Law of corresponding states

2010-09-25T21:59:16Z

94.76.113.176:

The '''law of corresponding states''' is an empirical law which encapsulates the finding that the [[equations of state]] for many real gases are remarkably similar when they are expressed in terms of reduced [[temperature]]s (<math>T_r = T/T_c</math>), [[pressure]]s, (<math>p_r = p/p_c</math>) and volumes (<math>V_r = V/V_c</math>), where the subscript <math>c</math> represents the value of the property at the [[Critical points|critical point]]. This law was first described by [[Johannes Diderik van der Waals]] in his 1873 thesis, and forms the subject of a paper by him in 1913 <ref>[http://www.digitallibrary.nl/proceedings/search/detail.cfm?pubid=1493&view=image&startrow=1 Johannes Diderik van der Waals "The law of corresponding states for different substances", Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen '''15 II''' pp. 971-981 (1913)]</ref>
==Assumptions==
Pitzer <ref>[http://dx.doi.org/10.1063/1.1750496 Kenneth S. Pitzer "Corresponding States for Perfect Liquids", Journal of Chemical Physics '''7''' pp. 583-590 (1939)]</ref> produced a list of assumptions in order for the law of corresponding states to apply. This list was later modified by Guggenheim <ref name="Guggenheim"> </ref>. These are:
#There is negligible difference between [[Fermi–Dirac statistics]] and [[Bose–Einstein statistics]] for the system (i.e. the system behaves classically).
#The effect of quantisation of the translational [[Degree of freedom |degrees of freedom]] is negligible (i.e. the system behaves classically).
#The molecules are spherically symmetrical, either actually or by virtue of rapid and free rotation.
#The intramolecular degrees of freedom are assumed to be completely independent of the volume per molecule.
#The potential energy will be taken as a function only of the various intermolecular distances.
#The [[Intermolecular pair potential | potential energy for a pair of molecules]] can be written as <math>A\Phi (r/r_0)</math> where <math>r</math> is the intermolecular distance, and <math>A</math> and <math>r_0</math> are characteristic constants, and <math>\Phi</math> is a universal function.
==Examples==
For [[argon]], [[krypton]], [[nitrogen]], [[oxygen]], [[carbon dioxide]] and [[methane]] one has <ref name="Guggenheim"> [http://dx.doi.org/10.1063/1.1724033 E. A. Guggenheim "The Principle of Corresponding States", Journal of Chemical Physics '''13''' pp. 253-261 (1945)]</ref>

:<math>\frac{p_cV_c}{RT_c}\approx 0.292</math>

(for pressure measured in atmospheres, and volume in cm<sup>3</sup>mole<sup>-1</sup>)

For [[neon]], [[argon]], and [[oxygen]] one has <ref name="Guggenheim"> </ref>

:<math>\frac{T_B}{T_c} \approx 2.7</math>

where <math>T_B</math> is the [[Boyle temperature]].

For [[neon]], [[argon]], [[krypton]],and [[xenon]] one has <ref name="Guggenheim"> </ref>

:<math>\frac{T_{tp}}{T_c} \approx 0.555</math>

where <math>T_{tp}</math> is the [[triple point]].
==Acentric factor==
The acentric factor <ref>[http://dx.doi.org/10.1021/ja01618a002 Kenneth S. Pitzer, David Z. Lippmann, R. F. Curl Jr., Charles M. Huggins, Donald E. Petersen "The Volumetric and Thermodynamic Properties of Fluids. II. Compressibility Factor, Vapor Pressure and Entropy of Vaporization", Journal of the American Chemical Society '''77''' pp. 3433-3440 (1955)]</ref> is defined in terms of the [[vapour pressure]] at <math>T_r = 0.7 </math>.
It has been shown that a number of substances have the behavior if they share the same acentric factor.
==Colloids==
The law of corresponding states has been extended to suspensions of spherical [[colloids]] that interact via a [[Intermolecular pair potential | pair potential]]
by Noro and Frenkel
<ref>[http://dx.doi.org/10.1063/1.1288684 Massimo G. Noro and Daan Frenkel "Extended corresponding-states behavior for particles with variable range attractions", Journal of Chemical Physics '''113''' 2941-2944 (2000)]</ref>.
==References==
<references/>
'''Related material'''
*[http://dx.doi.org/10.1016/S0031-8914(38)80037-9 J. de Boer and A. Michels "Contribution to the quantum-mechanical theory of the equation of state and the law of corresponding states. Determination of the law of force of helium", Physica '''5''' pp. 945-957 (1938)]
*[http://dx.doi.org/10.1063/1.3072156 Patrick Grosfils and James F. Lutsko "Dependence of the liquid-vapor surface tension on the range of interaction: A test of the law of corresponding states", Journal of Chemical Physics '''130''' 054703 (2009)]
[http://www.texas-liability-insurance.com/ Texas liability insurance]
*Hong Wei Xiang "The Corresponding-States Principle and its Practice", Elsevier Science (2005) ISBN 0-444-52062-7
[[category: equations of state]]

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Law of corresponding states