72.2212)
    R^2 of linearized fit: 0.999704




.. GENERATED FROM PYTHON SOURCE LINES 36-39

The Langmuir-limit check: with P0 pushed far above the working pressure
range (multilayer condensation never sets in), the BET curve should
collapse onto the Langmuir isotherm with the same K=C/P0.

.. GENERATED FROM PYTHON SOURCE LINES 39-49

.. code-block:: Python

    K = 2.0
    P0_large = 1.0e7
    C_large = K * P0_large
    Vm = 1.0
    P_scan = np.linspace(0.01, 5.0, 200)
    V_bet = BETIsotherm(Vm=Vm, C=C_large, P0=P0_large).loading(P_scan)
    theta_langmuir = langmuir_coverage(K, P_scan)
    max_relative_error = np.max(np.abs(V_bet - Vm * theta_langmuir) / (Vm * theta_langmuir))
    print(f"\nMax relative deviation of BET from Langmuir (P0/K=5e6): {max_relative_error:.2e}")





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

 .. code-block:: none


    Max relative deviation of BET from Langmuir (P0/K=5e6): 5.45e-07




.. GENERATED FROM PYTHON SOURCE LINES 50-67

.. code-block:: Python

    fig, axes = plt.subplots(1, 3, figsize=(14, 4))
    plot_isotherm(iso, P_max=8.0, P_data=P_data, q_data=V_data, ax=axes[0])

    x_lin = P_data / P0
    y_lin = x_lin / (V_data * (1.0 - x_lin))
    slope = (fit.C - 1.0) / (fit.Vm * fit.C)
    intercept = 1.0 / (fit.Vm * fit.C)
    plot_linearization(x_lin, y_lin, fit=(slope, intercept), ax=axes[1], xlabel="x = P/P0", ylabel="x / [V(1-x)]")

    axes[2].plot(P_scan, V_bet, label="BET (P0=1e7)")
    axes[2].plot(P_scan, Vm * theta_langmuir, "k--", label="Langmuir")
    axes[2].set_xlabel("P")
    axes[2].set_ylabel("loading")
    axes[2].set_title("BET -> Langmuir limit")
    axes[2].legend()
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/surface/bet/images/sphx_glr_plot_01_bet_isotherm_001.png
   :alt: BETIsotherm isotherm, Isotherm linearization, BET -> Langmuir limit
   :srcset: /api/gallery/surface/bet/images/sphx_glr_plot_01_bet_isotherm_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.094 seconds)


.. _sphx_glr_download_api_gallery_surface_bet_plot_01_bet_isotherm.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_01_bet_isotherm.ipynb <plot_01_bet_isotherm.ipynb>`

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

      :download:`Download Python source code: plot_01_bet_isotherm.py <plot_01_bet_isotherm.py>`

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

      :download:`Download zipped: plot_01_bet_isotherm.zip <plot_01_bet_isotherm.zip>`


.. only:: html

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

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