.. DO NOT EDIT.
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.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/analytical/chromatography/plot_06_purnell_resolution_equation.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
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        :ref:`Go to the end <sphx_glr_download_api_gallery_analytical_chromatography_plot_06_purnell_resolution_equation.py>`
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.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_analytical_chromatography_plot_06_purnell_resolution_equation.py:


Purnell's resolution equation: efficiency, selectivity, retention
====================================================================

:func:`~chemistrykit.analytical.purnell_resolution` writes resolution as
:math:`R_s=\frac{\sqrt N}{4}\,\frac{\alpha-1}{\alpha}\,\frac{k_2}{1+k_2}`.
We check it against the direct peak-width formula
:func:`~chemistrykit.analytical.resolution` on simulated chromatograms,
then compare the three levers: doubling `N` only multiplies
:math:`R_s` by :math:`\sqrt2`, whereas a small change in selectivity
:math:`\alpha` can do much more.

.. GENERATED FROM PYTHON SOURCE LINES 15-43

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.analytical import purnell_resolution, resolution, selectivity_factor, simulate_chromatogram

    t0 = 1.0
    k1 = 4.0


    def peaks(N, alpha):
        """Retention times, base widths, and direct resolution for a peak pair."""
        k2 = alpha * k1
        tR1, tR2 = t0 * (1 + k1), t0 * (1 + k2)
        w1, w2 = 4 * tR1 / np.sqrt(N), 4 * tR2 / np.sqrt(N)
        return tR1, tR2, resolution(tR1, tR2, w1, w2), k2


    cases = {
        "baseline: N=5000, alpha=1.05": (5000.0, 1.05),
        "double N: N=10000, alpha=1.05": (10000.0, 1.05),
        "better selectivity: N=5000, alpha=1.10": (5000.0, 1.10),
    }
    for label, (N, alpha) in cases.items():
        tR1, tR2, Rs_direct, k2 = peaks(N, alpha)
        print(
            f"{label:40s}: alpha = {selectivity_factor(k1, k2):.2f}, Rs (Purnell) = {purnell_resolution(N, alpha, k2):.3f}, Rs (from peak widths) = {Rs_direct:.3f}"
        )





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

 .. code-block:: none

    baseline: N=5000, alpha=1.05            : alpha = 1.05, Rs (Purnell) = 0.680, Rs (from peak widths) = 0.693
    double N: N=10000, alpha=1.05           : alpha = 1.05, Rs (Purnell) = 0.962, Rs (from peak widths) = 0.980
    better selectivity: N=5000, alpha=1.10  : alpha = 1.10, Rs (Purnell) = 1.309, Rs (from peak widths) = 1.360




.. GENERATED FROM PYTHON SOURCE LINES 44-64

.. code-block:: Python

    fig, axes = plt.subplots(1, 2, figsize=(12, 4))
    t = np.linspace(4.0, 6.0, 4000)
    for label, (N, alpha) in cases.items():
        tR1, tR2, Rs, k2 = peaks(N, alpha)
        axes[0].plot(t, simulate_chromatogram(t, centers=[tR1, tR2], N=N), label=f"{label} (Rs = {Rs:.2f})")
    axes[0].set_xlabel("time (min)")
    axes[0].set_ylabel("signal")
    axes[0].set_title("Same column, three levers")
    axes[0].legend(fontsize=7)

    N_grid = np.linspace(1000, 40000, 200)
    for alpha in (1.02, 1.05, 1.10):
        axes[1].plot(N_grid, purnell_resolution(N_grid, alpha, alpha * k1), label=rf"$\alpha$ = {alpha}")
    axes[1].axhline(1.5, color="gray", linestyle="--", linewidth=0.8, label="baseline separation, Rs = 1.5")
    axes[1].set_xlabel("plate count N")
    axes[1].set_ylabel(r"$R_s$")
    axes[1].set_title(r"$R_s$ grows only as $\sqrt{N}$")
    axes[1].legend()
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/analytical/chromatography/images/sphx_glr_plot_06_purnell_resolution_equation_001.png
   :alt: Same column, three levers, $R_s$ grows only as $\sqrt{N}$
   :srcset: /api/gallery/analytical/chromatography/images/sphx_glr_plot_06_purnell_resolution_equation_001.png
   :class: sphx-glr-single-img






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   **Total running time of the script:** (0 minutes 0.107 seconds)


.. _sphx_glr_download_api_gallery_analytical_chromatography_plot_06_purnell_resolution_equation.py:

.. only:: html

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