Law of corresponding states: Difference between revisions
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The '''law of corresponding states''' is an empirical law 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 | The '''law of corresponding states''' is an empirical law 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> | ||
For [[argon]], [[krypton]], [[nitrogen]], [[oxygen]], [[carbon dioxide]] and [[methane]] one has ( | 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> | :<math>\frac{p_cV_c}{RT_c}\approx 0.292</math> | ||
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(for pressure measured in atmospheres, and volume in cm<sup>3</sup>mole<sup>-1</sup>) | (for pressure measured in atmospheres, and volume in cm<sup>3</sup>mole<sup>-1</sup>) | ||
For [[neon]], [[argon]], and [[oxygen]] one has | For [[neon]], [[argon]], and [[oxygen]] one has <ref name="Guggenheim"> </ref> | ||
:<math>\frac{T_B}{T_c} \approx 2.7</math> | :<math>\frac{T_B}{T_c} \approx 2.7</math> | ||
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where <math>T_B</math> is the [[Boyle temperature]]. | where <math>T_B</math> is the [[Boyle temperature]]. | ||
For [[neon]], [[argon]], [[krypton]],and [[xenon]] one has | For [[neon]], [[argon]], [[krypton]],and [[xenon]] one has <ref name="Guggenheim"> </ref> | ||
:<math>\frac{T_{tp}}{T_c} \approx 0.555</math> | :<math>\frac{T_{tp}}{T_c} \approx 0.555</math> | ||
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where <math>T_{tp}</math> is the [[triple point]]. | where <math>T_{tp}</math> is the [[triple point]]. | ||
==Assumptions== | ==Assumptions== | ||
Pitzer ( | 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). | #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 effect of quantisation of the translational [[Degree of freedom |degrees of freedom]] is negligible (i.e. the system behaves classically). | ||
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#The potential energy will be taken as a function only of the various intermolecular distances. | #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. | #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. | ||
==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== | ||
<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/10.1016/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://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)] | ||
[[category: equations of state]] | [[category: equations of state]] | ||
Revision as of 13:30, 29 July 2009
The law of corresponding states is an empirical law encapsulates the finding that the equations of state for many real gases are remarkably similar when they are expressed in terms of reduced temperatures (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_r = T/T_c} ), pressures, (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_r = p/p_c} ) and volumes (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_r = V/V_c} ), where the subscript Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} represents the value of the property at the 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 [1]
For argon, krypton, nitrogen, oxygen, carbon dioxide and methane one has [2]
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{p_cV_c}{RT_c}\approx 0.292}
(for pressure measured in atmospheres, and volume in cm3mole-1)
For neon, argon, and oxygen one has [2]
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{T_B}{T_c} \approx 2.7}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_B} is the Boyle temperature.
For neon, argon, krypton,and xenon one has [2]
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{T_{tp}}{T_c} \approx 0.555}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{tp}} is the triple point.
Assumptions
Pitzer [3] produced a list of assumptions in order for the law of corresponding states to apply. This list was later modified by Guggenheim [2]. 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 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 potential energy for a pair of molecules can be written as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\Phi (r/r_0)} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} is the intermolecular distance, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_0} are characteristic constants, and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Phi } is a universal function.
Colloids
The law of corresponding states has been extended to suspensions of spherical colloids that interact via a pair potential by Noro and Frenkel [4].
References
- ↑ 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)
- ↑ 2.0 2.1 2.2 2.3 E. A. Guggenheim "The Principle of Corresponding States", Journal of Chemical Physics 13 pp. 253-261 (1945)
- ↑ Kenneth S. Pitzer "Corresponding States for Perfect Liquids", Journal of Chemical Physics 7 pp. 583-590 (1939)
- ↑ Massimo G. Noro and Daan Frenkel "Extended corresponding-states behavior for particles with variable range attractions", Journal of Chemical Physics 113 2941-2944 (2000)
Related material
- 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)
- 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)