.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/statmech/partition_functions/plot_04_sackur_tetrode_entropy.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_statmech_partition_functions_plot_04_sackur_tetrode_entropy.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_statmech_partition_functions_plot_04_sackur_tetrode_entropy.py:


The Sackur-Tetrode entropy of a monatomic gas
================================================

Sackur (1911) and Tetrode (1912) obtained the absolute entropy of an ideal
monatomic gas by dividing classical phase space into cells of size
:math:`h^3`:

.. math::

    S_m = R\left[\ln\left(\frac{k_BT}{P\Lambda^3}\right)+\frac52\right],
    \qquad \Lambda=\frac{h}{\sqrt{2\pi mk_BT}}

:func:`~chemistrykit.statmech.sackur_tetrode_entropy` evaluates it from
the atomic mass, temperature, and pressure alone. Below it is compared
with the standard molar entropies of the noble gases (Atkins & de Paula,
Table 13.1 lists argon at about 154.8 J/(mol K)), and its
:math:`\frac32R\ln m` mass dependence is shown directly.

.. GENERATED FROM PYTHON SOURCE LINES 22-40

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    import scipy.constants as sc

    from chemistrykit.constants import R
    from chemistrykit.statmech import sackur_tetrode_entropy

    # Standard molar entropies at 298.15 K and 1 bar, J/(mol K) (NIST-JANAF / CODATA key values)
    noble_gases = {"He": (4.0026, 126.15), "Ne": (20.180, 146.33), "Ar": (39.948, 154.85), "Kr": (83.798, 164.09), "Xe": (131.29, 169.69)}

    names = list(noble_gases)
    masses = np.array([noble_gases[n][0] for n in names])
    S_exp = np.array([noble_gases[n][1] for n in names])
    S_st = np.array([sackur_tetrode_entropy(mass=m * sc.atomic_mass, T=298.15, P=1.0e5) for m in masses])

    for name, s_calc, s_exp in zip(names, S_st, S_exp):
        print(f"{name:2s}: Sackur-Tetrode {s_calc:7.2f}  tabulated {s_exp:7.2f} J/(mol K)")





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

 .. code-block:: none

    He: Sackur-Tetrode  126.15  tabulated  126.15 J/(mol K)
    Ne: Sackur-Tetrode  146.33  tabulated  146.33 J/(mol K)
    Ar: Sackur-Tetrode  154.85  tabulated  154.85 J/(mol K)
    Kr: Sackur-Tetrode  164.09  tabulated  164.09 J/(mol K)
    Xe: Sackur-Tetrode  169.68  tabulated  169.69 J/(mol K)




.. GENERATED FROM PYTHON SOURCE LINES 41-64

.. code-block:: Python

    fig, axes = plt.subplots(1, 2, figsize=(11, 4.5))
    m_grid = np.logspace(0.3, 2.3, 200)
    axes[0].semilogx(m_grid, [sackur_tetrode_entropy(mass=m * sc.atomic_mass, T=298.15, P=1.0e5) for m in m_grid], color="steelblue", label="Sackur-Tetrode")
    axes[0].semilogx(masses, S_exp, "o", color="crimson", label="tabulated (298.15 K, 1 bar)")
    for name, m, s in zip(names, masses, S_exp):
        axes[0].annotate(name, (m, s), textcoords="offset points", xytext=(5, -12))
    axes[0].set_xlabel("atomic mass (u)")
    axes[0].set_ylabel(r"$S_m$ (J mol$^{-1}$ K$^{-1}$)")
    axes[0].set_title(r"Noble-gas entropies: slope $\frac{3}{2}R$ per $\ln m$")
    axes[0].legend()

    T = np.linspace(50.0, 1500.0, 200)
    argon = 39.948 * sc.atomic_mass
    S_T = np.array([sackur_tetrode_entropy(mass=argon, T=t, P=1.0e5) for t in T])
    axes[1].plot(T, S_T, color="steelblue", label="argon, 1 bar")
    axes[1].plot(T, S_T[0] + 2.5 * R * np.log(T / T[0]), color="gray", linestyle="--", label=r"$\frac{5}{2}R\ln T$ at constant $P$")
    axes[1].set_xlabel("T (K)")
    axes[1].set_ylabel(r"$S_m$ (J mol$^{-1}$ K$^{-1}$)")
    axes[1].set_title("Temperature dependence")
    axes[1].legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/statmech/partition_functions/images/sphx_glr_plot_04_sackur_tetrode_entropy_001.png
   :alt: Noble-gas entropies: slope $\frac{3}{2}R$ per $\ln m$, Temperature dependence
   :srcset: /api/gallery/statmech/partition_functions/images/sphx_glr_plot_04_sackur_tetrode_entropy_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.087 seconds)


.. _sphx_glr_download_api_gallery_statmech_partition_functions_plot_04_sackur_tetrode_entropy.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_04_sackur_tetrode_entropy.ipynb <plot_04_sackur_tetrode_entropy.ipynb>`

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

      :download:`Download Python source code: plot_04_sackur_tetrode_entropy.py <plot_04_sackur_tetrode_entropy.py>`

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

      :download:`Download zipped: plot_04_sackur_tetrode_entropy.zip <plot_04_sackur_tetrode_entropy.zip>`


.. only:: html

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

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