:orphan:

Examples
========

This gallery walks through every public feature of
``chemistrykit.electrochem``: the Nernst equation for standard and
concentration cells (with activity-coefficient corrections), a curated
standard-reduction-potential table with redox-couple balancing,
Butler-Volmer electrode kinetics and Tafel-plot linearization, Faraday's
laws of electrolysis and the galvanic-vs-electrolytic distinction, a
simplified constant-current battery discharge model with Peukert's-law
rate dependence, fuel-cell thermodynamics, Kohlrausch's conductivity
laws, and diffusion-limited electroanalytical currents.

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

- **nernst** -- the Nernst equation for standard cells and concentration
  cells, plus Debye-Huckel activity-coefficient-corrected reaction
  quotients.
- **standard_potentials** -- the curated standard-reduction-potential
  table, redox-couple electron balancing, and cell-potential combination.
- **butler_volmer** -- Butler-Volmer electrode kinetics and its
  high-overpotential Tafel-plot linearization, checked for convergence.
- **electrolysis** -- Faraday's laws of electrolysis, and galvanic vs.
  electrolytic cell operation.
- **battery** -- Volta's pile, a simplified constant-current battery
  discharge model, and Peukert's-law capacity-vs-rate dependence.
- **fuel_cell** -- Grove's hydrogen-oxygen gas battery and its
  thermodynamic voltage and efficiency limits.
- **conductivity** -- Kohlrausch's laws of electrolytic conductivity.
- **voltammetry** -- the Cottrell equation, polarography and the Ilkovič
  equation, and the Randles-Ševčík peak current.


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Battery discharge
-------------------

Volta's pile of series-stacked two-metal cells, a simplified constant-current battery discharge model and Peukert's-law
capacity-vs-rate dependence.


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    <div class="sphx-glr-thumbcontainer" tooltip="Peukert&#x27;s 1897 power law, t = C_p/I^k, says a lead-acid battery discharged faster delivers less total charge. Using effective_capacity, this example compares the ideal Peukert exponent k=1 (same capacity at every rate) with a realistic k=1.2, then draws discharge curves from ConstantCurrentBattery (a deliberately simplified model; see its docstring) whose runtimes shrink faster than 1/I.">

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  .. image:: /api/gallery/electrochem/battery/images/thumb/sphx_glr_plot_01_discharge_curves_thumb.png
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  :doc:`/api/gallery/electrochem/battery/plot_01_discharge_curves`

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      <div class="sphx-glr-thumbnail-title">Peukert's law: battery capacity falls at high discharge rates</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Volta&#x27;s 1800 pile was a stack of identical zinc | brine | silver (or copper) pairs. Each pair contributes the difference between the two metals&#x27; reduction potentials -- the &quot;contact&quot; driving force Volta insisted on against Galvani&#x27;s animal electricity -- and stacking pairs in series adds their voltages. This example computes the single-pair voltage with cell_potential, shows how the pile voltage grows linearly with the number of pairs, and compares Volta&#x27;s zinc-silver pairing with zinc-copper and a frog-leg-free &quot;same metal&quot; pair that produces nothing at all.">

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  .. image:: /api/gallery/electrochem/battery/images/thumb/sphx_glr_plot_02_voltaic_pile_thumb.png
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  :doc:`/api/gallery/electrochem/battery/plot_02_voltaic_pile`

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      <div class="sphx-glr-thumbnail-title">Volta's pile: stacking zinc-silver cells in series</div>
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Butler-Volmer kinetics
------------------------

Butler-Volmer electrode kinetics and its high-overpotential Tafel-plot
linearization.


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    <div class="sphx-glr-thumbcontainer" tooltip="Tafel (1905) found that far from equilibrium the overpotential of hydrogen evolution grows linearly with \log_{10} i: \eta = b\log_{10}(i/i_0). This example generates noisy &quot;measured&quot; high-overpotential data, draws the Tafel plot, and uses fit_tafel_plot to recover the Tafel slope b and exchange current density i_0 by linear regression -- the analysis Tafel&#x27;s successors still use, and compares the slope with tafel_slope&#x27;s textbook ~118 mV/decade for \alpha = 0.5, n = 1.">

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  .. image:: /api/gallery/electrochem/butler_volmer/images/thumb/sphx_glr_plot_01_tafel_thumb.png
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  :doc:`/api/gallery/electrochem/butler_volmer/plot_01_tafel`

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      <div class="sphx-glr-thumbnail-title">Tafel's law: overpotential linear in log current</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The Butler-Volmer equation writes the net electrode current as the difference of an anodic and a cathodic exponential. At zero overpotential the two partial currents are equal (each is i_0) and cancel -- a dynamic equilibrium; far from equilibrium one term dominates and the equation becomes Tafel&#x27;s straight line. This example plots both partial currents and butler_volmer_current_density, shows the effect of the transfer coefficient \alpha, and checks numerically how fast the Tafel limit is reached.">

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  .. image:: /api/gallery/electrochem/butler_volmer/images/thumb/sphx_glr_plot_02_butler_volmer_equation_thumb.png
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  :doc:`/api/gallery/electrochem/butler_volmer/plot_02_butler_volmer_equation`

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      <div class="sphx-glr-thumbnail-title">The Butler-Volmer equation: anodic and cathodic partial currents</div>
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Electrolytic conductivity
--------------------------

Kohlrausch's law of independent migration of ions and his square-root
law for molar conductivity.


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    <div class="sphx-glr-thumbcontainer" tooltip="Kohlrausch found that the molar conductivity of a strong electrolyte falls linearly with \sqrt{c}, \Lambda_m = \Lambda_m^\circ - K\sqrt{c}, and that the infinite-dilution value \Lambda_m^\circ is a sum of independent ionic contributions. This example extrapolates noisy KCl data to infinite dilution with fit_kohlrausch_law, and uses limiting_molar_conductivity for his classic trick: the limiting conductivity of the weak acid acetic acid, which cannot be extrapolated, from three strong electrolytes.">

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  .. image:: /api/gallery/electrochem/conductivity/images/thumb/sphx_glr_plot_01_kohlrausch_laws_thumb.png
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  :doc:`/api/gallery/electrochem/conductivity/plot_01_kohlrausch_laws`

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      <div class="sphx-glr-thumbnail-title">Kohlrausch's laws: independent migration and the square-root law</div>
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Electrolysis
-------------

Faraday's laws of electrolysis, Nicholson and Carlisle's electrolysis of
water, and Davy's electrolytic isolation of the alkali metals.


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    <div class="sphx-glr-thumbcontainer" tooltip="Faraday&#x27;s first law: the mass liberated at an electrode is proportional to the charge passed, m = QM/(nF). His second law: for a fixed charge, different substances are liberated in proportion to their equivalent weights M/n. This example verifies both with faradays_law_mass and mass_from_charge, recovering the single universal constant -- the Faraday -- that links charge to chemical change.">

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  .. image:: /api/gallery/electrochem/electrolysis/images/thumb/sphx_glr_plot_01_faradays_laws_thumb.png
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  :doc:`/api/gallery/electrochem/electrolysis/plot_01_faradays_laws`

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      <div class="sphx-glr-thumbnail-title">Faraday's laws of electrolysis</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Days after learning of Volta&#x27;s newly announced pile, William Nicholson and Anthony Carlisle connected one to a bowl of water in May 1800 and watched gas bubble from each wire -- the first electrolysis, and the first new discovery the pile made possible. This example reconstructs both halves of that observation from functions already in this package: the theoretical minimum decomposition voltage of water (~chemistrykit.electrochem.systems.electrolysis.minimum_applied_voltage_electrolytic, applied to the tabulated H+/H2 and O2/H2O standard potentials) comes out at the textbook 1.23 V, and Faraday&#x27;s law (~chemistrykit.electrochem.systems.electrolysis.moles_from_charge) recovers the 2:1 hydrogen-to-oxygen ratio Nicholson and Carlisle actually saw bubbling from the two wires, directly from electron-transfer stoichiometry.">

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  .. image:: /api/gallery/electrochem/electrolysis/images/thumb/sphx_glr_plot_02_water_electrolysis_thumb.png
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  :doc:`/api/gallery/electrochem/electrolysis/plot_02_water_electrolysis`

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      <div class="sphx-glr-thumbnail-title">Nicholson and Carlisle's electrolysis of water</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="In 1807 Humphry Davy passed current through molten potash and soda and isolated potassium and sodium -- metals no chemical reducing agent of the day could free, because their reduction potentials are among the most negative in the electrochemical series. This example ranks Davy&#x27;s two couples against the rest of STANDARD_REDUCTION_POTENTIALS, shows why the common reducing metals (Zn, Fe) cannot reduce K+ or Na+, computes the minimum electrolysis voltage for the molten chlorides with minimum_applied_voltage_electrolytic, and uses Faraday&#x27;s law to estimate how much potassium a current yields.">

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  :doc:`/api/gallery/electrochem/electrolysis/plot_03_davy_alkali_metals`

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      <div class="sphx-glr-thumbnail-title">Davy's electrolytic isolation of potassium and sodium</div>
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Fuel cells
-----------

Grove's gas battery and the thermodynamic voltage and efficiency limits
of the hydrogen-oxygen fuel cell.


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    <div class="sphx-glr-thumbcontainer" tooltip="In 1839-1842 William Grove ran electrolysis backwards: hydrogen and oxygen over platinum electrodes in dilute acid recombined to produce a current. This example computes the reversible voltage of that cell two ways -- from the standard potentials of the O2/H2O and H+/H2 couples and from \Delta G^\circ via reversible_cell_voltage -- its temperature dependence (~chemistrykit.electrochem.systems.fuel_cell.reversible_cell_voltage_at_temperature), and its thermodynamic efficiency limit \Delta G/\Delta H (~chemistrykit.electrochem.systems.fuel_cell.fuel_cell_efficiency_limit) compared with the Carnot limit of a heat engine.">

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  .. image:: /api/gallery/electrochem/fuel_cell/images/thumb/sphx_glr_plot_01_grove_gas_battery_thumb.png
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  :doc:`/api/gallery/electrochem/fuel_cell/plot_01_grove_gas_battery`

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      <div class="sphx-glr-thumbnail-title">Grove's gas battery: the hydrogen-oxygen fuel cell</div>
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The Nernst equation
---------------------

Standard and concentration cells via the Nernst equation, and
Debye-Huckel activity-coefficient-corrected reaction quotients.


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    <div class="sphx-glr-thumbcontainer" tooltip="nernst_potential implements E = E^\circ - \frac{RT}{nF}\ln Q. This example checks that it reduces to E^\circ at Q=1, plots the logarithmic dependence on the reaction quotient, and then builds a concentration cell (~chemistrykit.electrochem.systems.nernst.concentration_cell_potential) whose voltage comes entirely from a concentration difference -- about 59 mV per decade for a one-electron couple at 25 degC, as Nernst predicted.">

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  :doc:`/api/gallery/electrochem/nernst/plot_01_nernst_equation`

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      <div class="sphx-glr-thumbnail-title">The Nernst equation: cell potential versus concentration</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Debye and Hückel (1923) showed that each ion&#x27;s cloud of counter-ions lowers its activity below its concentration, by an amount growing with ionic strength and ionic charge. This example uses nernst_potential_with_activity (built on chemistrykit.solutions.systems.activity) to show how the potential of a Cu2+/Cu electrode drifts away from the ideal Nernst value as the solution gets more concentrated, and that the correction vanishes in the dilute limit.">

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  :doc:`/api/gallery/electrochem/nernst/plot_02_debye_huckel_activity`

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      <div class="sphx-glr-thumbnail-title">Debye-Hückel activity corrections to the Nernst equation</div>
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Standard reduction potentials
--------------------------------

The Daniell cell from tabulated standard reduction potentials, and the
1953 Stockholm sign convention behind the table.


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    <div class="sphx-glr-thumbcontainer" tooltip="Daniell&#x27;s 1836 two-fluid cell separated a zinc electrode in zinc (or acid) solution from a copper electrode in copper sulfate. Copper metal, not hydrogen gas, is deposited at the cathode, so the cell does not polarize. This example reproduces its standard potential with standard_cell_potential, balances the overall reaction with balance_redox_reaction, and uses the Nernst equation to show how gently the voltage falls as the cell discharges -- the steadiness that made it the first practical voltage standard.">

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      <div class="sphx-glr-thumbnail-title">The Daniell cell: a steady 1.10 V from zinc and copper</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Before IUPAC&#x27;s 1953 Stockholm meeting, one tradition tabulated oxidation potentials and another reduction potentials, so the same couple appeared with opposite signs in different books. This example shows both conventions for the table in STANDARD_REDUCTION_POTENTIALS (which follows Stockholm), and shows that the cathode-minus-anode rule of cell_potential only gives the right cell voltage and spontaneity when both half-reactions use the agreed reduction convention.">

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  .. image:: /api/gallery/electrochem/standard_potentials/images/thumb/sphx_glr_plot_02_stockholm_sign_convention_thumb.png
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  :doc:`/api/gallery/electrochem/standard_potentials/plot_02_stockholm_sign_convention`

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      <div class="sphx-glr-thumbnail-title">The 1953 Stockholm convention: electrode potentials are reduction potentials</div>
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Electroanalytical currents
---------------------------

Diffusion-limited currents: the Cottrell equation, Heyrovský's
polarography and the Ilkovič equation, and the Randles-Ševčík peak
current of cyclic voltammetry.


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    <div class="sphx-glr-thumbcontainer" tooltip="Cottrell (1903) solved the diffusion problem for a planar electrode whose potential is suddenly stepped so that the electroactive species is consumed at the surface: the current decays as t^{-1/2}, i = nFAC\sqrt{D/(\pi t)}. This example plots cottrell_current, checks the constant i\sqrt{t} signature, and recovers the diffusion coefficient from a noisy &quot;measured&quot; transient via the Cottrell plot of i against t^{-1/2}.">

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  :doc:`/api/gallery/electrochem/voltammetry/plot_01_cottrell_equation`

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      <div class="sphx-glr-thumbnail-title">The Cottrell equation: current decay after a potential step</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Heyrovský&#x27;s dropping mercury electrode (1922) records a sigmoidal &quot;polarographic wave&quot; for each reducible species: its position (the half-wave potential) identifies the species and its height (the diffusion current, given by Ilkovič&#x27;s 1934 equation) measures its concentration. This example builds a two-component polarogram from polarographic_wave_current with wave heights from ilkovic_diffusion_current, and shows the linear calibration of diffusion current against concentration.">

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  :doc:`/api/gallery/electrochem/voltammetry/plot_02_heyrovsky_polarography`

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      <div class="sphx-glr-thumbnail-title">Heyrovský's polarography and the Ilkovič equation</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Randles and Ševčík (1948) independently showed that the peak current of a reversible linear-sweep voltammogram grows with the square root of the scan rate, i_p = 0.4463\,nFAC\sqrt{nFvD/RT}. This example evaluates randles_sevcik_peak_current over a range of scan rates, and recovers the diffusion coefficient from the slope of the i_p vs. \sqrt{v} plot -- the standard diagnostic for a diffusion-controlled electrode reaction.">

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  :doc:`/api/gallery/electrochem/voltammetry/plot_03_randles_sevcik_cyclic_voltammetry`

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      <div class="sphx-glr-thumbnail-title">The Randles-Ševčík equation: peak current in cyclic voltammetry</div>
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.. toctree::
   :hidden:
   :includehidden:


   /api/gallery/electrochem/battery/index.rst
   /api/gallery/electrochem/butler_volmer/index.rst
   /api/gallery/electrochem/conductivity/index.rst
   /api/gallery/electrochem/electrolysis/index.rst
   /api/gallery/electrochem/fuel_cell/index.rst
   /api/gallery/electrochem/nernst/index.rst
   /api/gallery/electrochem/standard_potentials/index.rst
   /api/gallery/electrochem/voltammetry/index.rst



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