Law of corresponding states: Difference between revisions

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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>
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==
==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>
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>
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where <math>T_{tp}</math> is the [[triple point]].
where <math>T_{tp}</math> is the [[triple point]].
==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.
==Acentric factor==
==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>.
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>.

Revision as of 16:12, 20 April 2010

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 temperatures (), pressures, () and volumes (), where the subscript 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]

Assumptions

Pitzer [2] produced a list of assumptions in order for the law of corresponding states to apply. This list was later modified by Guggenheim [3]. These are:

  1. There is negligible difference between Fermi–Dirac statistics and Bose–Einstein statistics for the system (i.e. the system behaves classically).
  2. The effect of quantisation of the translational degrees of freedom is negligible (i.e. the system behaves classically).
  3. The molecules are spherically symmetrical, either actually or by virtue of rapid and free rotation.
  4. The intramolecular degrees of freedom are assumed to be completely independent of the volume per molecule.
  5. The potential energy will be taken as a function only of the various intermolecular distances.
  6. The potential energy for a pair of molecules can be written as where is the intermolecular distance, and and are characteristic constants, and is a universal function.

Examples

For argon, krypton, nitrogen, oxygen, carbon dioxide and methane one has [3]

(for pressure measured in atmospheres, and volume in cm3mole-1)

For neon, argon, and oxygen one has [3]

where is the Boyle temperature.

For neon, argon, krypton,and xenon one has [3]

where is the triple point.

Acentric factor

The acentric factor [4] is defined in terms of the vapour pressure at . 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 pair potential by Noro and Frenkel [5].

References

  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)
  2. ↑ Kenneth S. Pitzer "Corresponding States for Perfect Liquids", Journal of Chemical Physics 7 pp. 583-590 (1939)
  3. ↑ 3.0 3.1 3.2 3.3 E. A. Guggenheim "The Principle of Corresponding States", Journal of Chemical Physics 13 pp. 253-261 (1945)
  4. ↑ 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)
  5. ↑ Massimo G. Noro and Daan Frenkel "Extended corresponding-states behavior for particles with variable range attractions", Journal of Chemical Physics 113 2941-2944 (2000)

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