.. _sphx_glr_api_gallery_statmech_lattice_gas:

Lattice-gas adsorption
-----------------------

The lattice-gas derivation of the Langmuir adsorption isotherm, and
Boltzmann's :math:`S=k_B\ln W` counted directly on a lattice.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

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    <div class="sphx-glr-thumbcontainer" tooltip="Irving Langmuir&#x27;s 1918 picture of adsorption: a fixed number of independent, non-interacting surface sites, each either empty or holding exactly one molecule with binding energy \epsilon. LatticeGasAdsorption treats those sites in equilibrium with an ideal-gas reservoir and recovers exactly the Langmuir isotherm \theta=P/(P+P_0) -- linear in P at low pressure, saturating at a full monolayer at high pressure -- shown here for two binding strengths.">

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  .. image:: /api/gallery/statmech/lattice_gas/images/thumb/sphx_glr_plot_01_langmuir_adsorption_thumb.png
    :alt:

  :doc:`/api/gallery/statmech/lattice_gas/plot_01_langmuir_adsorption`

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      <div class="sphx-glr-thumbnail-title">Langmuir's adsorption isotherm from a lattice-gas model</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Boltzmann&#x27;s 1877 insight, written by Planck as S=k_B\ln W: entropy counts the microscopic arrangements W compatible with a macroscopic state. For N indistinguishable molecules on M lattice sites, W=\binom{M}{N}, and canonical_entropy evaluates k_B\ln W exactly via ln_binomial. The entropy is largest at half filling, where the number of arrangements peaks, and per site it converges to the Stirling-approximation mixing entropy -k_B[\theta\ln\theta+(1-\theta)\ln(1-\theta)] as M grows.">

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  .. image:: /api/gallery/statmech/lattice_gas/images/thumb/sphx_glr_plot_02_boltzmann_entropy_counting_thumb.png
    :alt:

  :doc:`/api/gallery/statmech/lattice_gas/plot_02_boltzmann_entropy_counting`

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      <div class="sphx-glr-thumbnail-title">Boltzmann's S = k ln W by counting lattice arrangements</div>
    </div>


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    </div>


.. toctree::
   :hidden:

   /api/gallery/statmech/lattice_gas/plot_01_langmuir_adsorption
   /api/gallery/statmech/lattice_gas/plot_02_boltzmann_entropy_counting