.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/surface/dubinin/plot_01_polanyi_dubinin_characteristic_curve.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_dubinin_plot_01_polanyi_dubinin_characteristic_curve.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_surface_dubinin_plot_01_polanyi_dubinin_characteristic_curve.py:


Polanyi's potential theory and the Dubinin-Radushkevich characteristic curve
==============================================================================

Polanyi (1914) described adsorption by the potential
:math:`A = RT\ln(P_0/P)`
(:func:`~chemistrykit.surface.systems.dubinin.polanyi_potential`) and
claimed that the filled pore volume depends on :math:`A` alone, through
one characteristic curve that does not change with temperature. Dubinin
and Radushkevich (1947) gave that curve for microporous carbons as
:math:`W = W_0\exp[-(A/E)^2]`
(:func:`~chemistrykit.surface.systems.dubinin.dubinin_radushkevich_loading`).
Below, isotherms measured at three temperatures look different against
:math:`P/P_0` but fall onto a single curve when replotted against
:math:`A`. A Dubinin-Radushkevich fit then recovers the pore volume
:math:`W_0` and the characteristic energy :math:`E`
(:func:`~chemistrykit.surface.systems.dubinin.fit_dubinin_radushkevich`).

.. GENERATED FROM PYTHON SOURCE LINES 21-45

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.constants import R
    from chemistrykit.surface.systems.dubinin import DubininRadushkevichIsotherm, fit_dubinin_radushkevich, polanyi_potential

    W0, E = 0.45, 12.0e3  # cm^3/g, J/mol
    dH_vap, T_ref, P0_ref = 30.0e3, 300.0, 10.0e3  # simple Clausius-Clapeyron vapor pressure, Pa


    def P0_of(T):
        return P0_ref * np.exp(-dH_vap / R * (1.0 / T - 1.0 / T_ref))


    temperatures = [273.0, 300.0, 330.0]
    x = np.logspace(-6, -0.01, 200)  # P/P0
    rng = np.random.default_rng(11)
    data = {}
    for T in temperatures:
        iso = DubininRadushkevichIsotherm(W0=W0, E=E, P0=P0_of(T), T=T)
        P = x * iso.P0
        W = iso.loading(P) * (1.0 + rng.normal(scale=0.005, size=P.shape))
        data[T] = (P, W, polanyi_potential(P, iso.P0, T))








.. GENERATED FROM PYTHON SOURCE LINES 46-52

.. code-block:: Python

    P, W, _ = data[300.0]
    mask = (P / P0_of(300.0)) < 0.2
    fit = fit_dubinin_radushkevich(P[mask], W[mask], P0=P0_of(300.0), T=300.0)
    print(f"True   (W0, E) = ({W0}, {E / 1e3:.1f} kJ/mol)")
    print(f"Fitted (W0, E) = ({fit.W0:.4f}, {fit.E / 1e3:.2f} kJ/mol), R^2 = {fit.r_squared:.5f}")





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

 .. code-block:: none

    True   (W0, E) = (0.45, 12.0 kJ/mol)
    Fitted (W0, E) = (0.4502, 12.00 kJ/mol), R^2 = 1.00000




.. GENERATED FROM PYTHON SOURCE LINES 53-74

.. code-block:: Python

    fig, axes = plt.subplots(1, 3, figsize=(15, 4))
    for T, (P, W, A) in data.items():
        axes[0].semilogx(P / P0_of(T), W, label=f"T = {T:.0f} K")
        axes[1].plot(A / 1e3, W, ".", markersize=3, label=f"T = {T:.0f} K")
        axes[2].plot((A / 1e3) ** 2, np.log(W), ".", markersize=3)
    axes[0].set_xlabel("P / P$_0$")
    axes[0].set_ylabel("W (cm$^3$/g)")
    axes[0].set_title("Isotherms at three temperatures")
    axes[0].legend()
    axes[1].set_xlabel("adsorption potential A (kJ/mol)")
    axes[1].set_ylabel("W (cm$^3$/g)")
    axes[1].set_title("Polanyi: one characteristic curve")
    axes[1].legend()
    A2 = np.linspace(0, 1600, 50)
    axes[2].plot(A2, np.log(fit.W0) - A2 / (fit.E / 1e3) ** 2, "k--", label="DR fit")
    axes[2].set_xlabel("A$^2$ (kJ/mol)$^2$")
    axes[2].set_ylabel("ln W")
    axes[2].set_title("Dubinin-Radushkevich linearization")
    axes[2].legend()
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/surface/dubinin/images/sphx_glr_plot_01_polanyi_dubinin_characteristic_curve_001.png
   :alt: Isotherms at three temperatures, Polanyi: one characteristic curve, Dubinin-Radushkevich linearization
   :srcset: /api/gallery/surface/dubinin/images/sphx_glr_plot_01_polanyi_dubinin_characteristic_curve_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.132 seconds)


.. _sphx_glr_download_api_gallery_surface_dubinin_plot_01_polanyi_dubinin_characteristic_curve.py:

.. only:: html

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

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