:orphan:

Sections
--------

- **equations_of_state** -- the ideal gas law, van der Waals continuity
  of liquid and gas, Redlich-Kwong, Peng-Robinson vapor pressures, and
  Lewis fugacity.
- **phase_equilibria** -- the Clausius-Clapeyron vapor-pressure curve and
  the Gibbs phase rule.
- **equilibrium** -- the law of mass action, Gibbs-energy minimization,
  the van't Hoff plot, and Le Chatelier's principle.
- **thermochemistry** -- Hess's law and Nernst's heat theorem.
- **mixtures** -- Raoult's-law P-x-y diagrams, Henry's law, van't Hoff's
  osmotic pressure, and Margules activity coefficients.


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Equations of state
-------------------

The ideal gas law, van der Waals, Redlich-Kwong, and Peng-Robinson
equations of state, and Lewis fugacities computed from them.


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    <div class="sphx-glr-thumbcontainer" tooltip="Clapeyron&#x27;s 1834 combined gas law unites Boyle&#x27;s law (PV constant at fixed T), Charles&#x27;s and Gay-Lussac&#x27;s law (V \propto T at fixed P), and Avogadro&#x27;s hypothesis into PV_m = RT. This example draws both classic views with IdealGas: hyperbolic Boyle isotherms, and straight Charles isobars that all extrapolate to zero volume at absolute zero.">

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  .. image:: /api/gallery/thermo/equations_of_state/images/thumb/sphx_glr_plot_01_ideal_gas_law_thumb.png
    :alt:

  :doc:`/api/gallery/thermo/equations_of_state/plot_01_ideal_gas_law`

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      <div class="sphx-glr-thumbnail-title">Clapeyron's ideal gas law, PV = RT</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Andrews&#x27;s CO2 isotherms showed a flat liquid-vapor plateau below a critical temperature and none above it. The van der Waals equation (~chemistrykit.thermo.systems.equations_of_state.VanDerWaals), built from CO2&#x27;s critical constants with from_critical_constants, reproduces the whole family: loops with three volume roots below T_c, an inflection at T_c, and single-valued isotherms above it, so gas can be turned into liquid continuously by going around the critical point.">

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  .. image:: /api/gallery/thermo/equations_of_state/images/thumb/sphx_glr_plot_02_van_der_waals_continuity_thumb.png
    :alt:

  :doc:`/api/gallery/thermo/equations_of_state/plot_02_van_der_waals_continuity`

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      <div class="sphx-glr-thumbnail-title">Andrews and van der Waals: continuity of the liquid and gas states</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Redlich and Kwong divided van der Waals&#x27;s attraction term by \sqrt{T}. This example draws CO2 P-V isotherms well above and just below the critical temperature from RedlichKwong, with VanDerWaals and IdealGas for comparison. Both real-gas equations are built from CO2&#x27;s critical temperature and pressure alone via from_critical_constants. Redlich-Kwong&#x27;s universal critical compressibility factor is 1/3, closer to real gases than van der Waals&#x27;s 3/8.">

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  .. image:: /api/gallery/thermo/equations_of_state/images/thumb/sphx_glr_plot_03_redlich_kwong_thumb.png
    :alt:

  :doc:`/api/gallery/thermo/equations_of_state/plot_03_redlich_kwong`

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      <div class="sphx-glr-thumbnail-title">Redlich-Kwong isotherms of CO2 against van der Waals and the ideal gas</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Peng and Robinson tuned the temperature dependence of their attraction term, \alpha(T), to Pitzer&#x27;s acentric factor \omega so that the equation reproduces pure-fluid vapor pressures. This example computes the CO2 vapor-pressure curve predicted by PengRobinson (\omega = 0.224) with saturation_pressure, and compares it with van der Waals and Redlich-Kwong built from the same critical constants. Only Peng-Robinson passes through Pitzer&#x27;s defining point, \log_{10}(P_{sat}/P_c) = -1 - \omega at T/T_c = 0.7.">

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  .. image:: /api/gallery/thermo/equations_of_state/images/thumb/sphx_glr_plot_04_peng_robinson_thumb.png
    :alt:

  :doc:`/api/gallery/thermo/equations_of_state/plot_04_peng_robinson`

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      <div class="sphx-glr-thumbnail-title">Peng-Robinson: vapor pressures from the acentric factor</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="G. N. Lewis replaced pressure by the fugacity f = \phi P so that the ideal-gas form \mu = \mu^\circ + RT\ln(f/P^\circ) stays exact for real fluids. The left panel shows the fugacity coefficient \phi of supercritical CO2 computed by fugacity_coefficient from three equations of state. The right panel shows Lewis&#x27;s phase-equilibrium criterion: below T_c the liquid and vapor fugacities from fugacity cross exactly at the saturation pressure.">

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  .. image:: /api/gallery/thermo/equations_of_state/images/thumb/sphx_glr_plot_05_lewis_fugacity_thumb.png
    :alt:

  :doc:`/api/gallery/thermo/equations_of_state/plot_05_lewis_fugacity`

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      <div class="sphx-glr-thumbnail-title">Lewis fugacity: the effective pressure of a real gas</div>
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Reaction equilibrium
--------------------

The law of mass action, Gibbs-energy minimization, the van't Hoff plot,
and Le Chatelier's principle.


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    <div class="sphx-glr-thumbcontainer" tooltip="For N_2O_4 \rightleftharpoons 2NO_2 the reaction quotient Q = x_{NO_2}^2/x_{N_2O_4} (at P = P^\circ), computed by reaction_quotient, rises monotonically with the extent of reaction. The law of mass action says the mixture stops changing where Q reaches the equilibrium constant K. Starting from different mixtures of reactant and product, the equilibrium compositions found by solve_equilibrium_composition all give the same Q = K.">

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  .. image:: /api/gallery/thermo/equilibrium/images/thumb/sphx_glr_plot_01_law_of_mass_action_thumb.png
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  :doc:`/api/gallery/thermo/equilibrium/plot_01_law_of_mass_action`

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      <div class="sphx-glr-thumbnail-title">Guldberg and Waage's law of mass action: Q = K at equilibrium</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Gibbs showed that at fixed temperature and pressure a system reaches equilibrium at the minimum of its total Gibbs energy G = \sum_i n_i\mu_i. For N_2O_4 \rightleftharpoons 2NO_2 this example plots G(\xi) from gibbs_energy_of_mixture at several temperatures and marks the composition returned by solve_equilibrium_composition, which sits at each minimum. The slope dG/d\xi = \sum_i\nu_i\mu_i (the reaction Gibbs energy) crosses zero there.">

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  .. image:: /api/gallery/thermo/equilibrium/images/thumb/sphx_glr_plot_02_gibbs_energy_minimization_thumb.png
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  :doc:`/api/gallery/thermo/equilibrium/plot_02_gibbs_energy_minimization`

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      <div class="sphx-glr-thumbnail-title">Gibbs's equilibrium criterion: minimum total Gibbs energy</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="van&#x27;t Hoff&#x27;s equation d\ln K/dT = \Delta H^\circ/(RT^2) makes \ln K a straight line in 1/T with slope -\Delta H^\circ/R and intercept \Delta S^\circ/R. This example generates noisy equilibrium constants for N_2O_4 \rightleftharpoons 2NO_2 from van_t_hoff_equilibrium_constant and recovers the reaction enthalpy and entropy with fit_van_t_hoff.">

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  .. image:: /api/gallery/thermo/equilibrium/images/thumb/sphx_glr_plot_03_van_t_hoff_plot_thumb.png
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  :doc:`/api/gallery/thermo/equilibrium/plot_03_van_t_hoff_plot`

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      <div class="sphx-glr-thumbnail-title">The van't Hoff plot: ln K against 1/T</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Two disturbances of N_2O_4 \rightleftharpoons 2NO_2, an endothermic reaction that increases the number of gas molecules:">

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  .. image:: /api/gallery/thermo/equilibrium/images/thumb/sphx_glr_plot_04_le_chatelier_thumb.png
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  :doc:`/api/gallery/thermo/equilibrium/plot_04_le_chatelier`

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      <div class="sphx-glr-thumbnail-title">Le Chatelier's principle: equilibrium shifts that oppose a disturbance</div>
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Mixtures and colligative properties
------------------------------------

Raoult's-law P-x-y diagrams, Henry's law for dilute solutes, van't Hoff's
osmotic pressure and the colligative properties, and Margules activity
coefficients for non-ideal solutions.


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    <div class="sphx-glr-thumbcontainer" tooltip="Raoult&#x27;s law, P_A = x_A P_A^* (~chemistrykit.thermo.systems.mixtures.raoult_vapor_pressure), makes each partial vapor pressure a straight line in liquid mole fraction. Adding the two lines (Dalton&#x27;s law) gives the total pressure, and BinaryIdealSolution computes the vapor composition, which is always richer in the more volatile component. That enrichment is the basis of fractional distillation.">

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  .. image:: /api/gallery/thermo/mixtures/images/thumb/sphx_glr_plot_01_raoult_law_pxy_thumb.png
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  :doc:`/api/gallery/thermo/mixtures/plot_01_raoult_law_pxy`

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      <div class="sphx-glr-thumbnail-title">Raoult's law and the P-x-y diagram of an ideal solution</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Henry&#x27;s law, P_B = x_B K_H (~chemistrykit.thermo.systems.mixtures.henry_law_pressure), says the vapor pressure of a dilute solute (equivalently, the amount of a gas that dissolves at a given pressure) is proportional to its mole fraction, with an empirical constant K_H that differs from the pure solute&#x27;s vapor pressure. Below, an illustrative real solute&#x27;s partial pressure follows the Henry&#x27;s-law line at high dilution and leaves it as the solute becomes concentrated, where Raoult&#x27;s law (~chemistrykit.thermo.systems.mixtures.raoult_vapor_pressure) takes over.">

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  .. image:: /api/gallery/thermo/mixtures/images/thumb/sphx_glr_plot_02_henry_law_thumb.png
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  :doc:`/api/gallery/thermo/mixtures/plot_02_henry_law`

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      <div class="sphx-glr-thumbnail-title">Henry's law: the dilute-solute limit of a real solution</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="van&#x27;t Hoff found that a dilute solution&#x27;s osmotic pressure (~chemistrykit.thermo.systems.mixtures.osmotic_pressure) equals the pressure the solute particles would exert as an ideal gas at the same concentration and temperature. The factor i counts particles per formula unit: 1 for sucrose, about 2 for NaCl. The same factor multiplies the other colligative effects, freezing-point depression and boiling-point elevation (~chemistrykit.thermo.systems.mixtures.freezing_point_depression, boiling_point_elevation).">

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  .. image:: /api/gallery/thermo/mixtures/images/thumb/sphx_glr_plot_03_osmotic_pressure_thumb.png
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  :doc:`/api/gallery/thermo/mixtures/plot_03_osmotic_pressure`

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      <div class="sphx-glr-thumbnail-title">van't Hoff's osmotic pressure law, Pi = iMRT</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Margules wrote the activity coefficients of a binary liquid as power series in mole fraction. The one-parameter form used by MargulesSolution, \ln\gamma_1 = A x_2^2 and \ln\gamma_2 = A x_1^2, turns Raoult&#x27;s straight lines into curves: A &gt; 0 (unlike molecules repel) bows the total pressure upward and can produce a maximum-pressure azeotrope, while A &lt; 0 bows it downward. The model still obeys the Gibbs-Duhem relation, and each component tends to Henry&#x27;s law at infinite dilution.">

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  .. image:: /api/gallery/thermo/mixtures/images/thumb/sphx_glr_plot_04_margules_activity_thumb.png
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  :doc:`/api/gallery/thermo/mixtures/plot_04_margules_activity`

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      <div class="sphx-glr-thumbnail-title">Margules activity coefficients: deviations from Raoult's law</div>
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Phase equilibria
----------------

The Clausius-Clapeyron liquid-vapor phase boundary, and the Gibbs phase
rule.


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    <div class="sphx-glr-thumbcontainer" tooltip="Water&#x27;s liquid-vapor phase boundary from the integrated Clausius-Clapeyron equation (~chemistrykit.thermo.systems.phase_equilibria.ClausiusClapeyron), anchored at the normal boiling point. Plotting \ln P against 1/T gives a straight line of slope -\Delta H_{vap}/R; from_two_points recovers \Delta H_{vap} from two measured points, and boiling_point inverts the curve to find the boiling point at altitude.">

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  .. image:: /api/gallery/thermo/phase_equilibria/images/thumb/sphx_glr_plot_01_clausius_clapeyron_thumb.png
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  :doc:`/api/gallery/thermo/phase_equilibria/plot_01_clausius_clapeyron`

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      <div class="sphx-glr-thumbnail-title">The Clausius-Clapeyron vapor-pressure curve of water</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Gibbs&#x27;s phase rule, F = C - P + 2 (~chemistrykit.thermo.systems.phase_equilibria.gibbs_phase_rule), says how many intensive variables can change while a set of phases coexists. On a one-component phase diagram, single-phase regions are areas (F=2), two-phase boundaries are curves (F=1), and three phases meet only at a point (F=0). The sketch below draws water&#x27;s diagram around its triple point (vaporization and sublimation curves from the Clausius-Clapeyron equation, melting line from the Clapeyron slope) and labels each feature with its degrees of freedom.">

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  .. image:: /api/gallery/thermo/phase_equilibria/images/thumb/sphx_glr_plot_02_gibbs_phase_rule_thumb.png
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  :doc:`/api/gallery/thermo/phase_equilibria/plot_02_gibbs_phase_rule`

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      <div class="sphx-glr-thumbnail-title">Gibbs's phase rule on the phase diagram of water</div>
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Thermochemistry
---------------

Hess's law of constant heat summation and Nernst's heat theorem: reaction
enthalpies built from steps or formation enthalpies, and the behavior of
reaction entropies and free energies as the temperature approaches
absolute zero.


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    <div class="sphx-glr-thumbcontainer" tooltip="Burning graphite straight to CO_2 releases the same heat as burning it first to CO and then burning the CO. The enthalpy-level diagram below shows both routes, summed with hess_law_enthalpy. The same principle lets reaction_enthalpy_from_formation compute any reaction enthalpy from tabulated enthalpies of formation, shown here for the combustion of methane.">

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  .. image:: /api/gallery/thermo/thermochemistry/images/thumb/sphx_glr_plot_01_hess_law_thumb.png
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  :doc:`/api/gallery/thermo/thermochemistry/plot_01_hess_law`

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      <div class="sphx-glr-thumbnail-title">Hess's law: reaction enthalpy does not depend on the path</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="For a reaction between perfect crystals, Nernst found that \Delta G and \Delta H approach each other with zero slope as T \to 0, so \Delta S = -\partial\Delta G/\partial T \to 0. With Debye T^3 heat capacities, \Delta C_p = \alpha T^3, the curves are \Delta H = \Delta H_0 + \alpha T^4/4, \Delta S = \alpha T^3/3, and \Delta G = \Delta H_0 - \alpha T^4/12. Planck&#x27;s stronger form, S \to 0 for every perfect crystal, gives absolute entropies from calorimetry, S(T) = \int_0^T C_p/T&#x27;\,dT&#x27;. Those absolute entropies underlie the standard Gibbs energies of formation that gibbs_energy_of_mixture takes as input.">

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  .. image:: /api/gallery/thermo/thermochemistry/images/thumb/sphx_glr_plot_02_nernst_heat_theorem_thumb.png
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  :doc:`/api/gallery/thermo/thermochemistry/plot_02_nernst_heat_theorem`

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      <div class="sphx-glr-thumbnail-title">Nernst's heat theorem: reaction entropy vanishes at absolute zero</div>
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.. toctree::
   :hidden:
   :includehidden:


   /api/gallery/thermo/equations_of_state/index.rst
   /api/gallery/thermo/equilibrium/index.rst
   /api/gallery/thermo/mixtures/index.rst
   /api/gallery/thermo/phase_equilibria/index.rst
   /api/gallery/thermo/thermochemistry/index.rst



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