:orphan:

Examples
========

This gallery walks through every public feature of
``chemistrykit.analytical``: redox and complexometric (EDTA)
titration-curve simulation with equivalence-point detection and Gran
plots (acid-base titrations are covered by ``chemistrykit.solutions``'s
gallery); chromatographic plate theory, the van Deemter equation,
retention indices, and resolution; linear-regression calibration curves
with limits of detection/quantitation; propagation of uncertainty;
Dixon's and Grubbs' outlier tests; Student's t intervals and the Horwitz
function; and Savitzky-Golay smoothing.

Each script in this gallery is self-contained and can be run directly
with ``python examples/analytical/<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
--------

- **titration** -- redox (Nernst) and EDTA titration curves with
  numerical equivalence-point detection, and the Gran plot.
- **chromatography** -- Tsvet's column separation, theoretical plates,
  the van Deemter and Golay equations, Kovats retention indices, and
  Purnell's resolution equation.
- **calibration** -- a least-squares calibration curve and its limits of
  detection/quantitation.
- **uncertainty** -- propagation-of-uncertainty formulas for sums,
  products, and powers, cross-checked against a general numerical
  implementation.
- **qtest** -- Dixon's Q-test and Grubbs' test for rejecting a suspect
  outlier from a small data set.
- **statistics** -- Student's t confidence intervals and the Horwitz
  interlaboratory-precision function.
- **smoothing** -- Savitzky-Golay smoothing and differentiation.


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Calibration curves
--------------------

Least-squares calibration lines, and the limits of detection and
quantitation derived from their residual scatter.


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    <div class="sphx-glr-thumbcontainer" tooltip="The method of least squares picks the straight line that minimizes the sum of squared residuals S(m,b)=\sum_i(y_i-mx_i-b)^2. fit_calibration performs exactly that fit for a set of calibration standards; here we check that its slope sits at the bottom of the S(m) parabola, and that the residuals it leaves behind scatter around zero.">

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  .. image:: /api/gallery/analytical/calibration/images/thumb/sphx_glr_plot_01_least_squares_calibration_thumb.png
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  :doc:`/api/gallery/analytical/calibration/plot_01_least_squares_calibration`

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      <div class="sphx-glr-thumbnail-title">Legendre and Gauss's least squares: fitting a calibration line</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Long and Winefordner argued that a detection limit should be tied to the calibration&#x27;s own residual scatter s_{y/x} and slope m. lod and loq use the widely adopted calibration-based convention \text{LOD}=3.3\,s_{y/x}/m and \text{LOQ}=10\,s_{y/x}/m. A noisier method has proportionally higher limits.">

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  .. image:: /api/gallery/analytical/calibration/images/thumb/sphx_glr_plot_02_limits_of_detection_and_quantitation_thumb.png
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  :doc:`/api/gallery/analytical/calibration/plot_02_limits_of_detection_and_quantitation`

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      <div class="sphx-glr-thumbnail-title">Limits of detection and quantitation from a calibration curve</div>
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Chromatography
----------------

Tsvet's column separation, theoretical plates, the van Deemter and Golay
plate-height equations, Kovats retention indices, and Purnell's
resolution equation.


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    <div class="sphx-glr-thumbcontainer" tooltip="In Tsvet&#x27;s experiment a pigment extract moves down a chalk column, each pigment at its own speed u/(1+k) set by how strongly it is retained (retention factor k). Snapshots of the column show the mixture resolving into separate colored bands; at the column outlet a detector records the same separation as a chromatogram, rendered here by simulate_chromatogram.">

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  .. image:: /api/gallery/analytical/chromatography/images/thumb/sphx_glr_plot_01_tsvet_column_chromatography_thumb.png
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  :doc:`/api/gallery/analytical/chromatography/plot_01_tsvet_column_chromatography`

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      <div class="sphx-glr-thumbnail-title">Tsvet's column: separating leaf pigments into colored bands</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Treating a column as N equilibration stages (&quot;theoretical plates&quot;) predicts a Gaussian peak whose width grows as t_R/\sqrt N. theoretical_plates inverts that: N=16(t_R/w_{base})^2=5.545(t_R/w_{1/2})^2, and plate_height gives the plate height H=L/N. Here we measure the widths of simulated peaks and recover the plate count they were built from.">

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  .. image:: /api/gallery/analytical/chromatography/images/thumb/sphx_glr_plot_02_martin_synge_theoretical_plates_thumb.png
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  :doc:`/api/gallery/analytical/chromatography/plot_02_martin_synge_theoretical_plates`

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      <div class="sphx-glr-thumbnail-title">Martin and Synge's theoretical plates from a peak's width</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="van_deemter_H evaluates H=A+B/u+Cu, the sum of eddy diffusion (`A`), longitudinal diffusion (`B/u`, dominant at low flow) and mass-transfer resistance (`Cu`, dominant at high flow). The competition between the last two gives a minimum plate height H_{min}=A+2\sqrt{BC} at u_{opt}=\sqrt{B/C} (~chemistrykit.analytical.minimum_plate_height, optimum_flow_velocity).">

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  .. image:: /api/gallery/analytical/chromatography/images/thumb/sphx_glr_plot_03_van_deemter_equation_thumb.png
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  :doc:`/api/gallery/analytical/chromatography/plot_03_van_deemter_equation`

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      <div class="sphx-glr-thumbnail-title">The van Deemter equation and the optimum flow velocity</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="An open capillary has no packing, so Golay&#x27;s equation for it is the van Deemter equation with no eddy-diffusion term: H=B/u+Cu. Using van_deemter_H with A=0 for a capillary and A&gt;0 for a packed column shows the lower plate height, and hence the many more plates a long capillary delivers.">

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  .. image:: /api/gallery/analytical/chromatography/images/thumb/sphx_glr_plot_04_golay_open_tubular_column_thumb.png
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  :doc:`/api/gallery/analytical/chromatography/plot_04_golay_open_tubular_column`

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      <div class="sphx-glr-thumbnail-title">Golay's open-tubular column: the van Deemter equation with A = 0</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Under isothermal gas chromatography, \log t&#x27; of the n-alkanes rises linearly with carbon number. Kováts placed every other compound on that ladder: kovats_retention_index interpolates \log t&#x27; between the bracketing alkanes, giving an index (n-octane = 800, n-nonane = 900, ...) that is far more transferable between instruments than a raw retention time.">

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  .. image:: /api/gallery/analytical/chromatography/images/thumb/sphx_glr_plot_05_kovats_retention_index_thumb.png
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  :doc:`/api/gallery/analytical/chromatography/plot_05_kovats_retention_index`

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      <div class="sphx-glr-thumbnail-title">Kováts retention indices from an n-alkane ladder</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="purnell_resolution writes resolution as 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 resolution on simulated chromatograms, then compare the three levers: doubling N only multiplies R_s by \sqrt2, whereas a small change in selectivity \alpha can do much more.">

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  .. image:: /api/gallery/analytical/chromatography/images/thumb/sphx_glr_plot_06_purnell_resolution_equation_thumb.png
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  :doc:`/api/gallery/analytical/chromatography/plot_06_purnell_resolution_equation`

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      <div class="sphx-glr-thumbnail-title">Purnell's resolution equation: efficiency, selectivity, retention</div>
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Outlier rejection
-------------------

Dixon's Q-test and Grubbs' test for rejecting a suspect outlier from a
small data set.


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    <div class="sphx-glr-thumbcontainer" tooltip="dixon_q_test compares the gap between a suspect value and its nearest neighbor, relative to the full data range, against a tabulated critical value (~chemistrykit.analytical.systems.qtest.Q_CRITICAL_TABLE) that depends on sample size and confidence level.">

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  .. image:: /api/gallery/analytical/qtest/images/thumb/sphx_glr_plot_01_dixon_q_test_thumb.png
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  :doc:`/api/gallery/analytical/qtest/plot_01_dixon_q_test`

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      <div class="sphx-glr-thumbnail-title">Dixon's Q-test for outlier rejection</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="grubbs_test computes G=\max_i|x_i-\bar x|/s and compares it with the critical value grubbs_critical_value, derived from Student&#x27;s t distribution so that it exists for any sample size. A Monte Carlo run on outlier-free normal data confirms that the test falsely rejects a point at the nominal rate \alpha.">

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  .. image:: /api/gallery/analytical/qtest/images/thumb/sphx_glr_plot_02_grubbs_outlier_test_thumb.png
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  :doc:`/api/gallery/analytical/qtest/plot_02_grubbs_outlier_test`

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      <div class="sphx-glr-thumbnail-title">Grubbs' outlier test: the extreme value in standard deviations</div>
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Signal smoothing
------------------

Savitzky-Golay smoothing and differentiation of a noisy analytical signal.


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    <div class="sphx-glr-thumbcontainer" tooltip="savitzky_golay fits a low-order polynomial by least squares to each moving window and keeps its value (or derivative) at the center, which reduces to a fixed convolution with weights from savitzky_golay_coefficients -- e.g. (-3,12,17,12,-3)/35 for 5 points and a quadratic. Unlike a plain moving average, it removes noise with little flattening of narrow peaks, and it also gives smooth derivatives.">

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  .. image:: /api/gallery/analytical/smoothing/images/thumb/sphx_glr_plot_01_savitzky_golay_filter_thumb.png
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  :doc:`/api/gallery/analytical/smoothing/plot_01_savitzky_golay_filter`

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      <div class="sphx-glr-thumbnail-title">Savitzky-Golay smoothing and derivatives of a noisy spectrum</div>
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Statistics of replicate measurements
--------------------------------------

Student's `t` confidence interval for a mean of a few replicates, and the
Horwitz function for expected interlaboratory precision.


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    <div class="sphx-glr-thumbcontainer" tooltip="With only n replicates the standard deviation s is itself uncertain, so the interval \bar x\pm t\,s/\sqrt n from t_confidence_interval uses Student&#x27;s t instead of the normal 1.96. We simulate many triplicate analyses and show that the t interval covers the true value 95% of the time, while the naive \pm1.96\,s/\sqrt n interval falls well short.">

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  .. image:: /api/gallery/analytical/statistics/images/thumb/sphx_glr_plot_01_student_t_confidence_interval_thumb.png
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  :doc:`/api/gallery/analytical/statistics/plot_01_student_t_confidence_interval`

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      <div class="sphx-glr-thumbnail-title">Student's t: confidence intervals from a few replicates</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="horwitz_rsd gives Horwitz&#x27;s empirical expected between-laboratory RSD, 2^{1-0.5\log_{10}C}, which doubles for every 100-fold drop in concentration. Plotted as \pm\text{RSD} against concentration it opens like a trumpet. horrat scores a method&#x27;s observed RSD against it.">

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  .. image:: /api/gallery/analytical/statistics/images/thumb/sphx_glr_plot_02_horwitz_trumpet_thumb.png
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  :doc:`/api/gallery/analytical/statistics/plot_02_horwitz_trumpet`

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      <div class="sphx-glr-thumbnail-title">The Horwitz trumpet: interlaboratory precision vs. concentration</div>
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Titration curves
------------------

Potentiometric redox (Nernst) and EDTA complexometric titration curves,
with equivalence points located by steepest change or by a Gran plot.


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    <div class="sphx-glr-thumbcontainer" tooltip="RedoxTitration computes the electrode potential during a redox titration from the Nernst equation, E=E^\circ-\frac{RT}{nF}\ln Q: before equivalence the analyte couple sets E, after it the titrant couple does. At half-equivalence E=E^\circ_1 exactly, at twice the equivalence volume E=E^\circ_2, and the endpoint is read, as in the laboratory, at the steepest rise of E.">

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  .. image:: /api/gallery/analytical/titration/images/thumb/sphx_glr_plot_01_nernst_redox_titration_thumb.png
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  :doc:`/api/gallery/analytical/titration/plot_01_nernst_redox_titration`

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      <div class="sphx-glr-thumbnail-title">The Nernst equation in a potentiometric redox titration</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="EDTATitration follows M+Y\rightleftharpoons MY through its conditional formation constant K_f&#x27;. The same EDTA titrant quantifies any metal it binds strongly enough; the size of the pM break at equivalence is set by K_f&#x27;, which is why Schwarzenbach&#x27;s titrations are run in a buffer at a pH where K_f&#x27; is large.">

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  .. image:: /api/gallery/analytical/titration/images/thumb/sphx_glr_plot_02_edta_complexometric_titration_thumb.png
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  :doc:`/api/gallery/analytical/titration/plot_02_edta_complexometric_titration`

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      <div class="sphx-glr-thumbnail-title">EDTA complexometric titration of a metal ion</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="gran_plot linearizes the buffer region of a weak-acid titration: V_b\,10^{-pH}=K_a(V_e-V_b). Plotting the left side against V_b gives a straight line whose x-intercept is the equivalence volume and whose slope is -K_a, so the endpoint follows from data taken well before it.">

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  .. image:: /api/gallery/analytical/titration/images/thumb/sphx_glr_plot_03_gran_plot_thumb.png
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  :doc:`/api/gallery/analytical/titration/plot_03_gran_plot`

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      <div class="sphx-glr-thumbnail-title">Gran plot: the equivalence point by linear extrapolation</div>
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Propagation of uncertainty
-----------------------------

Closed-form uncertainty-propagation formulas for sums, products, and
powers, cross-checked against a general numerical implementation.


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    <div class="sphx-glr-thumbcontainer" tooltip="The closed-form rules for sums (~chemistrykit.analytical.systems.uncertainty.propagate_sum), products/quotients (~chemistrykit.analytical.systems.uncertainty.propagate_product), and powers (~chemistrykit.analytical.systems.uncertainty.propagate_power) are each a special case of the general first-order propagation formula, which propagate_uncertainty evaluates directly via numerical partial derivatives -- useful when no simple closed form is at hand.">

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  .. image:: /api/gallery/analytical/uncertainty/images/thumb/sphx_glr_plot_01_error_propagation_thumb.png
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  :doc:`/api/gallery/analytical/uncertainty/plot_01_error_propagation`

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      <div class="sphx-glr-thumbnail-title">Propagation of uncertainty with the first-order (Ku) formula</div>
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.. toctree::
   :hidden:
   :includehidden:


   /api/gallery/analytical/calibration/index.rst
   /api/gallery/analytical/chromatography/index.rst
   /api/gallery/analytical/qtest/index.rst
   /api/gallery/analytical/smoothing/index.rst
   /api/gallery/analytical/statistics/index.rst
   /api/gallery/analytical/titration/index.rst
   /api/gallery/analytical/uncertainty/index.rst



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