.. _sphx_glr_api_gallery_thermodynamics_properties:

Properties
----------

Equations of state, vapour pressure and the properties of water and steam.


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    <div class="sphx-glr-thumbnails">

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    <div class="sphx-glr-thumbcontainer" tooltip="Boyle (1662) and Mariotte found that at fixed temperature a gas&#x27;s pressure times its volume stays constant. Gay-Lussac (1802) found that at fixed pressure every gas expands by the same fraction of its volume at 0 °C, about 1/267 per degree. In his 1834 memoir on Carnot&#x27;s theory, Émile Clapeyron combined the two as pv = R(267 + t). Regnault&#x27;s better measurements moved the 267 to 273, and with absolute temperature and the amount of gas the law reads pV = nRT (~engineeringkit.thermodynamics.ideal_gas_pressure, ideal_gas_temperature). The example rebuilds the absolute zero from a gas thermometer.">

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  .. image:: /api/gallery/thermodynamics/properties/images/thumb/sphx_glr_plot_01_clapeyron_ideal_gas_thumb.png
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  :doc:`/api/gallery/thermodynamics/properties/plot_01_clapeyron_ideal_gas`

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      <div class="sphx-glr-thumbnail-title">Clapeyron's ideal gas law (1834)</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Clapeyron (1834) applied Carnot&#x27;s reasoning to a small cycle of boiling and condensing and found that the slope of the vapour-pressure curve is fixed by the latent heat, \mathrm{d}p/\mathrm{d}T = h_{fg}/[T(v_g - v_f)]. Clausius (1850) rederived it from the two laws and, treating the vapour as an ideal gas and neglecting the liquid volume, integrated it: \ln p is linear in 1/T (~engineeringkit.thermodynamics.clausius_clapeyron_pressure). The example checks the exact slope against IAPWS-IF97 steam and uses the integrated form to find where water boils on a mountain.">

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  .. image:: /api/gallery/thermodynamics/properties/images/thumb/sphx_glr_plot_02_clausius_clapeyron_thumb.png
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  :doc:`/api/gallery/thermodynamics/properties/plot_02_clausius_clapeyron`

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      <div class="sphx-glr-thumbnail-title">The Clausius-Clapeyron relation (1834-1850)</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Joule&#x27;s free expansion of 1845 showed no temperature change, which was too coarse a test of whether a gas has internal energy apart from its temperature. From 1852 Joule and William Thomson forced gas steadily through a porous plug and measured the temperature on each side. The flow keeps its enthalpy, and real gases came out slightly cooler: air by about a quarter of a kelvin per atmosphere, carbon dioxide by about one. Hydrogen came out warmer. For a van der Waals gas at low pressure the coefficient is \mu_{JT} = (2a/RT - b)/c_{p,m} (~engineeringkit.thermodynamics.joule_thomson_coefficient_vdw). Attraction (a) cools the gas and the size of the molecules (b) warms it. The sign changes at the inversion temperature 2a/(Rb), which Linde&#x27;s air liquefier (1895) relied on.">

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  .. image:: /api/gallery/thermodynamics/properties/images/thumb/sphx_glr_plot_03_joule_thomson_thumb.png
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  :doc:`/api/gallery/thermodynamics/properties/plot_03_joule_thomson`

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      <div class="sphx-glr-thumbnail-title">The Joule-Thomson effect (1852)</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="In 1869 Thomas Andrews showed that carbon dioxide above 31 °C cannot be liquefied by any pressure, and below it liquefies at a sharp pressure. In his 1873 thesis Johannes van der Waals explained this with two corrections to the ideal gas: the molecules fill a volume b per mole, and they attract each other, which lowers the pressure by a/v^2: p = RT/(v - b) - a/v^2 (~engineeringkit.thermodynamics.van_der_waals_pressure). One equation then describes both gas and liquid, and has a critical point (~engineeringkit.thermodynamics.van_der_waals_critical_point). He received the 1910 Nobel Prize in Physics for it.">

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  .. image:: /api/gallery/thermodynamics/properties/images/thumb/sphx_glr_plot_04_van_der_waals_thumb.png
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  :doc:`/api/gallery/thermodynamics/properties/plot_04_van_der_waals`

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      <div class="sphx-glr-thumbnail-title">Van der Waals's equation of state (1873)</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="The integrated Clausius-Clapeyron relation makes \log p linear in 1/T. Measured vapour pressures bend away from that line because the latent heat falls as the temperature rises. In 1888 Charles Antoine, a French naval engineer, added a third constant to the temperature, \log_{10} p = A - B/(C + T) (~engineeringkit.thermodynamics.antoine_vapor_pressure). It fits most liquids to within a fraction of a percent over a range of 100 K, and tables of Antoine constants are still standard in chemical engineering. The example fits both forms to IAPWS-IF97 water and compares them.">

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  .. image:: /api/gallery/thermodynamics/properties/images/thumb/sphx_glr_plot_05_antoine_equation_thumb.png
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  :doc:`/api/gallery/thermodynamics/properties/plot_05_antoine_equation`

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      <div class="sphx-glr-thumbnail-title">Antoine's vapour-pressure equation (1888)</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Steam tables have been the working data of power engineering since Callendar and Mollier&#x27;s tables of the early 1900s. In 1997 the International Association for the Properties of Water and Steam adopted the Industrial Formulation IAPWS-IF97, led by Wolfgang Wagner. It splits the fluid into five regions, each with its own fast equation fitted to the scientific formulation IAPWS-95. In the compressed liquid (region 1) and the vapour (region 2) the equation is a dimensionless Gibbs free energy g(p, T)/(RT), and every property follows from its derivatives. Near the critical point (region 3) it is a Helmholtz free energy f(\rho, T)/(RT) of density, and region 4 is the saturation line (~engineeringkit.thermodynamics.steam_properties, saturation_pressure). The example checks the release&#x27;s own verification values and draws the T-s diagram.">

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  .. image:: /api/gallery/thermodynamics/properties/images/thumb/sphx_glr_plot_06_iapws_if97_thumb.png
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  :doc:`/api/gallery/thermodynamics/properties/plot_06_iapws_if97`

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      <div class="sphx-glr-thumbnail-title">IAPWS-IF97: the properties of water and steam (1997)</div>
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    </div>


.. toctree::
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   /api/gallery/thermodynamics/properties/plot_01_clapeyron_ideal_gas
   /api/gallery/thermodynamics/properties/plot_02_clausius_clapeyron
   /api/gallery/thermodynamics/properties/plot_03_joule_thomson
   /api/gallery/thermodynamics/properties/plot_04_van_der_waals
   /api/gallery/thermodynamics/properties/plot_05_antoine_equation
   /api/gallery/thermodynamics/properties/plot_06_iapws_if97