:orphan:

Examples
========

This gallery walks through every public feature of
``chemistrykit.surface``: Langmuir, Freundlich, and BET adsorption
isotherms with their standard linearizations for fitting parameters from
data; Langmuir-Hinshelwood single- and dual-site surface-reaction
kinetics; and a turnover-frequency/rate-enhancement catalysis model
built on ``chemistrykit.kinetics``'s Arrhenius equation; the Gibbs
adsorption equation; Polanyi/Dubinin-Radushkevich and Temkin isotherms;
Eley-Rideal kinetics; and temperature-programmed desorption.

Each script in this gallery is self-contained and can be run directly
with ``python examples/surface/<section>/<script>.py``. Every script
also carries an RST module docstring as its title/description and uses
``# %%`` markers to split narrative text from code, which is exactly what
Sphinx-Gallery renders into the pages below -- the script *is* the source
of truth for what you see, not a copy of it.

Sections
--------

- **langmuir** -- the Langmuir monolayer isotherm, its exact
  half-saturation point, and its linearization for fitting.
- **freundlich** -- the empirical Freundlich power-law isotherm and its
  linearization.
- **bet** -- the BET multilayer isotherm, and its numerical reduction to
  Langmuir in the appropriate limit.
- **langmuir_hinshelwood** -- single- and dual-site surface-reaction
  kinetics built on the Langmuir coverage.
- **catalysis** -- one example per catalysis milestone: Doebereiner and
  Berzelius, Arrhenius and Ostwald, Sabatier's hydrogenation, Constable's
  compensation effect, Balandin's volcano curve, and Boudart's turnover
  frequency.
- **gibbs_adsorption** -- surface excess from surface tension via the
  Gibbs adsorption equation.
- **dubinin** -- Polanyi's adsorption potential and the
  Dubinin-Radushkevich characteristic curve.
- **temkin** -- the logarithmic Temkin isotherm from a spread of site
  energies.
- **eley_rideal** -- Eley-Rideal kinetics contrasted with dual-site
  Langmuir-Hinshelwood kinetics.
- **tpd** -- temperature-programmed desorption and Redhead's analysis.


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BET isotherm
--------------

The BET multilayer adsorption isotherm, its linearization for fitting
(Vm, C), and its numerical reduction to the Langmuir isotherm when
multilayer condensation is suppressed (P0 much larger than the working
pressure range).


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    <div class="sphx-glr-thumbcontainer" tooltip="BETIsotherm generalizes Langmuir to multilayer adsorption. As the saturation vapor pressure P_0 \to \infty at fixed K=C/P_0, multilayer condensation is suppressed and BET reduces exactly to the Langmuir isotherm -- a reduction verified here numerically, not just asserted.">

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  .. image:: /api/gallery/surface/bet/images/thumb/sphx_glr_plot_01_bet_isotherm_thumb.png
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  :doc:`/api/gallery/surface/bet/plot_01_bet_isotherm`

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      <div class="sphx-glr-thumbnail-title">The BET multilayer adsorption isotherm</div>
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Catalysis
-----------

One example per milestone in the history of catalysis: Doebereiner's
platinum and Berzelius's naming of catalysis, Arrhenius and Ostwald's
kinetic definition of a catalyst, Sabatier's catalytic hydrogenation,
Constable's compensation effect, Balandin's volcano curve, and Boudart's
turnover frequency.


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    <div class="sphx-glr-thumbcontainer" tooltip="In 1823 Doebereiner found that spongy platinum ignites a jet of hydrogen in air at room temperature and comes out of the reaction unchanged; in 1835 Berzelius named the phenomenon catalysis. This example makes both halves of that observation quantitative with illustrative (not measured) parameters for 2H_2 + O_2 \to 2H_2O. Lowering the activation energy on platinum (~chemistrykit.surface.systems.catalysis.compare_catalyzed_rate) turns a reaction that is frozen at room temperature into a fast one, and the turnover_number keeps growing while the amount of platinum stays exactly the same. The catalyst is used again and again, never used up.">

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  .. image:: /api/gallery/surface/catalysis/images/thumb/sphx_glr_plot_01_doebereiner_berzelius_catalysis_thumb.png
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  :doc:`/api/gallery/surface/catalysis/plot_01_doebereiner_berzelius_catalysis`

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      <div class="sphx-glr-thumbnail-title">Doebereiner's platinum and Berzelius's catalysis: a catalyst that is not consumed</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Arrhenius&#x27;s k = Ae^{-E_a/RT} explains why a catalyst&#x27;s lower activation energy multiplies the rate by e^{\Delta E_a/RT} (~chemistrykit.surface.systems.catalysis.compare_catalyzed_rate). Ostwald&#x27;s definition adds the key condition that a catalyst cannot move the equilibrium. It lowers the barrier for the forward and the reverse reaction by the same amount, so both rate constants grow by the same factor and K = k_f/k_r does not change. Below, the reversible reaction A \rightleftharpoons B reaches equilibrium far sooner with the catalyst, and it reaches exactly the same equilibrium.">

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  .. image:: /api/gallery/surface/catalysis/images/thumb/sphx_glr_plot_02_arrhenius_ostwald_rate_enhancement_thumb.png
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  :doc:`/api/gallery/surface/catalysis/plot_02_arrhenius_ostwald_rate_enhancement`

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      <div class="sphx-glr-thumbnail-title">Arrhenius and Ostwald: a catalyst speeds a reaction but leaves its equilibrium alone</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Sabatier and Senderens showed from 1897 that finely divided nickel hydrogenates ethylene (C_2H_4 + H_2 \to C_2H_6) at temperatures where the gas-phase reaction does not run at all. This example uses illustrative Arrhenius parameters and compare_catalyzed_rate to compute the conversion reached in a one-second contact time as the temperature rises. Over nickel the reaction lights off hundreds of kelvin earlier. A second panel shows Sabatier&#x27;s rule of thumb from screening many metals: a metal that binds the reactant too weakly holds almost none of it on the surface, and one that binds it too strongly leaves no free sites. Only a moderate binder has both.">

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  .. image:: /api/gallery/surface/catalysis/images/thumb/sphx_glr_plot_03_sabatier_hydrogenation_thumb.png
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  :doc:`/api/gallery/surface/catalysis/plot_03_sabatier_hydrogenation`

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      <div class="sphx-glr-thumbnail-title">Sabatier and Senderens: catalytic hydrogenation over finely divided nickel</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Constable (1925) found that across a series of similar catalysts, a higher activation energy E_a came with a higher pre-exponential factor A, so \ln A rose linearly with E_a:">

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  .. image:: /api/gallery/surface/catalysis/images/thumb/sphx_glr_plot_04_constable_compensation_thumb.png
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  :doc:`/api/gallery/surface/catalysis/plot_04_constable_compensation`

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      <div class="sphx-glr-thumbnail-title">Constable's compensation effect across a catalyst series</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Balandin&#x27;s multiplet theory (1929) predicted that when catalytic activity is plotted against binding strength across a series of catalysts, it rises, peaks and falls again, tracing a &quot;volcano&quot;. Bligaard, Norskov and coworkers (2004) later derived that shape from the Bronsted-Evans-Polanyi relation. This example reproduces the volcano with the standard textbook toy model (Chorkendorff &amp; Niemantsverdriet, Concepts of Modern Catalysis and Kinetics, 2nd ed., Wiley-VCH, 2007, Ch. 8). The rate needs an occupied site to hold the intermediate and an empty site for the next step, so rate \propto\theta(1-\theta), with \theta the Langmuir coverage (~chemistrykit.surface.systems.langmuir.langmuir_coverage). The adsorption constant K stands in for binding strength. A hypothetical series of catalysts sits on the two legs of the volcano, and its summit is exactly Langmuir&#x27;s half-saturation point K = 1/P. This is not a literal reproduction of Balandin&#x27;s multiplet equations.">

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  .. image:: /api/gallery/surface/catalysis/images/thumb/sphx_glr_plot_05_balandin_volcano_thumb.png
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  :doc:`/api/gallery/surface/catalysis/plot_05_balandin_volcano`

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      <div class="sphx-glr-thumbnail-title">Balandin's volcano curve: catalytic activity peaks at intermediate binding</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Boudart argued that catalysts should be compared by their rate per active site, the turnover frequency (~chemistrykit.surface.systems.catalysis.turnover_frequency), and not by their rate per gram. Below, two samples of the same supported platinum differ only in dispersion, the fraction of Pt atoms exposed at the surface. The well-dispersed sample looks ten times more active per gram, yet both have the same TOF, because each exposed site works equally hard. A third, genuinely different metal does show a different TOF. The turnover_number gives the complementary lifetime measure of total cycles per site.">

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  .. image:: /api/gallery/surface/catalysis/images/thumb/sphx_glr_plot_06_boudart_turnover_frequency_thumb.png
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  :doc:`/api/gallery/surface/catalysis/plot_06_boudart_turnover_frequency`

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      <div class="sphx-glr-thumbnail-title">Boudart's turnover frequency: comparing catalysts per active site</div>
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Polanyi potential and Dubinin-Radushkevich
--------------------------------------------

Polanyi's adsorption potential, the temperature-independent
characteristic curve, and the Dubinin-Radushkevich micropore-filling
isotherm with its linearization.


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

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  .. image:: /api/gallery/surface/dubinin/images/thumb/sphx_glr_plot_01_polanyi_dubinin_characteristic_curve_thumb.png
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  :doc:`/api/gallery/surface/dubinin/plot_01_polanyi_dubinin_characteristic_curve`

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      <div class="sphx-glr-thumbnail-title">Polanyi's potential theory and the Dubinin-Radushkevich characteristic curve</div>
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Eley-Rideal kinetics
----------------------

The Eley-Rideal mechanism, in which a gas-phase molecule reacts directly
with an adsorbed one, and how its rate law differs from the dual-site
Langmuir-Hinshelwood mechanism.


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    <div class="sphx-glr-thumbcontainer" tooltip="In the Eley-Rideal mechanism (Eley and Rideal, 1940) only reactant A adsorbs. B reacts with adsorbed A straight from the gas phase, so the rate is k\theta_AP_B (~chemistrykit.surface.systems.eley_rideal.er_rate). A and B never compete for sites, so raising P_A can only help: the rate rises and levels off, and it stays first order in P_B. In the dual-site Langmuir-Hinshelwood mechanism (~chemistrykit.surface.systems.langmuir_hinshelwood.lh_rate_dual_site), too much A pushes B off the surface and the rate turns over. That difference is the standard kinetic test for telling the two mechanisms apart.">

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  .. image:: /api/gallery/surface/eley_rideal/images/thumb/sphx_glr_plot_01_eley_rideal_mechanism_thumb.png
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  :doc:`/api/gallery/surface/eley_rideal/plot_01_eley_rideal_mechanism`

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      <div class="sphx-glr-thumbnail-title">The Eley-Rideal mechanism: a gas molecule strikes an adsorbed one</div>
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Freundlich isotherm
----------------------

The empirical Freundlich power-law adsorption isotherm, and fitting
(Kf, n) from data via the standard log q vs. log P linearization.


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    <div class="sphx-glr-thumbcontainer" tooltip="FreundlichIsotherm models the empirical power law q=K_fP^{1/n}, useful for a heterogeneous surface where Langmuir&#x27;s single-site-energy assumption breaks down. Unlike Langmuir, it has no saturation limit, so extrapolating it far beyond the fitted pressure range is unphysical. fit_freundlich recovers (Kf, n) via the standard log q vs. log P linearization.">

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  .. image:: /api/gallery/surface/freundlich/images/thumb/sphx_glr_plot_01_freundlich_isotherm_thumb.png
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  :doc:`/api/gallery/surface/freundlich/plot_01_freundlich_isotherm`

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      <div class="sphx-glr-thumbnail-title">The Freundlich adsorption isotherm</div>
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Gibbs adsorption equation
---------------------------

Measuring how much solute is adsorbed at a liquid surface from surface
tension alone, via the Gibbs adsorption equation and the Szyszkowski
equation.


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    <div class="sphx-glr-thumbcontainer" tooltip="Gibbs showed that the amount of a dilute solute held at a liquid surface, the surface excess \Gamma, follows from how the surface tension changes with concentration: \Gamma = -(1/RT)\,d\gamma/d\ln c. So a surfactant that lowers \gamma must be concentrated at the surface, and no one has to look at the surface itself. This example takes Szyszkowski-type surface-tension data (~chemistrykit.surface.systems.gibbs_adsorption.szyszkowski_surface_tension) for an aqueous fatty-acid-like solute and applies gibbs_surface_excess. It recovers a saturating Langmuir curve for \Gamma and, from the plateau, the area each adsorbed molecule occupies.">

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  :doc:`/api/gallery/surface/gibbs_adsorption/plot_01_gibbs_adsorption_equation`

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      <div class="sphx-glr-thumbnail-title">The Gibbs adsorption equation: surface excess from surface tension</div>
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Langmuir isotherm
--------------------

The Langmuir monolayer adsorption isotherm: its exact half-saturation
point at P=1/K, and fitting (K, qmax) from data via the standard
1/q-vs-1/P linearization.


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    <div class="sphx-glr-thumbcontainer" tooltip="LangmuirIsotherm models monolayer adsorption on a fixed pool of identical surface sites. Its fractional coverage \theta=KP/(1+KP) reaches exactly one-half at the half-saturation pressure P=1/K, and fit_langmuir recovers (K, qmax) from noisy loading-vs-pressure data via the standard 1/q-vs-1/P linearization.">

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  .. image:: /api/gallery/surface/langmuir/images/thumb/sphx_glr_plot_01_langmuir_isotherm_thumb.png
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  :doc:`/api/gallery/surface/langmuir/plot_01_langmuir_isotherm`

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      <div class="sphx-glr-thumbnail-title">The Langmuir adsorption isotherm</div>
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Langmuir-Hinshelwood kinetics
--------------------------------

Single-site surface-reaction kinetics (rate as a function of Langmuir
coverage) and dual-site (competitive, bimolecular) kinetics, including
the non-monotonic rate-vs-pressure behavior of the dual-site mechanism.


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    <div class="sphx-glr-thumbcontainer" tooltip="Single-site Langmuir-Hinshelwood kinetics (~chemistrykit.surface.systems.langmuir_hinshelwood.lh_rate_single_site) transitions from first order (low pressure) to zero order (surface saturated) in the reactant&#x27;s pressure. Dual-site kinetics (~chemistrykit.surface.systems.langmuir_hinshelwood.lh_rate_dual_site) -- competitive adsorption of two reactants on the same sites -- shows the characteristic non-monotonic rate-vs-pressure behavior: too much of one reactant crowds the other off the surface entirely.">

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  .. image:: /api/gallery/surface/langmuir_hinshelwood/images/thumb/sphx_glr_plot_01_lh_kinetics_thumb.png
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  :doc:`/api/gallery/surface/langmuir_hinshelwood/plot_01_lh_kinetics`

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      <div class="sphx-glr-thumbnail-title">Langmuir-Hinshelwood surface-reaction kinetics</div>
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Temkin isotherm
-----------------

The Temkin isotherm: a logarithmic isotherm arising from a heat of
adsorption that falls with coverage, and its fit from data.


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    <div class="sphx-glr-thumbcontainer" tooltip="Temkin and Pyzhev (1940), modelling ammonia synthesis on iron, dropped Langmuir&#x27;s single site energy. In their picture the heat of adsorption falls linearly as the surface fills, which is the same as a uniform spread of site energies of width fRT. Averaging the Langmuir coverage over that spread (~chemistrykit.surface.systems.temkin.uniform_energy_coverage) gives an isotherm that is logarithmic over a wide middle range of coverage, \theta \approx (1/f)\ln(K_{max}P), the Temkin form. A single Langmuir curve covers only about two decades of pressure, while the Temkin curve rises steadily over many. Loading data in that range is a straight line in \ln P, which fit_temkin fits.">

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  .. image:: /api/gallery/surface/temkin/images/thumb/sphx_glr_plot_01_temkin_isotherm_thumb.png
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  :doc:`/api/gallery/surface/temkin/plot_01_temkin_isotherm`

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      <div class="sphx-glr-thumbnail-title">The Temkin isotherm: a heat of adsorption that falls with coverage</div>
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Temperature-programmed desorption
-----------------------------------

Simulated TPD spectra from the Polanyi-Wigner equation, and Redhead's
formula for turning a peak temperature into a desorption energy.


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    <div class="sphx-glr-thumbcontainer" tooltip="In a TPD experiment a covered surface is heated at a steady rate \beta while the desorbing gas is recorded, and each binding state shows up as a peak. simulate_tpd integrates the Polanyi-Wigner rate equation to produce these spectra. Redhead (1962) turned a first-order peak temperature T_p into a desorption energy: E_d = RT_p[\ln(\nu T_p/\beta) - 3.64] (~chemistrykit.surface.systems.tpd.redhead_desorption_energy). Below, the formula is checked against the exact peak condition (~chemistrykit.surface.systems.tpd.first_order_peak_temperature). First-order peaks shift with heating rate but not with coverage, while second-order peaks move to lower temperature as the coverage rises.">

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  .. image:: /api/gallery/surface/tpd/images/thumb/sphx_glr_plot_01_redhead_tpd_thumb.png
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  :doc:`/api/gallery/surface/tpd/plot_01_redhead_tpd`

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      <div class="sphx-glr-thumbnail-title">Redhead's analysis of temperature-programmed desorption</div>
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.. toctree::
   :hidden:
   :includehidden:


   /api/gallery/surface/bet/index.rst
   /api/gallery/surface/catalysis/index.rst
   /api/gallery/surface/dubinin/index.rst
   /api/gallery/surface/eley_rideal/index.rst
   /api/gallery/surface/freundlich/index.rst
   /api/gallery/surface/gibbs_adsorption/index.rst
   /api/gallery/surface/langmuir/index.rst
   /api/gallery/surface/langmuir_hinshelwood/index.rst
   /api/gallery/surface/temkin/index.rst
   /api/gallery/surface/tpd/index.rst



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