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High-pass filter

2015-01-05T10:58:13Z

89.168.254.224: /* Algorithmic implementation */

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Induction generator

2014-01-22T20:12:07Z

89.168.132.115: Corrected grammar

'''Molar conductivity''' is defined as the [[Conductivity (electrolytic)|conductivity]] of an [[electrolyte]] [[solution]] divided by the [[molar concentration]] of the electrolyte, and so measures the efficiency with which a given electrolyte conducts electricity in solution. Its units are [[Siemens (unit)|siemens]] per meter per [[molarity]], or siemens meter-squared per mole. The usual symbol is a capital lambda, Λ, or Λm.

== History ==

[[Friedrich Kohlrausch]] established that to a high accuracy in dilute solutions, molar conductivity is composed of individual contributions of ions. This is known as the ''law of independent migration of ions''.<ref>Castellan, G.W. ''Physical Chemistry''. Benjamin/Cummings, 1983.</ref>

== Description ==

From its definition, the molar conductivity is given by:<ref>''The best test preparation for the GRE Graduate Record Examination Chemistry Test.'' Published by the Research and Education Association, 2000, ISBN 0-8789-600-8. p. 149.</ref>

:<math>\Lambda_{\mathrm{m}} = \frac{\kappa}{c}</math>

where:
* κ is the measured conductivity
* ''c'' is the electrolyte concentration.

Two cases should be distinguished: strong electrolytes and weak electrolytes.

For [[strong electrolyte]]s, such as [[salt]]s, [[strong acid]]s and [[strong base]]s, the molar conductivity depends only ''weakly'' on concentration. Based on experimental data [[Friedrich Kohlrausch]] (around the year 1900) proposed the non-linear law for strong electrolytes:

:<math>\Lambda_{\mathrm{m}} =\Lambda_{\mathrm{m}}^\circ - K\sqrt{c}</math>

where
* <math>\Lambda_{\mathrm{m}}^\circ</math> is the molar conductivity at infinite dilution (or ''limiting molar conductivity'')
* ''K'' is the Kohlrausch coefficient, which depends mainly on the stoichiometry of the specific salt in solution.
This law is valid for low electrolyte concentrations only; it fits into the [[Debye-Hückel-Onsager equation]] :.<ref>{{cite book|last=Atkins|first=P. W.|title=The Elements of Physical Chemistry|publisher=Oxford University Press|year=2001|isbn=0-19-879290-5}}</ref>

For weak electrolytes (i.e. incompletely dissociated electrolytes), however, the molar conductivity ''strongly'' depends on concentration: The more dilute a solution, the greater its ''molar'' conductivity, due to increased [[ionic dissociation]]. (This, for example, is the case of SDS-coated proteins in the stacking gel of an [[SDS-PAGE]].)

The limiting molar conductivity can be decomposed into contributions from the different ions (Kohlrausch's law of independent migration of ions):

:<math>\Lambda_{\mathrm{m}}^\circ = \Sigma_i \nu_i \lambda_i</math>

where:
* <math>\lambda_i</math> is the molar ionic conductivity of ion ''i''.
* <math>\nu_i</math> is the number of ions ''i'' in the formula unit of the electrolyte (e.g. 2 and 1 for Na+ and SO42- in Na2SO4)

== Applications ==
[[Law of dilution|Ostwald's law of dilution]], which gives the dissociation constant of a weak electrolyte as a function of concentration, can be written in terms of molar conductivity. Thus, the [[pKa]] values of acids can be calculated by measuring the molar conductivity and extrapolating into zero concentration. Namely, pKa = p(K/(1 mol dm−3)) at the zero-concentration limit, where K is the dissociation constant from Ostwald's law.

==References==
<references />

[http://www.iupac.org/publications/pac/pdf/1994/pdf/6608x1739.pdf]

[[Category:Electrochemistry]]
[[Category:Physical chemistry]]

Neutron cross section

2014-01-15T21:43:31Z

89.168.178.8: /* Incident particle energy dependence */

[[Image:Cuboctahedron.png|thumb|A rectified cube is a [[cuboctahedron]] – edges reduced to vertices, and vertices expanded into new faces]]
[[Image:Dual Cube-Octahedron.svg|thumb|A ''birectified'' cube is an octahedron – faces are reduced to points and new faces are centered on the original vertices.]]
[[Image:Rectified cubic honeycomb.jpg|thumb|A [[rectified cubic honeycomb]] – edges reduced to vertices, and vertices expanded into new cells.]]
In [[Euclidean geometry]], '''rectification''' is the process of truncating a [[polytope]] by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting polytope will be bounded by the [[vertex figure]]s and the rectified facets of the original polytope.

== Example of rectification as a final truncation to an edge ==
Rectification is the final point of a truncation process. For example on a cube this sequence shows four steps of a continuum of truncations between the regular and rectified form:

[[Image:Cube truncation sequence.svg|520px]]

== Higher degree rectifications ==
Higher degree rectification can be performed on higher-dimensional regular polytopes. The highest degree of rectification creates the [[dual polytope]]. A rectification truncates edges to points. A birectification truncates faces to points. A trirectification truncates cells to points, and so on.

== Example of birectification as a final truncation to a face ==
This sequence shows a ''birectified cube'' as the final sequence from a cube to the dual where the original faces are truncated down to a single point:
:[[Image:Birectified cube sequence.png|480px]]

== In polygons ==

The dual of a polygon is the same as its rectified form. New vertices are placed at the center of the edges of the original polygon.

== In polyhedra and plane tilings ==
{{See|quasiregular polyhedron}}
Each [[platonic solid]] and its [[dual polyhedron|dual]] have the same rectified polyhedron. (This is not true of polytopes in higher dimensions.)

The rectified polyhedron turns out to be expressible as the intersection of the original platonic solid with an appropriated scaled concentric version of its dual. For this reason, its name is a combination of the names of the original and the dual:

# The rectified [[tetrahedron]], whose dual is the tetrahedron, is the ''tetratetrahedron'', better known as the [[octahedron]].
# The rectified [[octahedron]], whose dual is the [[cube]], is the [[cuboctahedron]].
# The rectified [[icosahedron]], whose dual is the [[dodecahedron]], is the [[icosidodecahedron]].
# A rectified [[square tiling]] is a [[square tiling]].
# A rectified [[triangular tiling]] or [[hexagonal tiling]] is a [[trihexagonal tiling]].

Examples
{| class="wikitable"
!Family
!Parent
!Rectification
!Dual
|-
!{{CDD|node|p|node|q|node}} [p,q]
!{{CDD|node_1|p|node|q|node}}
!{{CDD|node|p|node_1|q|node}}
!{{CDD|node|p|node|q|node_1}}
|- align=center
![3,3]
|[[Image:Uniform polyhedron-33-t0.png|75px]] [[Tetrahedron]]
|[[Image:Uniform polyhedron-33-t1.png|75px]] [[Tetratetrahedron|Octahedron]]
|[[Image:Uniform polyhedron-33-t2.png|75px]] [[Tetrahedron]]
|- align=center
![4,3]
|[[Image:Uniform polyhedron-43-t0.png|75px]] [[Cube]]
|[[Image:Uniform polyhedron-43-t1.png|75px]] [[Cuboctahedron]]
|[[Image:Uniform polyhedron-43-t2.png|75px]] [[Octahedron]]
|- align=center
![5,3]
|[[Image:Uniform polyhedron-53-t0.png|75px]] [[Dodecahedron]]
|[[Image:Uniform polyhedron-53-t1.png|75px]] [[Icosidodecahedron]]
|[[Image:Uniform polyhedron-53-t2.png|75px]] [[Icosahedron]]
|- align=center
![6,3]
|[[Image:Uniform tiling 63-t0.png|75px]] [[Hexagonal tiling]]
|[[Image:Uniform tiling 63-t1.png|75px]] [[Trihexagonal tiling]]
|[[Image:Uniform tiling 63-t2.png|75px]] [[Triangular tiling]]
|- align=center
![7,3]
|[[Image:Uniform tiling 73-t0.png|75px]] [[Order-3 heptagonal tiling]]
|[[Image:Uniform tiling 73-t1.png|75px]] [[Triheptagonal tiling]]
|[[Image:Uniform tiling 73-t2.png|75px]] [[Order-7 triangular tiling]]
|- align=center
![4,4]
|[[Image:Uniform tiling 44-t0.png|75px]] [[Square tiling]]
|[[Image:Uniform tiling 44-t1.png|75px]] [[Square tiling]]
|[[Image:Uniform tiling 44-t2.png|75px]] [[Square tiling]]
|- align=center
![5,4]
|[[Image:Uniform tiling 54-t0.png|75px]] [[Order-4 pentagonal tiling]]
|[[Image:Uniform tiling 54-t1.png|75px]] [[tetrapentagonal tiling]]
|[[Image:Uniform tiling 54-t2.png|75px]] [[Order-5 square tiling]]
|}

=== In nonregular polyhedra ===
If a polyhedron is not regular, the edge midpoints surrounding a vertex may not be coplanar. However, a form of rectification is still possible in this case: every polyhedron has a [[polyhedral graph]] as its [[n-skeleton|1-skeleton]], and from that graph one may form the [[medial graph]] by placing a vertex at each edge midpoint of the original graph, and connecting two of these new vertices by an edge whenever they belong to consecutive edges along a common face. The resulting medial graph remains polyhedral, so by [[Steinitz's theorem]] it can be represented as a polyhedron.

The [[Conway polyhedron notation]] equivalent to rectification is '''ambo''', represented by '''a'''. Applying twice '''aa''', (rectifying a rectification) is Conway's [[expansion (geometry)|'''expand''']] operation, '''e''', which is the same as Johnson's [[Cantellation (geometry)|cantellation]] operation, t0,2 generated from regular polyhedral and tilings.

== In polychora and 3d honeycomb tessellations ==

Each [[Convex_regular_4-polytope|convex regular polychoron]] has a rectified form as a [[uniform polychoron]].

A regular polychoron {p,q,r} has cells {p,q}. Its rectification will have two cell types, a rectified {p,q} polyhedron left from the original cells and {q,r} polyhedron as new cells formed by each truncated vertex.

A rectified {p,q,r} is not the same as a rectified {r,q,p}, however. A further truncation, called [[bitruncation (geometry)|bitruncation]], is symmetric between a polychoron and its dual. See [[Uniform_polychoron#Geometric_derivations]].

Examples
{| class="wikitable"
!Family
!Parent
!Rectification
!Birectification (Dual rectification)
!Trirectification (Dual)
|-
!{{CDD|node|p|node|q|node|r|node}} [p,q,r]
!{{CDD|node_1|p|node|q|node|r|node}}
!{{CDD|node|p|node_1|q|node|r|node}}
!{{CDD|node|p|node|q|node_1|r|node}}
!{{CDD|node|p|node|q|node|r|node_1}}
|- align=center
![3,3,3]
|[[Image:Schlegel wireframe 5-cell.png|120px]] [[5-cell]]
|[[Image:Schlegel half-solid rectified 5-cell.png|120px]] [[rectified 5-cell]]
|[[Image:Schlegel half-solid rectified 5-cell.png|120px]] [[rectified 5-cell]]
|[[Image:Schlegel wireframe 5-cell.png|120px]] [[5-cell]]
|- align=center
![4,3,3]
|[[Image:Schlegel wireframe 8-cell.png|150px]] [[tesseract]]
|[[Image:Schlegel half-solid rectified 8-cell.png|150px]] [[rectified tesseract]]
|[[Image:Schlegel half-solid rectified 16-cell.png|150px]] Rectified 16-cell ([[24-cell]])
|[[Image:Schlegel wireframe 16-cell.png|150px]] [[16-cell]]
|- align=center
![3,4,3]
|[[Image:Schlegel wireframe 24-cell.png|150px]] [[24-cell]]
|[[File:Schlegel half-solid cantellated 16-cell.png|150px]] [[rectified 24-cell]]
|[[File:Schlegel half-solid cantellated 16-cell.png|150px]] [[rectified 24-cell]]
|[[Image:Schlegel wireframe 24-cell.png|150px]] [[24-cell]]
|- align=center
![5,3,3]
|[[Image:Schlegel wireframe 120-cell.png|150px]] [[120-cell]]
|[[File:Rectified 120-cell schlegel halfsolid.png|150px]] [[rectified 120-cell]]
|[[File:Rectified_600-cell_schlegel_halfsolid.png|150px]] [[rectified 600-cell]]
|[[Image:Schlegel wireframe 600-cell vertex-centered.png|150px]] [[600-cell]]
|- align=center
![4,3,4]
|[[Image:Partial cubic honeycomb.png|150px]] [[Cubic honeycomb]]
|[[Image:Rectified cubic honeycomb.jpg|150px]] [[Rectified cubic honeycomb]]
|[[Image:Rectified cubic honeycomb.jpg|150px]] [[Rectified cubic honeycomb]]
|[[Image:Partial cubic honeycomb.png|150px]] [[Cubic honeycomb]]
|- align=center
![5,3,4]
|[[Image:Hyperbolic_orthogonal_dodecahedral_honeycomb.png|150px]] [[Order-4 dodecahedral honeycomb|Order-4 dodecahedral]]
|[[File:Rectified_order_4_dodecahedral_honeycomb.png|150px]] [[Rectified order-4 dodecahedral honeycomb|Rectified order-4 dodecahedral]]
|(No image) [[Rectified order-5 cubic honeycomb|Rectified order-5 cubic]]
|[[Image:Hyperb gcubic hc.png|150px]] [[Order-5 cubic honeycomb|Order-5 cubic]]
|}

== Degrees of rectification ==

A first rectification truncates edges down to points. If a polytope is [[Regular polytope|regular]], this form is represented by an extended [[Schläfli symbol]] notation t1{p,q,...}.

A second rectification, or '''birectification''', truncates [[Face (geometry)|faces]] down to points. If regular it has notation t2{p,q,...}. For [[polyhedron|polyhedra]], a birectification creates a [[dual polyhedron]].

Higher degree rectifications can be constructed for higher dimensional polytopes. In general an n-rectification truncates ''n-faces'' to points.

If an n-polytope is (n-1)-rectified, its [[Facet (mathematics)|facets]] are reduced to points and the polytope becomes its [[Dual polytope|dual]].

=== Notations and facets ===

There are different equivalent notations for each degree of rectification. These tables show the names by dimension and the two type of [[Facet (mathematics)|facet]]s for each.

==== Regular [[polygon]]s ====

[[Facet (mathematics)|Facet]]s are edges, represented as {2}.

{| class="wikitable"
!rowspan=2|name {p}
!rowspan=2|[[Coxeter-Dynkin diagram|Coxeter-Dynkin]]
!rowspan=2|t-notation [[Schläfli symbol]]
!colspan=3|Vertical [[Schläfli symbol]]
|-
!Name
!Facet-1
!Facet-2
|- align=center
|Parent
|{{CDD|node_1|p|node}}
|t0{p}
| {p}
| {2}
|
|- align=center
|Rectified
|{{CDD|node|p|node_1}}
|t1{p}
| {p}
|
| {2}
|}

==== Regular [[Uniform polyhedron|polyhedra]] and [[List of uniform tilings|tiling]]s ====

[[Facet (mathematics)|Facet]]s are regular polygons.
{| class="wikitable"
!rowspan=2|name {p,q}
!rowspan=2|[[Coxeter-Dynkin diagram|Coxeter-Dynkin]]
!rowspan=2|t-notation [[Schläfli symbol]]
!colspan=3|Vertical [[Schläfli symbol]]
|-
!Name
!Facet-1
!Facet-2
|- align=center
|Parent
|{{CDD|node_1|p|node|q|node}}
|t0{p,q}
| {p,q}
| {p}
|
|- align=center
|Rectified
|{{CDD|node|p|node_1|q|node}}
|t1{p,q}
| <math>\begin{Bmatrix} p \\ q \end{Bmatrix}</math> = r{p,q}
| {p}
| {q}
|- align=center
|Birectified
|{{CDD|node|p|node|q|node_1}}
|t2{p,q}
| {q,p}
|
| {q}
|}

==== Regular [[Uniform polychoron|polychora]] and [[honeycomb]]s ====
[[Facet (mathematics)|Facet]]s are regular or rectified polyhedra.
{| class="wikitable"
!rowspan=2|name {p,q,r}
!rowspan=2|[[Coxeter-Dynkin diagram|Coxeter-Dynkin]]
!rowspan=2|t-notation [[Schläfli symbol]]
!colspan=3|Extended [[Schläfli symbol]]
|-
!Name
!Facet-1
!Facet-2
|- align=center
|Parent
|{{CDD|node_1|p|node|q|node|r|node}}
|t0{p,q,r}
| {p,q,r}
| {p,q}
|
|- align=center
|Rectified
|{{CDD|node|p|node_1|q|node|r|node}}
|t1{p,q,r}
| <math>\begin{Bmatrix} p \ \ \\ q , r \end{Bmatrix}</math> = r{p,q,r}
| <math>\begin{Bmatrix} p \\ q \end{Bmatrix}</math> = r{p,q}
| {q,r}
|- align=center
|Birectified (Dual rectified)
|{{CDD|node|p|node|q|node_1|r|node}}
|t2{p,q,r}
| <math>\begin{Bmatrix} q , p \\ r \ \ \end{Bmatrix}</math> = r{r,q,p}
| {q,r}
|<math>\begin{Bmatrix} q \\ r \end{Bmatrix}</math> = r{q,r}
|- align=center
|Trirectified (Dual)
|{{CDD|node|p|node|q|node|r|node_1}}
|t3{p,q,r}
| {r,q,p}
|
| {r,q}
|}

==== Regular [[polyteron]]s and 4-space [[Honeycomb (geometry)|honeycombs]] ====
[[Facet (mathematics)|Facet]]s are regular or rectified polychora.
{| class="wikitable"
!rowspan=2|name {p,q,r,s}
!rowspan=2|[[Coxeter-Dynkin diagram|Coxeter-Dynkin]]
!rowspan=2|t-notation [[Schläfli symbol]]
!colspan=3|Extended [[Schläfli symbol]]
|-
!Name
!Facet-1
!Facet-2
|- align=center
|Parent
|{{CDD|node_1|p|node|q|node|r|node|s|node}}
|t0{p,q,r,s}
| {p,q,r,s}
| {p,q,r}
|
|- align=center
|Rectified
|{{CDD|node|p|node_1|q|node|r|node|s|node}}
|t1{p,q,r,s}
| <math>\begin{Bmatrix} p \ \ \ \ \ \\ q , r , s \end{Bmatrix}</math> = r{p,q,r,s}
|<math>\begin{Bmatrix} p \ \ \\ q , r \end{Bmatrix}</math> = r{p,q,r}
|{q,r,s}
|- align=center
|Birectified (Birectified dual)
|{{CDD|node|p|node|q|node_1|r|node|s|node}}
|t2{p,q,r,s}
| <math>\begin{Bmatrix} q , p \\ r , s \end{Bmatrix}</math> = 2r{p,q,r,s}
|<math>\begin{Bmatrix} q , p \\ r \ \ \end{Bmatrix}</math> = r{r,q,p}
|<math>\begin{Bmatrix} q \ \ \\ r , s \end{Bmatrix}</math> = r{q,r,s}
|- align=center
|Trirectified (Rectified dual)
|{{CDD|node|p|node|q|node|r|node_1|s|node}}
|t3{p,q,r,s}
| <math>\begin{Bmatrix} r , q , p \\ s \ \ \ \ \ \end{Bmatrix}</math> = r{s,r,q,p}
|{r,q,p}
|<math>\begin{Bmatrix} r , q \\ s \ \ \end{Bmatrix}</math> = r{s,r,q}
|- align=center
|Quadrirectified (Dual)
|{{CDD|node|p|node|q|node|r|node|s|node_1}}
|t4{p,q,r,s}
| {s,r,q,p}
|
| {s,r,q}
|}

==See also==
* [[Dual polytope]]
* [[Quasiregular polyhedron]]
* [[List of regular polytopes]]
* [[Truncation (geometry)]]
* [[Conway polyhedron notation]]

== References ==
* [[Coxeter|Coxeter, H.S.M.]] ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 (pp.145-154 Chapter 8: Truncation)
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
** [[Norman Johnson (mathematician)|N.W. Johnson]]: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966

== External links ==
* {{Mathworld | urlname=Rectification | title=Rectification }}
* {{GlossaryForHyperspace | anchor=Rectification | title=Rectification }}

{{Polyhedron_operators}}

[[Category:Polyhedra]]
[[Category:Polychora]]
[[Category:Polytopes]]

Quantal response equilibrium

2011-11-08T11:04:39Z

89.168.89.97:

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