.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/analytical/uncertainty/plot_01_error_propagation.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_analytical_uncertainty_plot_01_error_propagation.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_analytical_uncertainty_plot_01_error_propagation.py:


Propagation of uncertainty with the first-order (Ku) formula
===============================================================

The closed-form rules for sums (:func:`~chemistrykit.analytical.systems.uncertainty.propagate_sum`),
products/quotients (:func:`~chemistrykit.analytical.systems.uncertainty.propagate_product`),
and powers (:func:`~chemistrykit.analytical.systems.uncertainty.propagate_power`)
are each a special case of the general first-order propagation formula,
which :func:`~chemistrykit.analytical.systems.uncertainty.propagate_uncertainty`
evaluates directly via numerical partial derivatives -- useful when no
simple closed form is at hand.

.. GENERATED FROM PYTHON SOURCE LINES 15-22

.. code-block:: Python

    from chemistrykit.analytical.systems.uncertainty import (
        propagate_power,
        propagate_product,
        propagate_sum,
        propagate_uncertainty,
    )








.. GENERATED FROM PYTHON SOURCE LINES 23-25

A titration's net volume, Vb_final - Vb_initial, both read from a buret
with the same reading uncertainty:

.. GENERATED FROM PYTHON SOURCE LINES 25-29

.. code-block:: Python

    sigma_reading = 0.02  # mL
    sigma_volume = propagate_sum([sigma_reading, sigma_reading])
    print(f"Net volume uncertainty (two buret readings of {sigma_reading} mL each): {sigma_volume:.4f} mL")





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

 .. code-block:: none

    Net volume uncertainty (two buret readings of 0.02 mL each): 0.0283 mL




.. GENERATED FROM PYTHON SOURCE LINES 30-32

Molarity from mass, molar mass, and volume: M = m / (MW * V) -- a
product/quotient of three measured quantities.

.. GENERATED FROM PYTHON SOURCE LINES 32-44

.. code-block:: Python

    mass, sigma_mass = 0.2500, 0.0002  # g
    molar_mass, sigma_molar_mass = 58.44, 0.01  # g/mol, NaCl
    volume, sigma_volume_L = 0.1000, 0.0002  # L

    moles = mass / molar_mass
    molarity = moles / volume
    sigma_molarity = propagate_product(
        [mass, 1.0 / molar_mass, 1.0 / volume],
        [sigma_mass, sigma_molar_mass / molar_mass**2, sigma_volume_L / volume**2],
    )
    print(f"\nMolarity = {molarity:.6f} +/- {sigma_molarity:.6f} mol/L")





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

 .. code-block:: none


    Molarity = 0.042779 +/- 0.000092 mol/L




.. GENERATED FROM PYTHON SOURCE LINES 45-46

A cell's volume from a measured edge length, V = L^3:

.. GENERATED FROM PYTHON SOURCE LINES 46-52

.. code-block:: Python

    L, sigma_L = 2.000, 0.005  # cm
    V = L**3
    sigma_V = propagate_power(L, sigma_L, 3.0)
    print(f"\nVolume = {V:.4f} +/- {sigma_V:.4f} cm^3 (relative uncertainty tripled: {sigma_V / V:.4%} vs {sigma_L / L:.4%})")






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

 .. code-block:: none


    Volume = 8.0000 +/- 0.0600 cm^3 (relative uncertainty tripled: 0.7500% vs 0.2500%)




.. GENERATED FROM PYTHON SOURCE LINES 53-55

General numerical propagation reproduces the molarity closed-form
result for an arbitrary function of the same three variables:

.. GENERATED FROM PYTHON SOURCE LINES 55-62

.. code-block:: Python

    def molarity_func(m, mw, v):
        return m / (mw * v)


    sigma_molarity_numeric = propagate_uncertainty(molarity_func, [mass, molar_mass, volume], [sigma_mass, sigma_molar_mass, sigma_volume_L])
    print(f"\nMolarity uncertainty (closed form):  {sigma_molarity:.6f}")
    print(f"Molarity uncertainty (general/numeric): {sigma_molarity_numeric:.6f}")




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

 .. code-block:: none


    Molarity uncertainty (closed form):  0.000092
    Molarity uncertainty (general/numeric): 0.000092





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

   **Total running time of the script:** (0 minutes 0.001 seconds)


.. _sphx_glr_download_api_gallery_analytical_uncertainty_plot_01_error_propagation.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_01_error_propagation.ipynb <plot_01_error_propagation.ipynb>`

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

      :download:`Download Python source code: plot_01_error_propagation.py <plot_01_error_propagation.py>`

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

      :download:`Download zipped: plot_01_error_propagation.zip <plot_01_error_propagation.zip>`


.. only:: html

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

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