Breakthroughs in Chemical Thermodynamics
=========================================


.. include:: /_generated/nav/thermo.rst

.. epigraph::

   "Die Energie der Welt ist constant. Die Entropie der Welt strebt
   einem Maximum zu." ("The energy of the universe is constant. The
   entropy of the universe tends to a maximum.") -- Rudolf Clausius,
   "Über verschiedene für die Anwendung bequeme Formen der
   Hauptgleichungen der mechanischen Wärmetheorie," Annalen der Physik
   und Chemie 125, 353-400 (1865), closing lines.

Chemical thermodynamics is the nineteenth century's answer to a
deceptively simple question: which way does a chemical system go, and
how far? Not how fast -- that is kinetics's question -- but toward what
final state, and at what cost or benefit in heat, work, and the
irreducible statistical tendency Clausius named entropy that same year.
The systems in :mod:`chemistrykit.thermo` retrace that project from the
first quantitative gas and solution laws through Gibbs's still-
unsurpassed general framework of chemical equilibrium, to the modern
engineering equations of state real gases actually obey. 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

1803 -- Henry's Law of Gas Solubility
-----------------------------------------

William Henry, studying how much of a gas dissolves in a liquid at
equilibrium, found a strikingly simple proportionality: the amount of gas
absorbed is directly proportional to the pressure of that gas above the
liquid, at a fixed temperature.

.. math::

   P_B = x_B\,K_H

Unlike Raoult's law below, Henry's constant :math:`K_H` is not the pure
solute's own vapor pressure -- it is an empirical, solute-and-solvent-
specific number, because a dilute solute molecule surrounded entirely by
solvent experiences a completely different local environment than it
would in its own pure liquid. Henry's law and Raoult's law turn out to be
two different limiting laws of the very same real solution, exact in
opposite composition extremes (dilute solute versus nearly pure solute)
rather than competing descriptions of the same regime -- a picture only
fully appreciated once G. N. Lewis's early-twentieth-century activity/
fugacity framework explained why both idealizations, despite disagreeing
everywhere in between, are each exactly right at their own end of the
composition range.

*Implementation:* :func:`~chemistrykit.thermo.henry_law_pressure`
implements exactly this proportionality; the example below builds an
illustrative real (non-ideal) partial-pressure curve, using
:func:`~chemistrykit.thermo.raoult_vapor_pressure` for
the other limit, and shows it hugging the Henry's-law line at the dilute
end and the Raoult's-law line at the nearly-pure end of the same
composition range.

*References:* W. Henry, "Experiments on the Quantity of Gases Absorbed
by Water, at Different Temperatures, and under Different Pressures,"
Philos. Trans. R. Soc. 93, 29-274 (1803).

.. minigallery:: ../../examples/thermo/mixtures/plot_02_henry_law.py

1834 -- 1850 -- Clapeyron, Clausius, and the Ideal Gas and Vapor-Pressure Laws
------------------------------------------------------------------------------------

Benoit Paul Emile Clapeyron combined the empirical gas laws accumulated
piecemeal over the preceding century and a half -- Boyle's inverse
pressure-volume relation (1662), Gay-Lussac's and Charles's linear
temperature dependence (published 1802, with Charles's own unpublished
observations dating to the 1780s), and Avogadro's hypothesis (1811) that
equal volumes of gas at the same temperature and pressure contain equal
numbers of molecules -- into the single combined ideal gas law still
taught today.

.. math::

   PV_m = RT

The same 1834 memoir, reformulating Sadi Carnot's largely unnoticed 1824
theory of heat engines in the newer language of calculus, also derived
(via Carnot's caloric-theory reasoning, since the mechanical theory of
heat did not yet exist) an equation relating a phase boundary's slope to
the associated latent heat -- the relation now named for both Clapeyron
and Rudolf Clausius, who rederived it in 1850 on the rigorous footing of
the (by then fully mechanical) second law of thermodynamics, free of
Clapeyron's caloric assumptions, and specialized it -- assuming the vapor
behaves ideally and the condensed phase's volume is negligible -- to the
simpler exponential form actually used to predict a liquid's vapor
pressure at temperatures away from any single measured reference point:

.. math::

   \ln\frac{P(T)}{P_{ref}} = -\frac{\Delta H_{vap}}{R}
       \left(\frac{1}{T} - \frac{1}{T_{ref}}\right)

*Implementation:* :class:`chemistrykit.thermo.systems.equations_of_state.IdealGas`
implements Clapeyron's combined gas law directly (its ``molar_volume``
recovers the textbook 22.4 L/mol at standard temperature and pressure
exactly); :class:`chemistrykit.thermo.systems.phase_equilibria.ClausiusClapeyron`
implements the integrated Clausius-Clapeyron vapor-pressure equation, with
:meth:`~chemistrykit.thermo.ClausiusClapeyron.boiling_point`
inverting it to find the temperature at which a liquid boils under any
given pressure, and
:meth:`~chemistrykit.thermo.ClausiusClapeyron.from_two_points`
extracting an unknown enthalpy of vaporization from two measured
points on the boundary.

*References:* B. P. E. Clapeyron, "Mémoire sur la puissance motrice de la
chaleur," Journal de l'École Polytechnique 14, 153-190 (1834); R.
Clausius, "Über die bewegende Kraft der Wärme," Ann. Phys. Chem. 79,
368-397 and 500-524 (1850).

.. minigallery::
   ../../examples/thermo/equations_of_state/plot_01_ideal_gas_law.py
   ../../examples/thermo/phase_equilibria/plot_01_clausius_clapeyron.py

1840 -- Hess's Law of Constant Heat Summation
---------------------------------------------

Germain Henri Hess, a Swiss-born chemist working in St. Petersburg,
measured the heat released when sulfuric acid was diluted with water in
one step or in several, and when various bases were neutralized by
different routes. He found that the total heat was always the same:
the heat of a chemical change depends only on where it starts and where
it ends, not on the intermediate steps. His law predates the first law
of thermodynamics by several years, but it is exactly what the first law
later explained -- enthalpy is a state function, so the enthalpy changes
of any set of steps that add up to a reaction also add up:

.. math::

   \Delta H_{\text{overall}} = \sum_k c_k\,\Delta H_k

The law made thermochemistry a bookkeeping science. Enthalpies of
reactions that are hard or impossible to carry out cleanly -- such as
burning graphite only as far as carbon monoxide -- can be found from
reactions that are easy to measure, and a single table of standard
enthalpies of formation gives the enthalpy of any reaction:
:math:`\Delta_rH^\circ = \sum_i \nu_i\,\Delta_fH_i^\circ`.

*Implementation:* :func:`~chemistrykit.thermo.hess_law_enthalpy` sums a
weighted combination of step enthalpies (a negative weight reverses a
step), and :func:`~chemistrykit.thermo.reaction_enthalpy_from_formation`
applies the law to formation enthalpies. The tests check the two-step
combustion of graphite (-110.5 and -283.0 kJ/mol adding up to
-393.5 kJ/mol) and that both functions agree for random reaction
combinations.

*References:* G. H. Hess, "Thermochemische Untersuchungen," Ann. Phys.
Chem. 50, 385-404 (1840).

.. minigallery:: ../../examples/thermo/thermochemistry/plot_01_hess_law.py

1864 -- Guldberg and Waage's Law of Mass Action
-----------------------------------------------------

Cato Maximilian Guldberg and Peter Waage, working in Christiania (now
Oslo), proposed that a reaction's rate -- and hence, at equilibrium, the
balance between its forward and reverse rates -- depends on the reacting
substances' "active masses" (essentially their concentrations), raised to
powers reflecting how many molecules of each participate. Announced only
in Norwegian, in a venue with essentially no international readership,
the 1864 paper went unnoticed outside Scandinavia for years; a French
translation in 1867 fared little better. Guldberg and Waage's original
derivation also rested on a kinetic argument that was not fully
rigorous, and in 1879 the two authors themselves published a revised,
more careful treatment -- reframing the law explicitly as a *dynamic*
equilibrium between opposing forward and reverse reaction rates, rather
than the more static force-balance language of the 1864 original. In its
modern equilibrium form, the law states that the reaction quotient
built from the products' and reactants' activities, each raised to its
stoichiometric coefficient, equals a constant, the equilibrium constant
`K`, once the reaction has stopped proceeding net in either direction:

.. math::

   Q = \prod_i a_i^{\nu_i} \overset{\text{equilibrium}}{=} K

*Implementation:* :func:`~chemistrykit.thermo.reaction_quotient`
computes exactly this product directly from a set of activities and
signed stoichiometric coefficients; :func:`~chemistrykit.thermo.kp_from_kc`
and :func:`~chemistrykit.thermo.kc_from_kp` convert
between the pressure- and concentration-based forms of the same
equilibrium constant. The Gibbs-energy-minimization approach implemented
by :func:`~chemistrykit.thermo.solve_equilibrium_composition`
(see 1876, below) is the modern generalization of the same equilibrium
condition -- at its minimum, :math:`dG/d\xi_j = \sum_i \nu_{ij}\mu_i = 0`
for every reaction `j`, which is exactly :math:`Q_j = K_j` reached by a
different numerical route -- and the example below verifies this equality
directly, confirming that :func:`~chemistrykit.thermo.reaction_quotient`, evaluated at the
solver's own converged equilibrium composition, reproduces the target
equilibrium constant.

*References:* C. M. Guldberg and P. Waage, "Studier over Affiniteten,"
Forhandlinger: Videnskabs-Selskabet i Christiania (1864), 35; French
translation, "Études sur les affinités chimiques," (Christiania:
Brøgger & Christie, 1867); revised treatment: C. M. Guldberg and P.
Waage, J. Prakt. Chem. 19, 69-114 (1879).

.. minigallery:: ../../examples/thermo/equilibrium/plot_01_law_of_mass_action.py

1869 -- 1873 -- Andrews, van der Waals, and the Continuity of Liquid and Gas
------------------------------------------------------------------------------------

Thomas Andrews's painstaking experiments compressing carbon dioxide at
closely spaced temperatures established, in his 1869 Bakerian Lecture,
that liquid and gas are not two fundamentally different states of matter
at all: above a sharply defined critical temperature, no amount of
pressure can produce a distinct liquid phase, and a substance can be
taken continuously from unambiguously gaseous to unambiguously liquid
conditions by a path that never crosses a phase boundary, simply by
going around the critical point. Johannes Diderik van der Waals, in his
1873 Leiden doctoral thesis -- titled, in direct homage to Andrews,
*Over de Continuiteit van den Gas- en Vloeistoftoestand* ("On the
Continuity of the Gas and Liquid State") -- supplied the theory Andrews's
observation demanded: a single equation of state, modifying the ideal
gas law with just two extra terms for intermolecular attraction (`a`) and
finite molecular volume (`b`), that describes liquid and gas as the same
underlying equation's two possible branches rather than two separate
laws stitched together.

.. math::

   P = \frac{RT}{V_m - b} - \frac{a}{V_m^2}

Below the critical temperature the van der Waals cubic in :math:`V_m` has
three real positive roots -- liquid, unstable, and vapor branches -- and
at the critical temperature itself, exactly the triple root Andrews's
data demanded. Van der Waals received the 1910 Nobel Prize in Physics
for this equation.

*Implementation:* :class:`chemistrykit.thermo.systems.equations_of_state.VanDerWaals`
implements exactly this equation of state, solving its cubic for
:math:`V_m` via ``chemistrykit.thermo.utils.cubic_roots.real_positive_roots()``;
:meth:`~chemistrykit.thermo.VanDerWaals.from_critical_constants`
builds an instance directly from a substance's measured critical
temperature and pressure, and the resulting universal critical
compressibility factor :math:`Z_c = P_c V_c/(RT_c) = 3/8` -- independent
of which substance's constants are used -- is exactly the structural
signature of van der Waals's theory (real gases' actual :math:`Z_c`
values, typically 0.23-0.31, deviate from this universal prediction, one
of the equation's well-known quantitative limitations, addressed
empirically by Redlich-Kwong below).

*References:* T. Andrews, "On the Continuity of the Gaseous and Liquid
States of Matter," Philos. Trans. R. Soc. 159, 575-590 (1869); J. D. van
der Waals, *Over de Continuiteit van den Gas- en Vloeistoftoestand*
(doctoral thesis, Leiden, 1873).

.. minigallery:: ../../examples/thermo/equations_of_state/plot_02_van_der_waals_continuity.py

1876 -- 1878 -- Gibbs's Equilibrium of Heterogeneous Substances
-----------------------------------------------------------------

Josiah Willard Gibbs's "On the Equilibrium of Heterogeneous Substances,"
published in two parts across the *Transactions of the Connecticut
Academy of Arts and Sciences* -- a journal with essentially no
circulation among European chemists at the time -- laid out, in about a
hundred densely mathematical pages, most of the conceptual scaffolding
modern chemical thermodynamics still stands on: the chemical potential
:math:`\mu_i`, governing how a species' free energy changes as it is
added to a mixture or moves between phases; the free energy that bears
his name, whose minimization at fixed temperature and pressure is the
general condition for any chemical or phase equilibrium; and the phase
rule,

.. math::

   F = C - P + 2 - r

fixing how many independent intensive variables (temperature, pressure,
composition) can be varied at all while a given set of phases remains in
equilibrium. Precisely because it was so far ahead of contemporary
chemistry's mathematical fluency, and published in such an obscure
venue, Gibbs's work was largely unread in Europe for over a decade; only
after Wilhelm Ostwald's 1892 German translation did van't Hoff, Nernst,
and others recognize -- often to their own surprise -- that a problem
they were independently working out had already been solved, more
generally, by an American mathematical physicist working in relative
isolation at Yale.

*Implementation:* :func:`~chemistrykit.thermo.gibbs_phase_rule`
implements the phase rule directly; the ideal-gas chemical potential
:math:`\mu_i = \Delta G_{f,i}^\circ + RT\ln(x_i P/P^\circ)` used by
:func:`~chemistrykit.thermo.gibbs_energy_of_mixture`
and minimized by
:func:`~chemistrykit.thermo.solve_equilibrium_composition`
is exactly Gibbs's chemical-potential framework, applied here to solve
for a reacting mixture's equilibrium composition by direct free-energy
minimization rather than by solving :math:`Q=K` algebraically -- the
two routes coincide exactly (see 1864, above) because Gibbs's
equilibrium condition, :math:`\sum_i \nu_i \mu_i = 0` for every reaction,
*is* the mass-action law, derived from a more general variational
principle.

*References:* J. W. Gibbs, "On the Equilibrium of Heterogeneous
Substances," Trans. Conn. Acad. Arts Sci. 3, 108-248 (1876) and 343-524
(1878).

.. minigallery::
   ../../examples/thermo/equilibrium/plot_02_gibbs_energy_minimization.py
   ../../examples/thermo/phase_equilibria/plot_02_gibbs_phase_rule.py

1884 -- van't Hoff's Equation for the Temperature Dependence of K
-----------------------------------------------------------------------

Jacobus Henricus van't Hoff's *Études de Dynamique Chimique* -- the same
1884 book that gave chemical kinetics its concept of reaction order (see
:doc:`/history/kinetics_breakthroughs`) -- also supplied a companion
result for chemical equilibrium: combining the equilibrium condition
:math:`\Delta G^\circ=-RT\ln K` with the Gibbs-Helmholtz relation gives
an equilibrium constant's exact temperature dependence, the van't Hoff
isochore,

.. math::

   \frac{d\ln K}{dT} = \frac{\Delta H^\circ}{RT^2}

the same exponential-in-:math:`1/T` form Svante Arrhenius would, five
years later, put on a firmer molecular footing for *rate* constants
specifically (see :doc:`/history/kinetics_breakthroughs`). Assuming the
standard reaction enthalpy :math:`\Delta H^\circ` is constant over the
temperature range of interest, the isochore integrates to the same
two-point form already used above (1834-1850) for a phase boundary's
vapor pressure:

.. math::

   \ln\frac{K(T)}{K_{ref}} = -\frac{\Delta H^\circ}{R}
       \left(\frac{1}{T} - \frac{1}{T_{ref}}\right)

and a plot of :math:`\ln K` against :math:`1/T` -- the "van't Hoff plot,"
structurally identical to an Arrhenius plot -- linearizes the
relationship, letting :math:`\Delta H^\circ` and :math:`\Delta S^\circ`
be read off directly as its slope and intercept, exactly the strategy
Lineweaver and Burk would later apply to enzyme kinetics (see
:doc:`/history/kinetics_breakthroughs`).

*Implementation:* :func:`~chemistrykit.thermo.van_t_hoff_equilibrium_constant`
implements exactly the integrated two-point form -- used below to verify
Le Chatelier's qualitative temperature prediction directly;
:func:`~chemistrykit.thermo.fit_van_t_hoff`
implements the van't Hoff-plot linear regression itself, recovering
:math:`(\Delta H^\circ, \Delta S^\circ)` from synthetic noisy
equilibrium-constant-vs-temperature data and returning a
:class:`~chemistrykit.thermo.systems.equilibrium.VantHoffFit`, structurally
identical to :func:`chemistrykit.kinetics.fit_arrhenius`'s Arrhenius-plot
fit for rate constants (see :doc:`/history/kinetics_breakthroughs`).

*References:* J. H. van't Hoff, *Études de Dynamique Chimique* (Amsterdam:
Frederik Muller, 1884).

.. minigallery:: ../../examples/thermo/equilibrium/plot_03_van_t_hoff_plot.py

1884 -- Le Chatelier's Principle
------------------------------------

Henry Louis Le Chatelier proposed a single qualitative rule covering how
any system at equilibrium responds to being disturbed: it shifts in
whichever direction partially opposes the disturbance -- consuming heat
if heated, producing gas-phase moles if depressurized, consuming an
added reactant -- rather than amplifying it or staying put. Stated as a
free-standing postulate in 1884, the principle was refined and given a
more general thermodynamic-stability grounding a few years later by Karl
Ferdinand Braun (hence sometimes "Le Chatelier-Braun principle"); it is
now understood not as an independent law of nature but as a direct
consequence of the same equilibrium condition Gibbs had already
established -- a stable equilibrium sits at a genuine minimum of the free
energy, and a minimum, by definition, pushes back against any small
displacement away from it.

*Implementation:* :func:`~chemistrykit.thermo.van_t_hoff_equilibrium_constant`
demonstrates exactly Le Chatelier's temperature prediction: for an
endothermic reaction (positive :math:`\Delta H^\circ`), the equilibrium
constant provably increases as temperature rises, in the direction that
partially absorbs the added heat, and its docstring example verifies
this monotonic increase directly; the equilibrium-composition solver
:func:`~chemistrykit.thermo.solve_equilibrium_composition`
demonstrates the same qualitative shift-toward-relief behavior for
arbitrary reacting mixtures, since it is, by construction, always
sitting at the free-energy minimum Le Chatelier's principle describes
the response of.

*References:* H. Le Chatelier, "Sur un énoncé général des lois des
équilibres chimiques," C. R. Acad. Sci. 99, 786-789 (1884).

.. minigallery:: ../../examples/thermo/equilibrium/plot_04_le_chatelier.py

1886 -- van't Hoff's Osmotic Pressure Law and the Modern Theory of Solutions
----------------------------------------------------------------------------------

Jacobus Henricus van't Hoff, extending his 1884 kinetics work (see
:doc:`/history/kinetics_breakthroughs`) into the thermodynamics of
solutions, found that a dilute solution's osmotic pressure obeys an
equation formally *identical* to the ideal gas law -- a striking, and at
the time genuinely surprising, analogy between dissolved solute particles
and gas molecules that has no deeper physical necessity behind it (the
mechanism of osmosis, solvent flow through a semipermeable membrane, has
nothing literally to do with a gas's kinetic pressure), yet reproduces
the correct pressure to remarkable accuracy:

.. math::

   \Pi = iMRT

Van't Hoff's van't Hoff factor `i` -- the number of particles each solute
formula unit actually contributes in solution -- was itself indirect
evidence for electrolyte dissociation, since strong electrolytes like
NaCl were found to depress a solvent's freezing point and elevate osmotic
pressure by roughly twice the amount a non-dissociating solute of the
same molarity would, exactly as expected if each formula unit splits into
two independent ions in solution (a picture Svante Arrhenius would put on
a firmer theoretical footing the same decade with his theory of
electrolytic dissociation). Van't Hoff received the first Nobel Prize in
Chemistry in 1901, "in recognition of the extraordinary services he has
rendered by the discovery of the laws of chemical dynamics and osmotic
pressure in solutions."

*Implementation:* :func:`~chemistrykit.thermo.osmotic_pressure`
implements exactly this equation, and
:func:`~chemistrykit.thermo.freezing_point_depression`
and :func:`~chemistrykit.thermo.boiling_point_elevation`
implement the closely related colligative-property laws
:math:`\Delta T = iKb` sharing the same van't Hoff factor `i` convention,
using the tabulated solvent constants in
``CRYOSCOPIC_CONSTANTS``; the
example below computes all three colligative properties for an aqueous
NaCl solution with :math:`i=2`.

*References:* J. H. van't Hoff, "Die Rolle des osmotischen Druckes in der
Analogie zwischen Lösungen und Gasen," Z. Phys. Chem. 1, 481-508 (1887)
(the definitive German statement; first announced in Swedish the
previous year: K. Sven. Vetensk.-Akad. Handl. 21, No. 17 (1886)).

.. minigallery:: ../../examples/thermo/mixtures/plot_03_osmotic_pressure.py

1887 -- Raoult's Law of Vapor Pressure
-------------------------------------------

Francois-Marie Raoult, building on his own earlier cryoscopic
measurements of freezing-point depression through the early 1880s (the
experimental tool he had used, before van't Hoff's theory existed to
explain it, simply to determine unknown solutes' molecular weights),
turned to a mixture's vapor pressure directly and found an equally simple
proportionality: a component's partial vapor pressure above an ideal
mixture is just its mole fraction times its own pure-liquid vapor
pressure.

.. math::

   P_A = x_A\,P_A^*

Combined with Dalton's law of partial pressures across both components of
a binary mixture, Raoult's law predicts the full vapor-liquid equilibrium
curve for an ideal solution: the total vapor pressure as a function of
liquid composition, and -- since the more volatile component always
contributes disproportionately to the vapor -- the vapor's own,
systematically different composition. That vapor-enrichment effect is
the entire physical basis of fractional distillation, arguably chemistry's
single most economically important separation technique. Raoult's law
is, like van der Waals's equation above, an idealization: real mixtures
generally deviate from it except in the dilute-solvent limit, exactly the
composition regime in which Henry's law (1803, above) takes over
describing the *solute* instead.

*Implementation:* :func:`~chemistrykit.thermo.raoult_vapor_pressure`
implements this proportionality directly, and
:class:`~chemistrykit.thermo.systems.mixtures.BinaryIdealSolution`
combines it with Dalton's law for both components of a binary mixture,
via :meth:`~chemistrykit.thermo.BinaryIdealSolution.total_pressure`
and :meth:`~chemistrykit.thermo.BinaryIdealSolution.vapor_composition`,
to produce the full P-x-y vapor-liquid-equilibrium picture the example
below plots for an idealized benzene/toluene mixture.

*References:* F. M. Raoult, "Loi générale de la tension de vapeur des
dissolvants," C. R. Acad. Sci. 104, 1430-1433 (1887).

.. minigallery:: ../../examples/thermo/mixtures/plot_01_raoult_law_pxy.py

1895 -- Margules's Activity Coefficients for Non-Ideal Solutions
------------------------------------------------------------------

Raoult's law (1887, above) describes an ideal mixture, but most real
liquid mixtures deviate from it -- sometimes so strongly that the vapor
and liquid have the same composition at some point (an azeotrope), which
no ideal solution can do. Max Margules, a Viennese physicist and
meteorologist, showed how to describe these deviations while staying
consistent with thermodynamics. He wrote the logarithm of each
component's activity coefficient :math:`\gamma_i` as a power series in
mole fraction, constrained by the Gibbs-Duhem relation
:math:`x_1\,d\ln\gamma_1 + x_2\,d\ln\gamma_2 = 0`. Keeping only the first
term gives the one-parameter Margules model,

.. math::

   \ln\gamma_1 = A\,x_2^2, \qquad \ln\gamma_2 = A\,x_1^2, \qquad
   P_i = x_i\,\gamma_i\,P_i^*

with excess Gibbs energy :math:`G^E = RT A x_1 x_2`. A positive
:math:`A` raises the vapor pressure above Raoult's line, a negative one
lowers it, and a large enough :math:`A` of either sign produces an
azeotrope. The model also unites the two limiting laws above: each
component obeys Raoult's law as it becomes pure (:math:`\gamma_i \to 1`)
and Henry's law with :math:`K_H = P_i^* e^{A}` as it becomes dilute.
Margules's expansion is the ancestor of every activity-coefficient model
used in chemical engineering, from van Laar's to Wilson's and NRTL.

*Implementation:* :class:`~chemistrykit.thermo.MargulesSolution`
implements the one-parameter model, with activity coefficients, excess
Gibbs energy, partial and total pressures, vapor composition, and the
infinite-dilution Henry's-law constant. The tests check that
:math:`A = 0` reproduces
:class:`~chemistrykit.thermo.BinaryIdealSolution`, that the model
satisfies the Gibbs-Duhem relation, and that it tends to Henry's law at
infinite dilution.

*References:* M. Margules, "Über die Zusammensetzung der gesättigten
Dämpfe von Mischungen," Sitzungsber. Kais. Akad. Wiss. Wien,
Math.-Naturwiss. Kl., Abt. IIa, 104, 1243-1278 (1895).

.. minigallery:: ../../examples/thermo/mixtures/plot_04_margules_activity.py

1901 -- Lewis's Fugacity
------------------------

Gibbs's chemical potential (1876, above) has a simple form only for
ideal gases: :math:`\mu = \mu^\circ + RT\ln(P/P^\circ)`. Gilbert Newton
Lewis kept that simple form for every real substance by defining a new
quantity, the fugacity :math:`f` (from the Latin for "tendency to
escape"), to take the place of pressure:

.. math::

   \mu = \mu^\circ + RT\ln\frac{f}{P^\circ}, \qquad
   \lim_{P\to 0}\frac{f}{P} = 1

The ratio :math:`\phi = f/P`, the fugacity coefficient, measures how far
a real gas is from ideal, and it can be computed from any equation of
state. Lewis's definition also gives a practical test for phase
equilibrium: two phases of a substance coexist when their fugacities are
equal, which is how vapor pressures are now computed from equations of
state. Lewis later extended the same idea to solutions with the
activity (1907), and his 1923 textbook with Merle Randall,
*Thermodynamics and the Free Energy of Chemical Substances*, made both
concepts standard.

*Implementation:* :func:`~chemistrykit.thermo.fugacity_coefficient`
evaluates
:math:`\ln\phi = Z - 1 - \ln Z + \frac{1}{RT}\int_{V_m}^\infty (P - RT/V)\,dV`
by quadrature for any equation of state, and
:func:`~chemistrykit.thermo.fugacity` returns :math:`f = \phi P`.
:func:`~chemistrykit.thermo.saturation_pressure` applies Lewis's
equal-fugacity condition to a cubic equation of state to find the vapor
pressure. The tests compare against the closed-form van der Waals and
Redlich-Kwong fugacity coefficients and the known van der Waals
coexistence pressure :math:`P/P_c = 0.647` at :math:`T/T_c = 0.9`.

*References:* G. N. Lewis, "The Law of Physico-Chemical Change," Proc.
Am. Acad. Arts Sci. 37, 49-69 (1901); G. N. Lewis and M. Randall,
*Thermodynamics and the Free Energy of Chemical Substances* (New York:
McGraw-Hill, 1923).

.. minigallery:: ../../examples/thermo/equations_of_state/plot_05_lewis_fugacity.py

1906 -- Nernst's Heat Theorem and the Third Law
-----------------------------------------------------

Walther Nernst, investigating why calculated chemical affinities from
purely thermal (calorimetric) measurements kept disagreeing with directly
measured equilibrium constants at low temperature, proposed his "heat
theorem": as temperature approaches absolute zero, the entropy change of
any reaction between pure, perfectly crystalline condensed phases
approaches zero, :math:`\Delta S \to 0` as :math:`T \to 0`. Nernst's own
1906 statement was comparatively cautious -- a claim about *differences*
in entropy between reacting substances near absolute zero, not yet the
sweeping statement that every pure substance's entropy itself vanishes
there; it was Max Planck who, a few years later, generalized Nernst's
theorem into the stronger, now-standard third law -- :math:`S \to 0` as
:math:`T \to 0` for any perfect crystal -- a step Nernst himself accepted
only with some reservations, given how much further it reached beyond
what his own calorimetric evidence had actually shown. The third law's
practical payoff is enormous: it fixes an absolute zero-point for entropy
(and hence, combined with heat-capacity measurements, an absolute
standard entropy :math:`S^\circ` for any substance), which is exactly
what makes tabulated standard Gibbs energies of formation -- an absolute
reference scale rather than an arbitrary one -- meaningful quantities to
plug into an equilibrium calculation at all.

*Connection:* chemistrykit.thermo has no standalone third-law or
absolute-entropy calculation, but the standard Gibbs energies of
formation (``gibbs_formation``) that
:func:`~chemistrykit.thermo.gibbs_energy_of_mixture`
and :func:`~chemistrykit.thermo.solve_equilibrium_composition`
take as input are exactly the kind of absolute thermodynamic reference
quantity the third law makes well defined in the first place -- without
Nernst's and Planck's fixed entropy zero-point, "the" Gibbs energy of
formation of a substance would only ever be knowable up to an arbitrary,
unmeasurable additive constant per element.

*References:* W. Nernst, "Über die Berechnung chemischer Gleichgewichte
aus thermischen Messungen," Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl.
(1906), 1-40; M. Planck, *Thermodynamik*, 3rd ed. (Leipzig: Veit & Comp.,
1911), Sec. 282.

.. minigallery:: ../../examples/thermo/thermochemistry/plot_02_nernst_heat_theorem.py

1949 -- Redlich and Kwong's Equation of State
----------------------------------------------------

Otto Redlich and Joseph N. S. Kwong, seeking a cubic equation of state
accurate enough for real chemical-engineering design work without the
computational burden of a fully empirical multi-parameter fit, modified
van der Waals's attractive term to depend explicitly on temperature:

.. math::

   P = \frac{RT}{V_m - b} - \frac{a}{\sqrt{T}\,V_m(V_m+b)}

The extra :math:`\sqrt{T}` dependence, together with a different pair of
universal constants relating `a` and `b` to a substance's critical
temperature and pressure, gives noticeably better predictions of real
gas behavior -- particularly of vapor densities and fugacities away from
the critical point -- than van der Waals's original equation, at almost
no extra computational cost, and it became (along with its own later
modifications, such as Soave's 1972 Redlich-Kwong-Soave equation) a
standard workhorse of chemical process simulation for describing real
gas mixtures.

.. math::

   Z_c = \frac{P_c V_c}{RT_c} = \frac{1}{3}

*Implementation:* :class:`chemistrykit.thermo.systems.equations_of_state.RedlichKwong`
implements exactly this equation of state, again solving its cubic in
:math:`V_m` via ``chemistrykit.thermo.utils.cubic_roots.real_positive_roots()``;
:meth:`~chemistrykit.thermo.RedlichKwong.from_critical_constants`
builds an instance from a substance's critical constants and reproduces
Redlich-Kwong's own universal critical compressibility factor
:math:`Z_c = 1/3` -- a different, and for most real gases somewhat more
accurate, universal prediction than van der Waals's :math:`3/8` above.
The example below compares all three equations of state -- ideal gas, van
der Waals, and Redlich-Kwong -- on the same CO2 P-V isotherms, both well
above and just below the critical temperature.

*References:* O. Redlich and J. N. S. Kwong, "On the Thermodynamics of
Solutions. V. An Equation of State. Fugacities of Gaseous Solutions,"
Chem. Rev. 44, 233-244 (1949).

.. minigallery:: ../../examples/thermo/equations_of_state/plot_03_redlich_kwong.py

1976 -- Peng and Robinson's Equation of State
---------------------------------------------

Redlich-Kwong (1949, above) and Giorgio Soave's 1972 modification of it
predicted gas-phase properties well but liquid densities poorly. Ding-Yu
Peng and Donald B. Robinson, at the University of Alberta, changed the
denominator of the attraction term and made its strength depend on
temperature through Kenneth Pitzer's acentric factor :math:`\omega`,
a single number describing how far a molecule is from a simple
spherical one:

.. math::

   P = \frac{RT}{V_m - b} - \frac{a\,\alpha(T)}{V_m^2 + 2bV_m - b^2},
   \qquad
   \alpha(T) = \left[1 + \kappa\left(1 - \sqrt{T/T_c}\right)\right]^2

with :math:`\kappa = 0.37464 + 1.54226\,\omega - 0.26992\,\omega^2`.
Peng and Robinson fitted :math:`\kappa(\omega)` to measured vapor
pressures, so the equation passes through Pitzer's defining point,
:math:`\log_{10}(P_{sat}/P_c) = -1 - \omega` at :math:`T/T_c = 0.7`. Its
universal critical compressibility factor is about 0.307, closer to real
fluids than Redlich-Kwong's 1/3. With only :math:`T_c`, :math:`P_c`, and
:math:`\omega` as input, it became the most widely used equation of state
in the oil, gas, and chemical industries.

*Implementation:* :class:`~chemistrykit.thermo.PengRobinson` implements
the equation, built from critical constants and the acentric factor, and
solves its cubic in :math:`Z` for the liquid or vapor molar volume. The
tests check that its critical isotherm passes through :math:`P_c` with
zero slope at :math:`Z_c = 0.3074`, and that
:func:`~chemistrykit.thermo.saturation_pressure` recovers the acentric
factor it was built with.

*References:* D.-Y. Peng and D. B. Robinson, "A New Two-Constant
Equation of State," Ind. Eng. Chem. Fundam. 15, 59-64 (1976); K. S.
Pitzer, D. Z. Lippmann, R. F. Curl, C. M. Huggins, and D. E. Petersen,
"The Volumetric and Thermodynamic Properties of Fluids. II.
Compressibility Factor, Vapor Pressure and Entropy of Vaporization,"
J. Am. Chem. Soc. 77, 3433-3440 (1955).

.. minigallery:: ../../examples/thermo/equations_of_state/plot_04_peng_robinson.py

See Also
--------

- :doc:`/api/thermo`
- :doc:`/history/kinetics_breakthroughs`