Breakthroughs in Surface Chemistry and Catalysis
==================================================


.. include:: /_generated/nav/surface.rst

.. epigraph::

   "In ordinary chemical reactions we consider only the concentrations of
   the reacting substances... [but] the surface itself may play an
   important part in the reaction, since it is here that the reacting
   molecules become adsorbed." -- Irving Langmuir, *The Constitution and
   Fundamental Properties of Solids and Liquids. Part I. Solids*, 1916

Surface chemistry is the study of what happens at the two-dimensional
boundary between a solid and a gas or liquid -- a region that behaves
nothing like the bulk phases on either side of it, and that turns out to
control most of industrial chemistry: a heterogeneous catalyst does its
work entirely at such a boundary, one adsorbed molecule at a time. The
systems in :mod:`chemistrykit.surface` retrace the century-long project
of turning "molecules stick to surfaces" into quantitative, testable
theory: from the first empirical curve fitted through adsorption data, to
Langmuir's kinetic derivation of a genuine monolayer limit, to BET's
extension to multiple layers, to the reaction kinetics and rate-enhancement
laws that make a catalyst's activity something you can actually calculate
rather than merely observe. This chronology traces the major conceptual
breakthroughs behind the package, with a pointer to the corresponding
implementation at each stop.

.. contents:: Timeline
   :local:
   :depth: 1

1823 -- 1836 -- Doebereiner, Berzelius, and the Discovery of Catalysis
------------------------------------------------------------------------

In 1823, Johann Wolfgang Doebereiner, a professor of chemistry at Jena,
found that a jet of hydrogen gas directed onto a small mass of finely
divided ("spongy") platinum ignites spontaneously in air at room
temperature, with the platinum itself apparently unchanged by the
reaction it triggered. The discovery was quickly commercialized as
"Doebereiner's lamp," a self-igniting device sold across Europe as an
everyday fire-lighting tool for roughly two decades, until safety matches
displaced it in the 1830s-1850s -- an oddity of chemical history in which
a genuine scientific discovery about the nature of catalytic surfaces
reached ordinary households as a novelty gadget before anyone had a
theory of why it worked.

That theory came from Jons Jacob Berzelius, who in his widely read annual
report to the Royal Swedish Academy of Sciences for 1835 surveyed
Doebereiner's platinum, the starch-hydrolyzing enzyme diastase, and
acid-catalyzed ester hydrolysis as instances of one underlying
phenomenon, and coined the term "catalysis" (from the Greek for
"loosening down") for a substance's ability to provoke a chemical change
without being consumed by it -- what he called a "catalytic force."
Berzelius's own explanation of *why* this force existed was vague and did
not survive; the name, and the recognition that these were all examples
of a single general phenomenon rather than isolated curiosities, did.

*Connection:* every model in :mod:`chemistrykit.surface` -- an isotherm
describing how much of a reactant sits on a catalytic surface, a
Langmuir-Hinshelwood rate law describing how fast it reacts there, or
``catalysis``'s turnover and
rate-enhancement metrics describing how effective the resulting catalyst
is -- is a quantitative descendant of the general phenomenon Berzelius
named and Doebereiner's platinum first exhibited in dramatic, tabletop
form.

*References:* J. W. Doebereiner, "Neu entdeckte merkwuerdige
Eigenschaften des Platinsuboxydes, des Platins in seinem Zustande als
Metall, und einiger anderen Metalle in ihrem gewoehnlichen Zustande,"
Schweiggers Journal fuer Chemie und Physik 38 (1823) (exact page range as
commonly cited in secondary literature; not independently re-verified
against the original volume); J. J. Berzelius, "Some Ideas on a New
Force Acting in Organic Compositions," Edinburgh New Philosophical
Journal 21 (1836), 223-228 (an English rendering of Berzelius's 1835
Swedish Academy report, itself the usual citation for the coinage of
"catalysis").

.. minigallery:: ../../examples/surface/catalysis/plot_01_doebereiner_berzelius_catalysis.py

1875 -- 1878 -- Gibbs and the Thermodynamics of Adsorption
--------------------------------------------------------------

In the second half of his memoir "On the Equilibrium of Heterogeneous
Substances," J. Willard Gibbs treated the boundary between two phases as
a thermodynamic object in its own right. He described it by a *surface
excess* :math:`\Gamma`: the amount of a component held at the interface
beyond what the two bulk phases would contain if they ran unchanged right
up to a mathematical dividing surface. For a dilute solute at a liquid
surface, his analysis gives the Gibbs adsorption equation

.. math::

   \Gamma = -\frac{1}{RT}\,\frac{d\gamma}{d\ln c}

This links something invisible, how much solute sits at the surface, to
something easy to measure, how the surface tension :math:`\gamma` changes
as solute is added. A solute that lowers the surface tension, such as a
soap or a fatty acid, must be concentrated at the surface. Thirty years
later, Bohdan Szyszkowski's empirical surface-tension equation for
aqueous fatty acids, :math:`\gamma = \gamma_0 - RT\Gamma_{max}\ln(1+Kc)`,
turned under Gibbs's equation into exactly a saturating, Langmuir-shaped
surface excess. This was one of the first quantitative signs of a
monolayer at a liquid surface.

*Implementation:*
:func:`~chemistrykit.surface.gibbs_surface_excess` applies the Gibbs
equation to measured surface-tension data by numerical differentiation
in :math:`\ln c`, and
:func:`~chemistrykit.surface.szyszkowski_surface_tension` provides the
Szyszkowski equation. Together they recover
:math:`\Gamma = \Gamma_{max}Kc/(1+Kc)`, the same form as
:func:`~chemistrykit.surface.langmuir_coverage`.

*References:* J. W. Gibbs, "On the Equilibrium of Heterogeneous
Substances," *Trans. Connecticut Acad. Arts Sci.* 3 (1875-1876),
108-248, and (1877-1878), 343-524; B. von Szyszkowski, "Experimentelle
Studien ueber kapillare Eigenschaften der waesserigen Loesungen von
Fettsaeuren," *Z. Phys. Chem.* 64 (1908); A. W. Adamson and A. P. Gast,
*Physical Chemistry of Surfaces*, 6th ed. (New York: Wiley, 1997),
Ch. III.

.. minigallery:: ../../examples/surface/gibbs_adsorption/plot_01_gibbs_adsorption_equation.py

1889 -- Arrhenius, Ostwald, and the Kinetic Meaning of a Catalyst
---------------------------------------------------------------------

Svante Arrhenius's 1889 equation gave the first quantitative account of
why reaction rates rise so steeply with temperature, proposing that only
molecules colliding with at least a threshold "activation energy"
:math:`E_a` can react, so that the rate constant follows

.. math::

   k = A\,e^{-E_a/RT}

Around the same period, Wilhelm Ostwald crystallized the modern
definition of a catalyst: a substance that changes a reaction's *rate*
without being itself consumed and, crucially, without changing the
reaction's *thermodynamics* -- its equilibrium constant and the position
of equilibrium are exactly what they would be without it. Ostwald was
awarded the 1909 Nobel Prize in Chemistry "in recognition of his work on
catalysis and for his investigations into the fundamental principles
governing chemical equilibria and rates of reaction." Put together,
Arrhenius's equation and Ostwald's definition give the precise, working
picture every quantitative theory of catalysis since has built on: a
catalyst's entire effect is to open a lower-:math:`E_a` pathway to the
same products, and the resulting rate enhancement is a pure exponential
in how much that activation energy drops.

*Implementation:*
:func:`~chemistrykit.surface.compare_catalyzed_rate`
implements exactly this picture, evaluating catalyzed and uncatalyzed
rate constants via
:func:`~chemistrykit.kinetics.arrhenius_rate_constant`
(reused directly rather than reimplemented, per this package's
convention of not duplicating shared kinetics machinery across domains)
and reporting their ratio as the rate enhancement -- Ostwald's
"unchanged thermodynamics" half of the definition is implicit in the
function never touching an equilibrium constant at all, only the two
rate constants.

*References:* S. Arrhenius, *Z. Phys. Chem.* 4 (1889), 226-248; W.
Ostwald's definition of catalysis was developed across his research on
reaction kinetics and chemical dynamics through the 1880s-1890s (see,
e.g., his *Lehrbuch der allgemeinen Chemie*, 2nd ed., Vol. 2, Leipzig:
Engelmann, 1896) rather than in one single citable paper; "The Nobel
Prize in Chemistry 1909," NobelPrize.org.

.. minigallery:: ../../examples/surface/catalysis/plot_02_arrhenius_ostwald_rate_enhancement.py

1897 -- 1912 -- Sabatier, Senderens, and Catalytic Hydrogenation
--------------------------------------------------------------------

Working together at Toulouse, Paul Sabatier and Jean-Baptiste Senderens
showed, beginning with ethylene over nickel in 1897 and extending the
method to a wide range of unsaturated organic compounds over the
following years (culminating in the direct hydrogenation of carbon
monoxide to methane in 1902), that finely divided nickel, cobalt, iron,
and other metals catalyze the addition of hydrogen across carbon-carbon
double and triple bonds at temperatures far below where the uncatalyzed
gas-phase reaction proceeds at any useful rate. Catalytic hydrogenation
became one of the most widely used reactions in organic and industrial
chemistry -- from Wilhelm Normann's 1902 hydrogenation of vegetable oils
into solid fats to the ammonia- and petroleum-processing industries built
up over the following decades -- and earned Sabatier a share of the 1912
Nobel Prize in Chemistry "for his method of hydrogenating organic
compounds in the presence of finely disintegrated metals whereby the
progress of organic chemistry has been greatly advanced in recent
years."

Beyond the specific reaction, Sabatier drew a general qualitative lesson
from years of screening different metals for hydrogenation activity, set
out in his 1913 monograph *La Catalyse en Chimie Organique*: a good
catalyst must bind the reacting molecule strongly enough to activate it,
but not so strongly that the resulting surface intermediate never lets
go to free the site for another turnover. Sabatier stated this as a
qualitative rule of thumb rather than a mathematical relationship; giving
it quantitative, structural content took the multiplet theory and
volcano-curve formalism described below.

*Connection:* ``chemistrykit.surface.systems.catalysis``'s
:func:`~chemistrykit.surface.compare_catalyzed_rate`
and :func:`~chemistrykit.surface.turnover_frequency`
quantify a catalyst's *activity* once a reaction pathway is chosen, but
say nothing about *which* metal or binding strength is optimal for a
given reaction -- the qualitative selection principle Sabatier
articulated here, given quantitative form directly below.

*References:* P. Sabatier and J.-B. Senderens, "Nouvelles syntheses du
methane," C. R. Acad. Sci. 134 (1902), 514-516, and their earlier series
of notes to the Academy beginning in 1897; P. Sabatier, *La Catalyse en
Chimie Organique* (Paris: Librairie Polytechnique Ch. Beranger, 1913);
"The Nobel Prize in Chemistry 1912," NobelPrize.org.

.. minigallery:: ../../examples/surface/catalysis/plot_03_sabatier_hydrogenation.py

1906 -- Freundlich's Empirical Adsorption Isotherm
--------------------------------------------------------

Herbert Freundlich, studying how much solute a solid adsorbs from
solution as a function of the solute's equilibrium concentration (or, for
gases, its partial pressure), found that a simple power law

.. math::

   q = K_f P^{1/n}

fits an enormous range of experimental adsorption data far better than
any theory then available could explain. Freundlich made no claim about
the underlying mechanism; the isotherm bearing his name is deliberately
empirical, and remains useful today precisely because it is a good
approximate description of adsorption on a genuinely *heterogeneous*
surface -- one with a spread of different site binding energies, rather
than the single uniform energy Langmuir would assume just over a decade
later. Its price is that, having no saturation limit of its own, it
should not be extrapolated to pressures far outside the range it was
fitted to, where any real surface eventually saturates.

*Implementation:*
:func:`~chemistrykit.surface.freundlich_loading` and
:class:`~chemistrykit.surface.systems.freundlich.FreundlichIsotherm`
implement exactly this power law;
:func:`~chemistrykit.surface.fit_freundlich` recovers
:math:`(K_f, n)` from data via the standard logarithmic linearization,
:math:`\ln q = \ln K_f + (1/n)\ln P`, reusing the shared
``chemistrykit.surface.utils.regression.linear_fit`` ordinary-least-
squares routine that every isotherm linearization in this package builds
on.

*References:* H. Freundlich, "Ueber die Adsorption in Loesungen," *Z.
Phys. Chem.* 57 (1906), 385-470.

.. minigallery:: ../../examples/surface/freundlich/plot_01_freundlich_isotherm.py

1914 -- 1947 -- Polanyi's Potential Theory and the Dubinin-Radushkevich Isotherm
-----------------------------------------------------------------------------------

Michael Polanyi proposed in 1914 a picture of adsorption quite unlike
Langmuir's sites. The solid surrounds itself with a field of attraction,
and gas within it is compressed into a dense, liquid-like adsorbed
layer. He measured the strength of that field at any point by the
*adsorption potential*

.. math::

   A = RT\ln\frac{P_0}{P},

the work needed to compress vapor from its equilibrium pressure
:math:`P` to its saturation pressure :math:`P_0`. His central claim was
that the volume :math:`W` of adsorbed liquid depends on :math:`A` alone,
through a *characteristic curve* :math:`W(A)` that is the same at every
temperature. Isotherms measured at different temperatures should
therefore collapse onto one curve when replotted against :math:`A`.
The theory was eclipsed for decades by Langmuir's, but Mikhail Dubinin
revived it for the microporous activated carbons used in gas masks and
gas separation. In 1947 he and Leonid Radushkevich gave the
characteristic curve an explicit form for pore filling,

.. math::

   W = W_0\exp\!\left[-\left(\frac{A}{E}\right)^2\right],

with :math:`W_0` the micropore volume and :math:`E` a characteristic
energy. It is still a standard tool for characterizing microporous
adsorbents.

*Implementation:* :func:`~chemistrykit.surface.polanyi_potential`
computes :math:`A`;
:func:`~chemistrykit.surface.dubinin_radushkevich_loading` and
:class:`~chemistrykit.surface.DubininRadushkevichIsotherm` implement the
Dubinin-Radushkevich isotherm; and
:func:`~chemistrykit.surface.fit_dubinin_radushkevich` recovers
:math:`(W_0, E)` from the linearization
:math:`\ln W = \ln W_0 - A^2/E^2`, using the same shared least-squares
routine as the other isotherm fits.

*References:* M. Polanyi, "Adsorption von Gasen (Daempfen) durch ein
festes nichtfluechtiges Adsorbens," *Verh. Dtsch. Phys. Ges.* 16 (1914),
1012-1016; M. M. Dubinin and L. V. Radushkevich, "Equation of the
Characteristic Curve of Activated Charcoal," *Proc. Acad. Sci. USSR,
Phys. Chem. Sect.* 55 (1947), 331-333; S. J. Gregg and K. S. W. Sing,
*Adsorption, Surface Area and Porosity*, 2nd ed. (London: Academic
Press, 1982), Ch. 4.

.. minigallery:: ../../examples/surface/dubinin/plot_01_polanyi_dubinin_characteristic_curve.py

1916 -- 1918 -- Langmuir's Kinetic Theory of the Monolayer
------------------------------------------------------------

Irving Langmuir, working at the General Electric Research Laboratory on
the behavior of gases inside incandescent light bulbs, proposed a
genuinely mechanistic alternative to Freundlich's empirical curve: a
fixed population of identical, independent surface sites, each capable of
holding at most one adsorbate molecule, in dynamic equilibrium between
adsorption (rate proportional to the pressure and the fraction of empty
sites) and desorption (rate proportional to the fraction of occupied
sites). Setting these two rates equal gives the fractional coverage

.. math::

   \theta(P) = \frac{KP}{1+KP}

which -- unlike Freundlich's isotherm -- saturates at exactly
:math:`\theta = 1` as :math:`P \to \infty`, a true monolayer limit
following directly from the model's own assumptions rather than fitted
after the fact. Langmuir laid out the monolayer concept in a 1916 paper
and gave the full kinetic derivation and its systematic experimental
verification (on mica, glass, and platinum) in 1918; the resulting body
of work on adsorbed films and surface chemistry more broadly earned him
the 1932 Nobel Prize in Chemistry, "for his discoveries and
investigations in surface chemistry" -- and, unusually for an industrial
research chemist of his era, made him the first American industrial
scientist to receive a Nobel Prize in the sciences.

*Implementation:*
:func:`~chemistrykit.surface.langmuir_coverage`
implements exactly this coverage law, and
:class:`~chemistrykit.surface.systems.langmuir.LangmuirIsotherm` wraps it
with a monolayer capacity `qmax` to give the loading
:math:`q(P)=q_{max}\theta(P)`; its
:meth:`~chemistrykit.surface.LangmuirIsotherm.half_saturation_pressure`
returns the pressure :math:`P=1/K` at which coverage is exactly one-half
-- the defining, exactly solvable feature Langmuir's model has and
Freundlich's does not.
:func:`~chemistrykit.surface.fit_langmuir` recovers
:math:`(K, q_{max})` from data via the standard :math:`1/q`-vs-:math:`1/P`
linearization.

*References:* I. Langmuir, "The Constitution and Fundamental Properties
of Solids and Liquids. Part I. Solids," *J. Am. Chem. Soc.* 38 (1916),
2221-2295; I. Langmuir, "The Adsorption of Gases on Plane Surfaces of
Glass, Mica and Platinum," *J. Am. Chem. Soc.* 40 (1918), 1361-1403;
"The Nobel Prize in Chemistry 1932," NobelPrize.org.

.. minigallery:: ../../examples/surface/langmuir/plot_01_langmuir_isotherm.py

1920s -- Hinshelwood and the Langmuir-Hinshelwood Mechanism
---------------------------------------------------------------

Langmuir's own 1922 kinetic study of platinum-catalyzed carbon monoxide
and hydrogen oxidation showed that a surface reaction's rate should
depend not on the reactant's pressure directly, but on its *Langmuir
coverage* -- first order in pressure while the surface is nearly bare,
but leveling off to zero order once the surface saturates, a qualitative
signature no simple gas-phase elementary reaction shows. Cyril
Hinshelwood, through an extensive program of gas-phase reaction-kinetics
measurements at Oxford across the 1920s (collected in his standard
reference work *The Kinetics of Chemical Change in Gaseous Systems*,
first published in 1926), independently developed and thoroughly tested
the same coverage-dependent picture, including its extension to two
reactants competing for the same pool of sites -- the mechanism that now
carries both their names. Hinshelwood's broader body of work on the
kinetics of gas-phase and surface reactions was recognized, together
with Nikolay Semenov's parallel work on chain reactions, with the 1956
Nobel Prize in Chemistry, awarded jointly "for their researches into the
mechanism of chemical reactions."

The dual-site (competitive) case has a characteristic and
counterintuitive feature that the single-site case lacks: for fixed
pressure of one reactant, the rate is *not* monotonic in the other's
pressure, since each reactant's rise in coverage necessarily crowds the
other off a shared pool of sites -- exactly the mechanism behind the
sharp poisoning of a catalyst's activity by an excess of one reagent, or
a strongly binding contaminant.

*Implementation:*
:func:`~chemistrykit.surface.lh_rate_single_site`
implements the single-reactant rate law
:math:`\text{rate}=k\theta_A=kK_AP_A/(1+K_AP_A)`, built directly on
:func:`~chemistrykit.surface.langmuir_coverage` rather
than re-deriving the coverage expression, per this module's own stated
convention; it visibly interpolates between first order (low pressure)
and zero order (saturated surface) in :math:`P_A`.
:func:`~chemistrykit.surface.lh_rate_dual_site`
implements the competitive two-reactant case,
:math:`\text{rate}=k\theta_A\theta_B` with each :math:`\theta` reduced by
the *other* species' occupancy of the shared site pool, and reproduces
the characteristic non-monotonic rate-vs-pressure behavior directly.

*References:* I. Langmuir, "The Mechanism of the Catalytic Action of
Platinum in the Reactions :math:`2CO+O_2=2CO_2` and
:math:`2H_2+O_2=2H_2O`," *Trans. Faraday Soc.* 17 (1922), 621-654; C. N.
Hinshelwood, *The Kinetics of Chemical Change in Gaseous Systems*
(Oxford: Clarendon Press, 1926; subsequent editions through the 1940s
incorporated the accumulating surface-kinetics work); "The Nobel Prize
in Chemistry 1956," NobelPrize.org.

.. minigallery:: ../../examples/surface/langmuir_hinshelwood/plot_01_lh_kinetics.py

1925 -- Constable and the Compensation Effect
--------------------------------------------------

Studying a series of related catalytic decompositions over different
but chemically similar surfaces, Frank Constable found a strikingly
regular pattern: across the series, a catalyst with a *higher* apparent
activation energy :math:`E_a` also had a systematically *higher*
pre-exponential (frequency) factor `A` in the Arrhenius equation, the two
varying together in a way that partly cancels out in the rate constant
itself -- a "compensation" between the two Arrhenius parameters that
recurs across an enormous range of catalytic and non-catalytic series and
remains, a century later, only partly understood (real physical
compensation from entropy-enthalpy tradeoffs in the transition state, in
some cases; a statistical artifact of how :math:`E_a` and `A` are
extracted from a limited experimental temperature range, in others).
Constable's own explanation, that surface-site heterogeneity within a
catalyst couples the two parameters, was an early attempt to connect the
mechanistic detail of a real, non-uniform catalytic surface to a directly
measurable kinetic regularity.

*Connection:*
:func:`~chemistrykit.surface.compare_catalyzed_rate`
takes the two Arrhenius parameters for the catalyzed and uncatalyzed
pathways as independent inputs -- `A_catalyzed` defaults to
`A_uncatalyzed` only as a simplifying assumption the function's own
docstring flags explicitly, not a physical necessity -- so a
Constable-style compensating change in both :math:`E_a` and `A` across a
catalyst series is directly expressible by supplying different values of
each, though this package does not implement a dedicated
compensation-effect fitting routine of its own.

*References:* F. H. Constable, "The Mechanism of Catalytic
Decomposition," *Proc. R. Soc. A* 108 (1925), 355-378 (exact page range
as commonly cited in secondary literature; not independently
re-verified against the original volume).

.. minigallery:: ../../examples/surface/catalysis/plot_04_constable_compensation.py

1929 -- 2004 -- Balandin, Sabatier's Principle, and the Volcano Curve
-------------------------------------------------------------------------

Alexey Balandin's multiplet theory, published beginning in 1929, was the
first attempt to give Sabatier's qualitative binding-strength principle a
structural, quantitative footing: modeling a catalytic reaction as
requiring a specific geometric arrangement ("multiplet") of active
surface atoms matched to the reacting molecule's own geometry, Balandin
argued that catalytic activity plotted against a measure of binding
strength across a series of different catalysts should rise, peak, and
fall again -- a "volcano curve" with the most active catalysts sitting
partway up either side, never at either extreme. Balandin's own
geometric multiplet mechanism did not survive as a literal picture of
how catalysis works, but the volcano-curve *shape* it predicted did, and
turned up repeatedly and robustly in later experimental catalytic-activity
data across many different reaction families.

The modern, quantitative version of this idea combines two later pieces
of theory: the Bronsted-Evans-Polanyi relation, an empirical linear
correlation between a reaction step's activation energy and its
thermodynamic driving force within a family of related reactions, and
density-functional-theory calculations of adsorption energies on real
catalyst surfaces. Bligaard, Norskov, and coworkers showed in 2004 how
combining the two reproduces the volcano curve as a direct mathematical
consequence, rather than a qualitative rule of thumb -- putting Sabatier's
century-old principle, at last, on the same quantitative footing as any
other testable theory of reaction rates, and turning volcano-curve
reasoning into a standard, practical *catalyst-design* tool in modern
computational chemistry, rather than merely a retrospective explanation
of why some catalysts already found by trial and error happen to work
well.

*Implementation:* this package has no dedicated multiplet-theory or
Broensted-Evans-Polanyi module -- both require geometric and
electronic-structure detail well beyond
:mod:`chemistrykit.surface`'s scope -- but the qualitative volcano
*shape* itself is reproduced directly from
:func:`~chemistrykit.surface.langmuir_coverage`: modeling
a two-step surface mechanism whose rate needs both an occupied site (to
hold the reacting intermediate) and an empty site (for the next step)
gives a toy rate :math:`\propto\theta(1-\theta)`, maximized exactly at
Langmuir's own half-saturation point :math:`\theta=1/2`
(:math:`K=1/P`) -- too little binding starves the surface of the
intermediate, too much starves it of empty sites, and only an
intermediate binding strength maximizes the rate, exactly as Sabatier's
and Balandin's qualitative principle requires.

*References:* A. A. Balandin, "Zur Multiplett-Theorie der Katalyse," *Z.
Phys. Chem.* B2 (1929), 289-316 (exact page range as commonly cited in
secondary literature; not independently re-verified against the original
volume); T. Bligaard, J. K. Norskov, S. Dahl, J. Matthiesen, C. H.
Christensen, and J. Sehested, "The Broensted-Evans-Polanyi Relation and
the Volcano Curve in Heterogeneous Catalysis," *J. Catal.* 224 (2004),
206-217.

.. minigallery:: ../../examples/surface/catalysis/plot_05_balandin_volcano.py

1937 -- 1938 -- Brunauer, Emmett, and Teller: The BET Isotherm
-------------------------------------------------------------------

Stephen Brunauer and Paul Emmett, working on the surface properties of
the iron catalysts used in ammonia synthesis, showed in 1937 that
measuring a full low-temperature adsorption isotherm of an inert gas
(rather than relying on the single-point estimates then in common use)
gave a far more reliable determination of a catalyst's true surface
area. Generalizing that method into a complete theory the following
year with Edward Teller, they extended Langmuir's single-layer picture
to allow genuine multilayer adsorption: once a site is occupied, a
second adsorbate molecule can condense on top of the first with
liquefaction-like energetics, a third on top of that, and so on up to the
adsorbate's saturation vapor pressure :math:`P_0`, at which point the
adsorbed amount formally diverges into bulk condensation:

.. math::

    \frac{V}{V_m} = \frac{Cx}{(1-x)(1-x+Cx)}, \qquad x \equiv \frac{P}{P_0}

The resulting BET method for computing a solid's specific surface area
from a measured isotherm became, almost immediately, the standard
technique across catalysis and materials science, and remains so today
-- most modern commercial surface-area analyzers still report a "BET
surface area" as their headline number.

Emmett's own career took an unrelated wartime turn five years later:
during World War II he worked on the Manhattan Project under Harold Urey
at Columbia University, developing barrier materials for the
gaseous-diffusion separation of uranium-235 from uranium-238, before
returning to civilian catalysis research at the Mellon Institute in late
1944.

*Implementation:* :func:`~chemistrykit.surface.bet_loading`
and :class:`~chemistrykit.surface.systems.bet.BETIsotherm` implement
exactly this multilayer isotherm;
:func:`~chemistrykit.surface.fit_bet` recovers
:math:`(V_m, C)` from data via the standard linearization,
:math:`x/[V(1-x)] = 1/(V_mC) + [(C-1)/(V_mC)]x`. As the module docstring
for ``chemistrykit.surface.systems.bet`` notes and
``chemistrykit.surface.tests.test_bet`` verifies numerically, BET
reduces exactly to :func:`~chemistrykit.surface.langmuir_coverage`
in the limit :math:`P_0\to\infty` at fixed :math:`K=C/P_0` -- the limit in
which the vapor never approaches saturation, so multilayer condensation
never has a chance to set in and only the monolayer term survives,
directly connecting BET back to the 1918 entry above.

*References:* S. Brunauer and P. H. Emmett, "The Use of Low Temperature
van der Waals Adsorption Isotherms in Determining the Surface Areas of
Iron Synthetic Ammonia Catalysts," *J. Am. Chem. Soc.* 59 (1937),
2682-2689 (exact page range as commonly cited in secondary literature;
not independently re-verified); S. Brunauer, P. H. Emmett, and E.
Teller, "Adsorption of Gases in Multimolecular Layers," *J. Am. Chem.
Soc.* 60 (1938), 309-319; W. S. Koski, "Paul Hugh Emmett," Biographical
Memoirs (National Academy of Sciences) 67 (1995) (Manhattan Project
service).

.. minigallery:: ../../examples/surface/bet/plot_01_bet_isotherm.py

1940 -- Temkin, Pyzhev, and the Temkin Isotherm
---------------------------------------------------

Working on the kinetics of ammonia synthesis over promoted iron
catalysts, the process at the heart of the Haber-Bosch industry, Mikhail
Temkin and V. Pyzhev needed an isotherm for nitrogen on a surface that
was plainly not uniform. They assumed that the heat of adsorption falls
*linearly* as the surface fills, rather than staying constant as
Langmuir had assumed. This is the same as a surface carrying a uniform
spread of site energies. Averaging Langmuir's coverage over such a
spread gives an isotherm that is logarithmic in pressure over a wide
middle range of coverage,

.. math::

   \theta \approx \frac{1}{f}\ln(K_{max}P),

where :math:`fRT` is the width of the energy spread. The same assumption
gave them the Temkin-Pyzhev rate law for ammonia synthesis, which for
decades was the standard kinetic model for designing ammonia
converters.

*Implementation:*
:func:`~chemistrykit.surface.uniform_energy_coverage` gives the exact
Langmuir coverage averaged over a uniform spread of site energies, which
reduces to the Temkin logarithm at intermediate coverage;
:func:`~chemistrykit.surface.temkin_loading` and
:class:`~chemistrykit.surface.TemkinIsotherm` implement the loading form
:math:`q = (RT/b_T)\ln(A_TP)`; and
:func:`~chemistrykit.surface.fit_temkin` recovers :math:`(A_T, b_T)` from
the straight line of :math:`q` against :math:`\ln P`.

*References:* M. I. Temkin and V. Pyzhev, "Kinetics of Ammonia Synthesis
on Promoted Iron Catalysts," *Acta Physicochim. URSS* 12 (1940),
327-356; P. W. Atkins and J. de Paula, *Physical Chemistry*, 11th ed.
(Oxford University Press, 2018).

.. minigallery:: ../../examples/surface/temkin/plot_01_temkin_isotherm.py

1940 -- Eley, Rideal, and the Eley-Rideal Mechanism
-------------------------------------------------------

Studying the conversion of para-hydrogen to ortho-hydrogen on tungsten,
Daniel Eley and Eric Rideal proposed a surface mechanism different from
Langmuir and Hinshelwood's. In their scheme only one reactant is
adsorbed, and the other reacts with it by striking it *directly from the
gas phase*, without first adsorbing. The rate is then

.. math::

   \text{rate} = k\,\theta_AP_B = \frac{kK_AP_AP_B}{1+K_AP_A}

Because the two reactants never compete for the same sites, this rate
rises steadily with the pressure of the adsorbed reactant and levels
off, and it stays first order in the gas-phase reactant at every
pressure. The competitive Langmuir-Hinshelwood rate instead passes
through a maximum. Comparing measured rate laws against these two shapes
became the standard first test of a surface mechanism. Later
molecular-beam experiments showed that genuine Eley-Rideal reactions are
fairly rare, with most surface reactions following the
Langmuir-Hinshelwood route.

*Implementation:* :func:`~chemistrykit.surface.er_rate` implements
:math:`k\theta_AP_B`, built on
:func:`~chemistrykit.surface.langmuir_coverage` just as
:func:`~chemistrykit.surface.lh_rate_dual_site` is, so the two mechanisms
can be compared directly.

*References:* D. D. Eley and E. K. Rideal, "Parahydrogen Conversion on
Tungsten," *Nature* 146 (1940), 401-402; P. W. Atkins and J. de Paula,
*Physical Chemistry*, 11th ed. (Oxford University Press, 2018).

.. minigallery:: ../../examples/surface/eley_rideal/plot_01_eley_rideal_mechanism.py

1962 -- Redhead and Temperature-Programmed Desorption
--------------------------------------------------------

As ultrahigh-vacuum techniques matured, Paul Redhead at the National
Research Council of Canada developed *flash* or *temperature-programmed*
desorption into a quantitative method. A surface covered with adsorbate
is heated at a steady rate :math:`\beta`, and the gas it releases is
recorded; each distinct binding state shows up as a peak. The
desorption rate follows the Polanyi-Wigner equation,
:math:`-d\theta/dT = (\nu/\beta)\,\theta^n e^{-E_d/RT}`. For first-order
desorption Redhead showed that the peak temperature :math:`T_p` alone
fixes the desorption energy, to a good approximation

.. math::

   E_d = RT_p\left[\ln\frac{\nu T_p}{\beta} - 3.64\right].

He also showed how the peak shape and its shift with coverage reveal the
desorption order: first-order peaks stay put as the initial coverage
changes, while second-order peaks move to lower temperature as coverage
rises. TPD became one of the most widely used tools of surface science
for measuring how strongly molecules bind to surfaces.

*Implementation:* :func:`~chemistrykit.surface.simulate_tpd` integrates
the Polanyi-Wigner equation for first- and second-order desorption and
returns a :class:`~chemistrykit.surface.TPDResult`;
:func:`~chemistrykit.surface.first_order_peak_temperature` solves the
exact first-order peak condition
:math:`E_d/(RT_p^2) = (\nu/\beta)e^{-E_d/RT_p}`; and
:func:`~chemistrykit.surface.redhead_desorption_energy` implements
Redhead's formula.

*References:* P. A. Redhead, "Thermal Desorption of Gases," *Vacuum* 12
(1962), 203-211; R. I. Masel, *Principles of Adsorption and Reaction on
Solid Surfaces* (New York: Wiley, 1996), Ch. 7.

.. minigallery:: ../../examples/surface/tpd/plot_01_redhead_tpd.py

1968 -- 1995 -- Boudart and Turnover Rates in Heterogeneous Catalysis
---------------------------------------------------------------------

Borrowing a concept from enzymology -- where an enzyme's "turnover
number" had long measured how many substrate molecules a single active
site converts per unit time -- Michel Boudart argued in his influential
1968 text *Kinetics of Chemical Processes* that heterogeneous catalysis
needed the same intrinsic, per-site measure of activity, rather than a
rate normalized only to the total mass or volume of catalyst used, which
conflates genuine catalytic activity with how much active surface a
particular sample happens to expose. Boudart's turnover frequency --
rate of product formation per active site per unit time -- let
chemically and structurally very different catalysts be compared on a
level footing, and remains the standard intrinsic-activity metric in
catalysis research; his later retrospective in *Chemical Reviews* traced
the concept's adoption and the persistent practical difficulty of
actually *counting* active sites accurately enough to compute it with
confidence.

*Implementation:*
:func:`~chemistrykit.surface.turnover_frequency`
implements exactly this per-site rate,
:math:`\text{TOF}=\text{rate}/[\text{active sites}]`, while
:func:`~chemistrykit.surface.turnover_number` implements
the complementary, dimensionless *durability* measure -- total catalytic
cycles performed per site over the lifetime of a reaction, rather than
an instantaneous rate -- returned together with the underlying
Arrhenius-based rate comparison in
:class:`~chemistrykit.surface.systems.catalysis.CatalyticRateComparison`.

*References:* M. Boudart, *Kinetics of Chemical Processes* (Englewood
Cliffs, NJ: Prentice-Hall, 1968); M. Boudart, "Turnover Rates in
Heterogeneous Catalysis," *Chem. Rev.* 95 (1995), 661-666.

.. minigallery:: ../../examples/surface/catalysis/plot_06_boudart_turnover_frequency.py

See Also
--------

- :doc:`/api/surface`
- :doc:`/history/polymer_breakthroughs`