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A '''regular polyhedron''' is a [[polyhedron]] whose [[symmetry group]] acts transitively on its [[Flag (geometry)|flag]]s. A regular polyhedron is highly symmetrical, being all of [[edge-transitive]], [[vertex-transitive]] and [[face-transitive]]. In classical contexts, many different equivalent definitions are used; a common one is that faces are [[Congruence (geometry)|congruent]] [[regular polygons]] which are assembled in the same way around each [[vertex (geometry)|vertex]].
[[File:Autoprotolyse eau.svg|thumb|upright=1.5]]
{{Acids and bases}}
The '''self-ionization of water''' (also '''autoionization of water''', and '''autodissociation of water''') is an [[ionization]] reaction in [[properties of water|pure water]] or an [[aqueous solution]], in which a water molecule, H<sub>2</sub>O, [[deprotonation|loses the nucleus of one of its hydrogen atoms]] to become a [[hydroxide]] ion, OH<sup>−</sup>. The [[hydron (chemistry)|hydrogen nucleus, H<sup>+</sup>]], immediately [[protonation|protonates]] another water molecule to form [[hydronium]], H<sub>3</sub>O<sup>+</sup>. It is an example of [[autoprotolysis]], and exemplifies the [[amphoteric]] nature of water.


A regular polyhedron is identified by its [[Schläfli symbol]] of the form {''n'', ''m''}, where ''n'' is the number of sides of each face and ''m'' the number of faces meeting at each vertex. There are 5 finite convex regular polyhedra, known as the [[Platonic solid]]s. These are the: [[tetrahedron]] {3, 3}, [[cube]] {4, 3}, [[octahedron]] {3, 4}, [[dodecahedron]] {5, 3} and [[regular icosahedron|icosahedron]] {3, 5}. There are also four regular star polyhedra, making nine regular polyhedra in all.
==Equilibrium constant==
Chemically pure water has an electrical [[Conductivity (electrolytic)|conductivity]] of 0.055&nbsp;&micro;[[Siemens (unit)|S]]∙cm<sup>−1</sup>. According to the theories of [[Svante Arrhenius]], this must be [[ionic conductor|due to the presence of ions]]. The ions are produced by the self-ionization reaction
:H<sub>2</sub>O + H<sub>2</sub>O {{eqm}} H<sub>3</sub>O<sup>+</sup> + OH<sup>−</sup>
This equilibrium applies to pure water and any aqueous solution.


== The regular polyhedra ==
Approximating [[activity (chemistry)|activities]] by concentrations, the chemical [[equilibrium constant]], ''K''<sub>eq</sub>, for this reaction is given by:
:<math>K_{\rm{eq}} = \frac{[{\rm{H_3O^+}}] [{\rm{OH^-}}]}{[{\rm{H_2O}}]^2}</math>
If the [[concentration (chemistry)|concentration]] of dissolved [[solution|solutes]] is low, the concentration [H<sub>2</sub>O] can be taken as being constant at c. 55.5M.<ref>McMurry, John. (2004) Organic Chemistry, p. 44</ref>


There are five [[Convex polygon|convex]] regular polyhedra, known as the '''[[Platonic solid]]s''', and four regular [[star polyhedron|star polyhedra]], the '''[[Kepler-Poinsot polyhedra]]''':
Expressed with activities {{mvar|a}}, instead of concentrations, the thermodynamic equilibrium constant for the water ionization reaction is:
:<math>K_{\rm w} = \frac{a_{\rm{H_3O^+}} \cdot a_{\rm{OH^-}}}{a_{\rm{H_2O}}^2}</math>


=== Platonic solids ===
which is numerically equal to the more traditional thermodynamic equilibrium constant written as:
{{Main|Platonic solid}}
:<math>K_{\rm w} = \frac{a_{\rm{H^+}} \cdot a_{\rm{OH^-}}}{a_{\rm{H_2O}}}</math>


:{| class="wikitable"
under the assumption that the sum of the chemical potentials of H<sup>+</sup> and H<sub>3</sub>O<sup>+</sup> is formally equal to twice the chemical potential of H<sub>2</sub>O at the same temperature and pressure.<ref name="rel">[http://www.iapws.org/relguide/Ionization.pdf "Release on the Ionization Constant of H<sub>2</sub>O"] The International Association for the Properties of Water and Steam, Lucerne, Switzerland, August 2007.</ref>
|- align=center
|[[Image:Tetrahedron.png|100px]]
|[[Image:Hexahedron.png|100px]]
|[[Image:Octahedron.png|100px]]
|[[Image:Dodecahedron.png|100px]]
|[[Image:Icosahedron.png|100px]]
|- align=center
|[[Tetrahedron]] {3,&nbsp;3}||[[Cube]] {4,&nbsp;3}||[[Octahedron]] {3,&nbsp;4}||[[Dodecahedron]] {5,&nbsp;3}||[[Regular icosahedron|Icosahedron]] {3,&nbsp;5}
|}


=== Kepler-Poinsot polyhedra ===
In infinitely dilute aqueous solution, the activity of water solvent is unity.


{{Main|Kepler-Poinsot polyhedra}}
The '''ionization constant''', '''dissociation constant''', '''self-ionization constant''', or '''ionic product''' of water, symbolized by ''K''<sub>w</sub> may be given by:
:<math>K_{\rm w}=[{\rm{H_3O^+}}][{\rm{OH^-}}] = K_{\rm{eq}} \cdot [{\rm{H_2O}}]^2 </math>
where [H<sub>3</sub>O<sup>+</sup>] is the concentration of hydrogen or [[hydronium ion]], and [OH<sup>−</sup>] is the concentration of [[hydroxide]] ion.


:{| class="wikitable"
The ionic product of water can be also expressed on activity basis as:<ref>IUPAC. Compendium of Chemical Terminology, 2nd ed. (the "Gold Book"). Compiled by A. D. McNaught and A. Wilkinson. Blackwell Scientific Publications, Oxford (1997). XML on-line corrected version: http://goldbook.iupac.org (2006–) created by M. Nic, J. Jirat, B. Kosata; updates compiled by A. Jenkins. ISBN 0-9678550-9-8. doi:10.1351/goldbook.    [http://goldbook.iupac.org/A00532.html Entry: autoprotolysis constant].</ref>
|- align=center
:<math>K_{\rm w}=a_{\rm{H_3O^+}} \cdot a_{\rm{OH^-}} </math>
|[[Image:Small stellated dodecahedron.png|100px]]
|[[Image:Great dodecahedron.png|100px]]
|[[Image:Great stellated dodecahedron.png|100px]]
|[[Image:Great icosahedron.png|100px]]
|- align=center
|[[Small stellated dodecahedron]]<br>{5/2,&nbsp;5}
||[[Great dodecahedron]]<br>{5,&nbsp;5/2}
||[[Great stellated dodecahedron]]<br>{5/2,&nbsp;3}
||[[Great icosahedron]]<br>{3,&nbsp;5/2}
|}


== Characteristics ==
At 25&nbsp;°C ''K''<sub>w</sub> is equal to {{val|1.0|e=-14}}.
===Equivalent properties===


The property of having a similar arrangement of faces around each vertex can be replaced by any of the following equivalent conditions in the definition:
==Dependence on temperature, pressure and ionic strength==
{|
|[[Image:Temperature dependence water ionization.svg|thumb|240px|Temperature dependence of the water ionization constant at 25 MPa]]
|[[Image:Pressure dependence water ionization pKw on P.svg|thumb|300px|Pressure dependence of the water ionization constant at 25&nbsp;°C]]
|[[File:Kw vs I.png|thumb|270px|Variation of pK<sub>w</sub> with ionic strength of NaCl solutions at 25&nbsp;°C]]
|}
The dependence of the water ionization on temperature and pressure has been investigated thoroughly.<ref>[http://www.iapws.org/ International Association for the Properties of Water and Steam (IAPWS)]</ref> The value of pK<sub>w</sub> decreases as temperature increases from the melting point of ice to a minimum at c.&nbsp;250&nbsp;°C, after which it increases up to the [[critical point (thermodynamics)|critical point]] of water c.&nbsp;374&nbsp;°C. It  decreases with increasing pressure.


*The vertices of the polyhedron all lie on a [[sphere]].
{| class="wikitable" style="text-align: center;"
*All the [[dihedral angle]]s of the polyhedron are equal
|+pK<sub>w</sub> values for liquid water.<ref>{{cite journal | last1 = Bandura | first1 = Andrei V. | last2 = Lvov | first2 = Serguei N. | year = 2006 | title = The Ionization Constant of Water over Wide Ranges of Temperature and Density | journal = Journal of Physical and Chemical Reference Data | volume = 35 | issue = 1 | pages = 15–30 | doi = 10.1063/1.1928231 | url = http://www.nist.gov/data/PDFfiles/jpcrd696.pdf }}</ref>
*All the [[vertex figure]]s of the polyhedron are [[regular polygon]]s.
|-
*All the [[solid angle]]s of the polyhedron are congruent. (Cromwell, 1997)
! Temperature
 
! Pressure<ref>0.1 MPa for T &lt; 100°C. [[Vapor pressure of water|Saturation pressure]] for T &ge; 100°C.</ref>
===Concentric spheres===
! pK<sub>w</sub>
 
|-
A regular polyhedron has all of three related spheres (other polyhedra lack at least one kind) which share its centre:
|0°C
* An [[insphere]], tangent to all faces.
|0.10 MPa
* An intersphere or [[midsphere]], tangent to all edges.
|14.95
* A [[circumsphere]], tangent to all vertices.
|-
 
|25°C
===Symmetry===
|0.10 MPa
 
|13.99
The regular polyhedra are the most [[symmetry|symmetrical]] of all the polyhedra. They lie in just three [[symmetry group]]s, which are named after them:
|-
*Tetrahedral
|50°C
*Octahedral (or cubic)
|0.10 MPa
*Icosahedral (or dodecahedral)
|13.26
 
|-
Any shapes with icosahedral or octahedral symmetry will also contain tetrahedral symmetry.
|75°C
 
|0.10 MPa
===Euler characteristic===
|12.70
 
|-
The five Platonic solids have an [[Euler characteristic]] of 2. Some of the regular stars have a [[Kepler-Poinsot polyhedra#The Euler characteristic|different value]].
|100°C
 
|0.10 MPa
===Interior points===
|12.25
 
The sum of the distances from any point in the interior of a regular polyhedron to the sides is independent of the location of the point. (This is an extension of [[Viviani's theorem]].) However, the converse does not hold, not even for [[tetrahedra]].<ref>Chen, Zhibo, and Liang, Tian. "The converse of Viviani's theorem", ''[[The College Mathematics Journal]]'' 37(5), 2006, pp. 390–391.</ref>
 
== Duality of the regular polyhedra ==
In a [[dual polyhedron|dual]] pair of polyhedra, the vertices of one polyhedron correspond to the faces of the other, and vice versa.
 
The regular polyhedra show this duality as follows:
* The [[tetrahedron]] is self-dual, i.e. it pairs with itself.
* The [[cube]] and [[octahedron]] are dual to each other.
* The [[regular icosahedron|icosahedron]] and [[dodecahedron]] are dual to each other.
* The [[small stellated dodecahedron]] and [[great dodecahedron]] are dual to each other.
* The [[great stellated dodecahedron]] and [[great icosahedron]] are dual to each other.
 
The Schläfli symbol of the dual is just the original written backwards, for example the dual of {5, 3} is {3, 5}.
 
==History==<!-- This section is linked from [[Polyhedron]] -->
 
'''See also''' [[Regular polytope#History of discovery|''Regular polytope: History of discovery'']].
 
=== Prehistory ===
 
Stones carved in shapes resembling clusters of spheres or knobs have been found in [[Scotland]] and may be as much as 4,000 years old. Some of these stones show not only the symmetries of the five Platonic solids, but also some of the relations of duality amongst them (that is, that the centres of the faces of the cube gives the vertices of an octahedron). Examples of these stones are on display in the John Evans room of the [[Ashmolean Museum]] at [[Oxford University]]. Why these objects were made, or how their creators gained the inspiration for them, is a mystery. There is doubt regarding the mathematical interpretation of these objects, as many have non-platonic forms, and perhaps only one has been found to be a true icosahedron, as opposed to a reinterpretation of the icosahedron dual, the dodecahedron.<ref>[http://www.neverendingbooks.org/index.php/the-scottish-solids-hoax.html The Scottish Solids Hoax], </ref>
 
It is also possible that the [[Etruscan civilization|Etruscans]] preceded the Greeks in their awareness of at least some of the regular polyhedra, as evidenced by the discovery near [[Padua]] (in Northern [[Italy]]) in the late 19th century of a [[dodecahedron]] made of [[soapstone]], and dating back more than 2,500 years (Lindemann, 1987).
 
=== Greeks ===
 
The earliest known ''written'' records of the regular convex solids originated from Classical Greece. When these solids were all discovered and by whom is not known, but [[Theaetetus (mathematician)|Theaetetus]], (an [[Athens|Athenian]]), was the first to give a mathematical description of all five (Van der Waerden, 1954), (Euclid, book XIII). [[H.S.M. Coxeter]] (Coxeter, 1948, Section 1.9) credits [[Plato]] (400 BC) with having made models of them, and mentions that one of the earlier [[Pythagoreans]], [[Timaeus of Locri]], used all five in a correspondence between the polyhedra and the nature of the universe as it was then perceived - this correspondence is recorded in Plato's dialogue [[Timaeus (dialogue)|''Timaeus'']]. Euclid's reference to Plato led to their common description as the ''Platonic solids''.
 
One might characterise the Greek definition as follows:
*A regular polygon is a ([[Convex polygon|convex]]) planar figure with all edges equal and all corners equal
*A regular polyhedron is a solid (convex) figure with all faces being congruent regular polygons, the same number arranged all alike around each vertex.
 
This definition rules out, for example, the [[square pyramid]] (since although all the faces are regular, the square base is not congruent to the triangular sides), or the shape formed by joining two tetrahedra together (since although all faces of that [[triangular bipyramid]] would be equilateral triangles, that is, congruent and regular, some vertices have 3 triangles and others have 4).
 
This concept of a regular polyhedron would remain unchallenged for almost 2000 years.
 
=== Regular star polyhedra ===
 
Regular star polygons such as the [[pentagram]] (star pentagon) were also known to the ancient Greeks - the [[pentagram]] was used by the [[Pythagoreans]] as their secret sign, but they did not use them to construct polyhedra. It was not until the early 17th century that [[Johannes Kepler]] realised that pentagrams could be used as the faces of regular [[star polyhedron|star polyhedra]]. Some of these star polyhedra may have been discovered by others before Kepler's time, but Kepler was the first to recognise that they could be considered "regular" if one removed the restriction that regular polyhedra be convex. Two hundred years later [[Louis Poinsot]] also allowed star [[vertex figure]]s (circuits around each corner), enabling him to discover two new regular star polyhedra along with rediscovering Kepler's. These four are the only regular star polyhedra, and have come to be known as the [[Kepler-Poinsot polyhedra]]. It was not until the mid-19th century, several decades after Poinsot published, that Cayley gave them their modern English names: (Kepler's) [[small stellated dodecahedron]] and [[great stellated dodecahedron]], and (Poinsot's) [[great icosahedron]] and [[great dodecahedron]].
 
The Kepler-Poinsot polyhedra may be constructed from the Platonic solids by a process called [[stellation]]. The reciprocal process to stellation is called [[facetting]] (or faceting). Every stellation of one polyhedron is [[Dual polyhedron|dual]], or reciprocal, to some facetting of the dual polyhedron. The regular star polyhedra can also be obtained by facetting the Platonic solids. This was first done by Bertrand around the same time that Cayley named them.
 
By the end of the 19th century there were therefore nine regular polyhedra - five convex and four star.
 
== Regular polyhedra in nature ==
 
Each of the Platonic solids occurs naturally in one form or another.
 
The tetrahedron, cube, and octahedron all occur as [[crystal]]s. These by no means exhaust the numbers of possible forms of crystals (Smith, 1982, p212), of which there are 48. Neither the [[regular icosahedron]] nor the [[regular dodecahedron]] are amongst them, although one of their forms appears as a [[quasicrystal]]: the [[pyritohedron]] has twelve non-regular pentagonal faces arranged in the same pattern as the regular dodecahedron.
 
A more recent discovery is of a series of new types of [[carbon]] molecule, known as the [[fullerene]]s (see Curl, 1991). Although C<sub>60</sub>, the most easily produced fullerene, looks more or less spherical, some of the larger varieties (such as C<sub>240</sub>, C<sub>480</sub> and C<sub>960</sub>) are hypothesised to take on the form of slightly rounded icosahedra, a few nanometres across.
 
[[File:Circogoniaicosahedra ekw.jpg|left|frame|Circogonia icosahedra, a species of [[Radiolaria]].]]
 
Polyhedra appear in biology as well. In the early 20th century, [[Ernst Haeckel]] described a number of species of [[Radiolaria]], some of whose skeletons are shaped like various regular polyhedra (Haeckel, 1904). Examples include ''Circoporus octahedrus'', ''Circogonia icosahedra'', ''Lithocubus geometricus'' and ''Circorrhegma dodecahedra''; the shapes of these creatures are indicated by their names. The outer protein shells of many [[virus]]es form regular polyhedra. For example, [[HIV]] is enclosed in a regular icosahedron.
 
In ancient times the [[Pythagoreanism|Pythagoreans]] believed that there was a harmony between the regular polyhedra and the orbits of the [[planet]]s. In the 17th century, [[Johannes Kepler]] studied data on planetary motion compiled by [[Tycho Brahe]] and for a decade tried to establish the Pythagorean ideal by finding a match between the sizes of the polyhedra and the sizes of the planets' orbits. His search failed in its original objective, but out of this research came Kepler's discoveries of the Kepler solids as regular polytopes, the realisation that the orbits of planets are not circles, and [[Kepler's laws of planetary motion|the laws of planetary motion]] for which he is now famous. In Kepler's time only five planets (excluding the earth) were known, nicely matching the number of Platonic solids. Kepler's work, and the discovery since that time of [[Uranus]] and [[Neptune]], have invalidated the Pythagorean idea.
 
Around the same time as the Pythagoreans, Plato described a theory of matter in which the five elements (earth, air, fire, water and spirit) each comprised tiny copies of one of the five regular solids. Matter was built up from a mixture of these polyhedra, with each substance having different proportions in the mix. Two thousand years later [[Dalton's atomic theory]] would show this idea to be along the right lines, though not related directly to the regular solids.
 
== Further generalisations ==
The 20th century saw a succession of generalisations of the idea of a regular polyhedron, leading to several new classes.
 
=== Regular skew polyhedra ===
 
{{Main|Regular skew polyhedron}}
 
In the first decades, Coxeter and Petrie allowed "saddle" vertices with alternating ridges and valleys, enabling them to construct three infinite folded surfaces which they called [[regular skew polyhedron|regular skew polyhedra]].<ref>[[Coxeter]], ''The Beauty of Geometry: Twelve Essays'', Dover Publications, 1999, ISBN 0-486-40919-8 (Chapter 5: Regular Skew Polyhedra in three and four dimensions and their topological analogues, Proceedings of the London Mathematics Society, Ser. 2, Vol 43, 1937.)</ref> Coxeter offered a modified [[Schläfli symbol]] {l,m|n} for these figures, with {l,m} implying the [[vertex figure]], with ''m'' regular ''l''-gons around a vertex. The ''n'' defines ''n''-gonal ''holes''. Their vertex figures are [[regular skew polygon]]s, vertices zig-zagging between two planes.
 
{| class="wikitable"
!colspan=3| Infinite regular skew polyhedra in 3-space (partially drawn)
|-
|-
|align=center|[[Image:Six-square skew polyhedron.png|210px]]<br>{4,6|4}
|150°C
|align=center|[[Image:Four-hexagon skew polyhedron.png|210px]]<br>{6,4|4}
|0.47 MPa
|align=center|[[Image:Six-hexagon skew polyhedron.png|210px]]<br>{6,6|3}
|11.64
|}
 
Finite regular skew polyhedra exist in 4-space. These finite regular skew polyhedra in 4-space can be seen as a subset of the faces of [[uniform 4-polytope]]s. Two dual solutions are related to the [[5-cell]], two dual solutions are related to the [[24-cell]], and an infinite set of self-dual [[duoprism]]s generate regular skew polyhedra as {4, 4 &#124; n}. In the infinite limit these approach a [[duocylinder]] and look like a [[torus]] in their [[stereographic projection]]s into 3-space.
 
{| class=wikitable
|+ Finite regular skew polyhedra in 4-space
|-
|-
!colspan=4|Orthogonal [[Coxeter plane]] projections
|200°C
!rowspan=2|[[Stereographic projection]]
|1.5 MPa
|11.31
|-
|-
!colspan=2| A<sub>4</sub>
|250°C
!colspan=2| F<sub>4</sub>
|4.0 MPa
|11.20
|-
|-
|[[File:4-simplex t03.svg|150px]]
|300°C
|[[File:4-simplex t12.svg|150px]]
|8.7 MPa
|[[File:24-cell t03 F4.svg|150px]]
|11.34
|[[File:24-cell t12 F4.svg|150px]]
|[[File:Clifford-torus.gif|150px]]
|-
|-
![[Runcinated 5-cell#Related skew polyhedron|{4, 6 &#124; 3}]]
|350°C
![[Bitruncated 5-cell#Related skew polyhedron|{6, 4 &#124; 3}]]
|17 MPa
![[Runcinated 24-cell#Related skew polyhedron|{4, 8 &#124; 3}]]
|11.92
![[Bitruncated 24-cell#Related skew polyhedron|{8, 4 &#124; 3}]]
![[Duoprism#Related polytopes|{4, 4 &#124; n}]]
|}
|}


=== Regular polyhedra in non-Euclidean spaces ===
With [[electrolyte]] solutions, the value of pK<sub>w</sub> is dependent on [[ionic strength]] of the electrolyte. Values for [[sodium chloride]] are typical for a 1:1 electrolyte. With 1:2 electrolytes, MX<sub>2</sub>, pK<sub>w</sub> decreases with increasing ionic strength.<ref>{{cite book|last1=Harned|first1=H.S. |last2=Owen, |first2=B.B. |title=The Physical Chemistry of Electrolytic Solutions|edition=£rd.|year=1958|publisher=Reinhold Publishing Corp., |location=New York|pages=634–649, 752–754}}</ref>
Studies of non-Euclidean [[hyperbolic space|hyperbolic]], [[elliptic space|elliptic]] and [[complex space|complex]] spaces, discovered over the preceding century, led to the  discovery of more new polyhedra such as [[complex polytope|complex polyhedra]] which could only take regular geometric form in those spaces.


=== Abstract regular polyhedra ===
The value of ''K''<sub>w</sub> is usually of interest in the [[liquid phase]]. Example values for [[superheated steam]] (gas) and [[supercritical water]] fluid are given in the table.
{{Main|Abstract regular polyhedron}}


By now, polyhedra were firmly understood as three-dimensional examples of more general ''[[polytope]]s'' in any number of dimensions. The second half of the century saw the development of abstract algebraic ideas such as [[Polyhedral combinatorics]], culminating in the idea of an [[abstract polytope]] as a [[partially ordered set]] (poset) of elements. The elements of an abstract polyhedron are its body (the maximal element), its faces, edges, vertices and the ''null polytope'' or empty set. These abstract elements can be mapped into ordinary space or ''realised'' as geometrical figures. Some abstract polyhedra have well-formed or ''faithful'' realisations, others do not. A ''flag'' is a connected set of elements of each dimension - for a polyhedron that is the body, a face, an edge of the face, a vertex of the edge, and the null polytope. An abstract polytope is said to be ''regular'' if its combinatorial symmetries are transitive on its flags - that is to say, that any flag can be mapped onto any other under a symmetry of the polyhedron. Abstract regular polytopes remain an active area of research.
:{| class="wikitable" style="text-align:center"
 
|+Comparison of pK<sub>w</sub> values for liquid water, superheated steam, and  supercritical water.<ref name="rel"/>
Five such regular abstract polyhedra, which can not be realised faithfully, were identified by [[H. S. M. Coxeter]] in his book ''[[Regular Polytopes]]'' (1977) and again by [[J. M. Wills]] in his paper "The combinatorially regular polyhedra of index 2" (1987).<ref>[http://homepages.wmich.edu/~drichter/regularpolyhedra.htm The Regular Polyhedra (of index two)], David A. Richter</ref> They are all topologically equivalent to [[toroid]]s. Their construction, by arranging ''n'' faces around each vertex, can be repeated indefinitely as tilings of the [[Hyperbolic geometry#Models of the hyperbolic plane|hyperbolic plane]]. In the diagrams below, the hyperbolic tiling images have colors corresponding to those of the polyhedra images.
|-
 
! scope="row" |T/°C
:{| class="wikitable" width=400
| 350 || 400 || 450 || 500 || 600 || 800
|- align=center
|-  
! Polyhedron
! scope="row" |0.1 MPa
|[[Image:DU36 medial rhombic triacontahedron.png|100px]]<br>[[Medial rhombic triacontahedron]]
|11.920 (liquid)<sup>a</sup> || 47.961<sup>b</sup> || 47.873<sup>b</sup> || 47.638 <sup>b</sup> || 46.384 <sup>b</sup> ||40.785 <sup>b</sup>
|[[Image:Dodecadodecahedron.png|100px]]<br>[[Dodecadodecahedron]]
|-
|[[Image:DU41 medial triambic icosahedron.png|100px]]<br>[[Medial triambic icosahedron]]
! scope="row" |25 MPa
|[[Image:Ditrigonal dodecadodecahedron.png|100px]]<br>[[Ditrigonal dodecadodecahedron]]
|11.551 (liquid)<sup>c</sup> ||16.566||18.135||18.758||19.425||20.113
|[[Image:Excavated dodecahedron.png|100px]]<br>[[Excavated dodecahedron]]
|-
|- align=center
! scope="row" |100 MPa
![[Vertex figure]]
|10.600 (liquid)<sup>c</sup> ||10.744||11.005||11.381||12.296||13.544
|{5}, {5/2}<br>[[File:Regular polygon 5.svg|40px]][[File:Pentagram green.svg|40px]]
|-
|(5.5/2)<sup>2</sup><br>[[File:Dodecadodecahedron vertfig.png|60px]]
! scope="row" |1000 MPa
|{5}, {5/2}<br>[[File:Regular polygon 5.svg|40px]][[File:Pentagram green.svg|40px]]
|8.311 (liquid)<sup>c</sup> ||8.178||8.084||8.019||7.952||7.957
|(5.5/3)<sup>3</sup><br>[[File:Ditrigonal dodecadodecahedron vertfig.png|60px]]
|[[File:Medial triambic icosahedron face.png|60px]]
|- align=center valign=top
!Faces
|30 rhombi<br>[[File:Rhombus definition2.svg|60px]]
|12 pentagons<br>12 pentagram<br>[[File:Regular polygon 5.svg|40px]][[File:Pentagram green.svg|40px]]
|20 hexagons<br>[[File:Medial triambic icosahedron face.png|60px]]
|12 pentagons<br>12 pentagrams<br>[[File:Regular polygon 5.svg|40px]][[File:Pentagram green.svg|40px]]
|20 hexagons<br>[[File:Star hexagon face.png|60px]]
|- align=center
! Tiling
|[[Image:Uniform tiling 45-t0.png|100px]]<br>[[Order-5 square tiling|{4, 5}]]
|[[Image:Uniform tiling 552-t1.png|100px]]<br>[[Order-4 pentagonal tiling|{5, 4}]]
|[[Image:Uniform tiling 65-t0.png|100px]]<br>[[Order-5 hexagonal tiling|{6, 5}]]
|[[Image:Uniform tiling 553-t1.png|100px]]<br>[[Order-6 pentagonal tiling|{5, 6}]]
|[[Image:Uniform tiling 66-t2.png|100px]]<br>[[Order-6 hexagonal tiling|{6, 6}]]
|}
|}
''Notes to the Table. The values are for supercritical fluid except those marked: <sup>a</sup> at saturation pressure corresponding to 350 °C. <sup>b</sup> superheated steam. <sup>c</sup> subcooled liquid.''


These occur as dual pairs as follows:
==Isotope effects==
*The [[medial rhombic triacontahedron]] and [[dodecadodecahedron]] are dual to each other.
[[Heavy water]], D<sub>2</sub>O, self-ionizes less than normal water, H<sub>2</sub>O;
*The [[medial triambic icosahedron]] and [[ditrigonal dodecadodecahedron]] are dual to each other.
:D<sub>2</sub>O + D<sub>2</sub>O {{eqm}} D<sub>3</sub>O<sup>+</sup> + OD<sup>−</sup>
*The [[excavated dodecahedron]] is self-dual.
This is attributed to oxygen forming a slightly stronger bond to [[deuterium]] because the larger mass of deuterium difference results in a lower [[zero-point energy]], a quantum mechanical effect.


==== Tilings of the real projective plane ====
The dissociation constant and the ionic product of heavy water is given by:
Another group of regular abstract polyhedra comprise tilings of the [[real projective plane]]. These include the [[Hemi-cube (geometry)|hemi-cube]], [[hemi-octahedron]], [[hemi-dodecahedron]], and [[hemi-icosahedron]]. They are (globally) [[projective polyhedra]], and are the projective counterparts of the [[Platonic solid]]s. The tetrahedron does not have a projective counterpart as it does not have parallel faces which can be identified as one, as the other four Platonic solids do.
:<math>K_{\rm{eq}} = \frac{[{\rm{D_3O^+}}] [{\rm{OD^-}}]}{[{\rm{D_2O}}]^2}</math>
 
:<math>K_{\rm w}=[{\rm{D_3O^+}}][{\rm{OD^-}}] = K_{\rm{eq}} \times [{\rm{D_2O}}]^2 </math>
{| class=wikitable
The following table compares the values of pK<sub>w</sub> for H<sub>2</sub>O and D<sub>2</sub>O.<ref name=crc>{{cite book | author=Lide, D. R. (Ed.) | title=CRC Handbook of Chemistry and Physics (70th Edn.) | publisher=Boca Raton (FL):CRC Press | year=1990}}</ref>
|- align=center
:{| class="wikitable" style="text-align:center"
|[[File:Hemicube2.PNG|150px]]<br>[[Hemicube (geometry)|Hemi-cube]]<br>{4,3}
|+pK<sub>w</sub> values for pure water
|[[File:Hemioctahedron.png|150px]]<br>[[Hemi-octahedron]]<br>{3,4}
|-
|[[File:Hemi-Dodecahedron2.PNG|150px]]<br>[[Hemi-dodecahedron]]<br>{3,5}
! scope="row" |T/°C
|[[File:Hemi-icosahedron.png|150px]]<br>[[Hemi-icosahedron]]<br>{5,3}
|10||20|| 25||30|| 40 || 50 
|-
! scope="row" |H<sub>2</sub>O
|14.535 || 14.167|| 13.997|| 13.830|| 13.535 ||13.262
|-
! scope="row" |D<sub>2</sub>O
|15.439||15.049||14.869||14.699||14.385|| 14.103
|}
|}


These occur as dual pairs in the same way as the original Platonic solids do.
===Ionization equilibria in water - heavy water mixtures===
In water - heavy water mixtures equilibria several species are involved:H<sub>2</sub>O, HDO, D<sub>2</sub>O, H<sub>3</sub>O<sup>+</sup>, D<sub>3</sub>O<sup>+</sup>, H<sub>2</sub>D</sub>O<sup>+</sup>, HD<sub>2</sub>O<sup>+</sup>, HO<sup>-</sup>, DO<sup>-</sup>.


=== Spherical polyhedra ===
==Mechanism==
{{main|Spherical polyhedron}}


The usual nine regular polyhedra can also be represented as spherical tilings (tilings of the [[sphere]]):
The [[reaction rate|rate of reaction]] for the dissociation
: H<sub>2</sub>O → H<sup>+</sup> + OH<sup>−</sup>
depends on the [[activation energy]], ΔE<sup>‡</sup>. According to the [[Boltzmann distribution]] the proportion of water molecules that have sufficient energy, due to thermal population, is given by


{| class=wikitable
:<math>\frac{N}{N_0} = e^{-\frac{\Delta E^\ddagger}{kT}}</math>
|- align=center
where {{mvar|k}} is the [[Boltzmann constant]]. Thus some dissociation can occur because sufficient thermal energy is available. The following sequence of events has been proposed on the basis of [[electric field]] fluctuations in liquid water.<ref>{{cite journal | author = Geissler, P. L.; Dellago, C.; Chandler, D.; Hutter, J.; Parrinello, M. | year = 2001 | title = Autoionization in liquid water | journal = [[Science (journal)|Science]] | volume = 291 | pages = 2121–2124 | doi = 10.1126/science.1056991 | pmid = 11251111 | issue = 5511|bibcode = 2001Sci...291.2121G }}</ref> Random fluctuations in molecular motions occasionally (about once every 10 hours per water molecule<ref>{{cite journal | author = Eigen, M.; de Maeyer, L. | year = 1955 | title = Untersuchungen über die Kinetik der Neutralisation I | journal = Z. Elektrochem. | volume = 59 | pages = 986}}</ref>) produce an electric field strong enough to break an oxygen–hydrogen [[covalent bond|bond]], resulting in a hydroxide (OH<sup>−</sup>) and hydronium ion (H<sub>3</sub>O<sup>+</sup>); the hydrogen nucleus of the hydronium ion travels along water molecules by the [[Grotthuss mechanism]] and a change in the [[hydrogen bond]] network in the solvent isolates the two ions, which are stabilized by solvation. Within 1&nbsp;[[picosecond]], however, a second reorganization of the hydrogen bond network allows rapid proton transfer down the electric potential difference and subsequent recombination of the ions. This timescale is consistent with the time it takes for hydrogen bonds to reorientate themselves in water.<ref>{{cite journal | author = Stillinger, F. H. | year = 1975 | journal = Adv. Chem. Phys. | volume = 31 | pages = 1 | doi = 10.1002/9780470143834.ch1 | title = Theory and Molecular Models for Water}}</ref><ref>{{cite journal | author = Rapaport, D. C. | year = 1983 | journal = [[Mol. Phys.]] | volume = 50 | pages = 1151 | doi = 10.1080/00268978300102931 | title = Hydrogen bonds in water | issue = 5|bibcode = 1983MolPh..50.1151R }}</ref><ref>{{cite journal | author = Chen, S.-H. &amp; Teixeira, J. | year = 1986 | journal = Adv. Chem. Phys | volume = 64 | pages = 1 | doi = 10.1002/9780470142882.ch1 | title = Structure and Dynamics of Low-Temperature Water as Studied by Scattering Techniques}}</ref>
|[[File:Uniform tiling 332-t0-1-.png|100px]]<br>[[Tetrahedron]]<br>{3,3}
|[[File:Uniform tiling 432-t0.png|100px]]<br>[[Cube]]<br>{4,3}
|[[File:Uniform tiling 432-t2.png|100px]]<br>[[Octahedron]]<br>{3,4}
|[[File:Uniform tiling 532-t0.png|100px]]<br>[[Dodecahedron]]<br>{5,3}
|[[File:Uniform tiling 532-t2.png|100px]]<br>[[regular icosahedron|Icosahedron]]<br>{3,5}
|}
{| class=wikitable
|- align=center
|[[File:Small stellated dodecahedron tiling.png|100px]]<br>[[Small stellated dodecahedron]]<br>{5/2,5}
|[[File:Great dodecahedron tiling.png|100px]]<br>[[Great dodecahedron]]<br>{5,5/2}
|[[File:Great stellated dodecahedron tiling.png|100px]]<br>[[Great stellated dodecahedron]]<br>{5/2,3}
|[[File:Great icosahedron tiling.png|100px]]<br>[[Great icosahedron]]<br>{3,5/2}
|}
 
==== Regular polyhedra that can only exist as spherical polyhedra ====
{{see also|Hosohedron|Dihedron}}
For a regular polyhedron whose Schläfli symbol is {''m'',&nbsp;''n''}, the number of polygonal faces may be found by:
 
:<math>N_2=\frac{4n}{2m+2n-mn}</math>
 
The [[Platonic solid]]s known to antiquity are the only integer solutions for ''m'' ≥ 3 and ''n'' ≥ 3. The restriction ''m'' ≥ 3 enforces that the polygonal faces must have at least three sides.
 
When considering polyhedra as a [[spherical tiling]], this restriction may be relaxed, since [[digon]]s (2-gons) can be represented as spherical lunes, having non-zero [[area (geometry)|area]]. Allowing ''m'' = 2 admits a new infinite class of regular polyhedra, which are the [[hosohedron|hosohedra]]. On a spherical surface, the regular polyhedron {2,&nbsp;''n''} is represented as n abutting lunes, with interior angles of 2π/''n''. All these lunes share two common vertices.<ref name=cox>Coxeter, ''Regular polytopes'', p. 12</ref>
 
A regular [[dihedron]], {''n'',&nbsp;2}<ref name=cox></ref> (2-hedron) in three-dimensional [[Euclidean space]] can be considered a [[degeneracy (mathematics)|degenerate]] [[Prism (geometry)|prism]] consisting of two (planar) ''n''-sided [[polygon]]s connected "back-to-back", so that the resulting object has no depth. However, as a [[spherical tiling]], a dihedron can exist as nondegenerate form, with two ''n''-sided faces covering the sphere, each face being a [[Sphere|hemisphere]], and vertices around a [[great circle]]. It is ''regular'' if the vertices are equally spaced.


{| class=wikitable
== Relationship with the neutral point of water ==
|- align=center
Water molecules dissociate into equal amounts of H<sub>3</sub>O<sup>+</sup> and OH<sup>−</sup>, so their concentrations are equal to {{val|1.00|e=-7|u=mol∙dm<sup>−3</sup>}} at 25 °C. A solution in which the H<sub>3</sub>O<sup>+</sup> and OH<sup>−</sup> concentrations equal each other is considered a '''neutral''' solution.  In general, the pH of the neutral point is numerically equal to pK<sub>w</sub>/2.
|[[File:Hengonal dihedron.png|100px]]<br>[[Henagon]]al dihedron<br>{1,2}
|[[File:Digonal dihedron.png|100px]]<br>[[Digon]]al dihedron<br>{2,2}
|[[File:Trigonal dihedron.png|100px]]<br>[[Triangle|Trigonal]] dihedron<br>{3,2}
|...
|[[File:Hexagonal dihedron.png|100px]]<br>[[Hexagon]]al dihedron<br>{6,2}
|...
|{''n'',2}
|- align=center
|[[File:Henagonal hosohedron.png|100px]]<br>Henagonal hosohedron<br>{2,1}
|[[File:Digonal dihedron.png|100px]]<br>Digonal hosohedron<br>{2,2}
|[[File:Trigonal hosohedron.png|100px]]<br>Trigonal hosohedron<br>{2,3}
|...
|[[File:Hexagonal hosohedron.png|100px]]<br>Hexagonal hosohedron<br>{2,6}
|...
|{2,''n''}
|}


The hosohedron {2,''n''} is dual to the dihedron {''n'',2}. Note that when ''n'' = 2, we obtain the polyhedron {2,2}, which is both a hosohedron and a dihedron.
Pure water is neutral, but most water samples contain impurities. If an impurity is an [[acid]] or [[base (chemistry)|base]] this will affect the concentrations of hydronium ion and hydoxide ion. Water samples which are exposed to air will absorb the acid [[carbon dioxide]] and the concentration of H<sub>3</sub>O<sup>+</sup> will increase. The concentration of OH<sup>−</sup> will decrease in such a way that the product [H<sub>3</sub>O<sup>+</sup>][OH<sup>−</sup>] remains constant for fixed temperature and pressure.


Additionally, there is also the ''[[henagonal henahedron]]'', {1,1}, which consists of a single vertex on the sphere, no edges and a single face as the sphere outside the vertex.{{fact|date=January 2014|reason=this figure is excluded by most definitions of a polyhedron, whether spherical or not}} The ''henagonal henahedron'' is [[self-dual]], i.e. the point and face center can be swapped creating itself as a [[central inversion]]. {''n'',1} and {1,''n''} do not exist for ''n'' ≥ 3.
==See also==
{{Portal|Water}}
*[[Chemical equilibrium]]
*[[Acid–base reaction]]
*[[Standard hydrogen electrode]]


{| class=wikitable
==References==
|- align=center
<references/>
|[[File:Spherical henagonal henahedron.png|100px]]<br>Henagonal henahedron<br>{1,1}
|}


== See also ==
==External links==
* [[Quasiregular polyhedron]]
* [http://www.vias.org/genchem/acidbase_equ_12591_04.html General Chemistry]&nbsp;– Autoionization of Water
* [[Semiregular polyhedron]]
* [[Uniform polyhedron]]
* [[Regular polytope]]


== References ==
{{Chemical equilibria}}
{{reflist}}
* [[Joseph Louis François Bertrand|Bertrand, J.]] (1858). Note sur la théorie des polyèdres réguliers, ''Comptes rendus des séances de l'Académie des Sciences'', '''46''', pp.&nbsp;79–82.
*{{cite book
  | last = Cromwell | first = Peter R. | title = Polyhedra | publisher = Cambridge University Press | year = 1997 | page = 77 | isbn = 0-521-66405-5 }}
* Haeckel, E. (1904). ''[[Kunstformen der Natur]]''. Available as Haeckel, E. ''Art forms in nature'', Prestel USA (1998), ISBN 3-7913-1990-6, or online at http://caliban.mpiz-koeln.mpg.de/~stueber/haeckel/kunstformen/natur.html
*Smith, J. V. (1982). ''Geometrical And Structural Crystallography''. John Wiley and Sons.
* [[Duncan MacLaren Young Sommerville|Sommerville, D. M. Y.]] (1930). ''An Introduction to the Geometry of n Dimensions.'' E. P. Dutton, New York. (Dover Publications edition, 1958). Chapter X: The Regular Polytopes.
*[[H.S.M. Coxeter|Coxeter, H.S.M.]]; Regular Polytopes (third edition). Dover Publications Inc. ISBN 0-486-61480-8


== External links ==
{{DEFAULTSORT:Self-Ionization Of Water}}
* {{MathWorld |urlname=RegularPolyhedron |title=Regular Polyhedron}}
[[Category:Water chemistry]]
[[Category:Equilibrium chemistry]]
[[Category:Acid-base chemistry|Water]]


{{DEFAULTSORT:Regular Polyhedron}}
[[de:Protolyse#Autoprotolyse]]
[[Category:Regular polyhedra| ]]

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my blog post ipad repair hanover park The self-ionization of water (also autoionization of water, and autodissociation of water) is an ionization reaction in pure water or an aqueous solution, in which a water molecule, H2O, loses the nucleus of one of its hydrogen atoms to become a hydroxide ion, OH−. The hydrogen nucleus, H+, immediately protonates another water molecule to form hydronium, H3O+. It is an example of autoprotolysis, and exemplifies the amphoteric nature of water.

Equilibrium constant

Chemically pure water has an electrical conductivity of 0.055 µS∙cm−1. According to the theories of Svante Arrhenius, this must be due to the presence of ions. The ions are produced by the self-ionization reaction

H2O + H2O Template:Eqm H3O+ + OH−

This equilibrium applies to pure water and any aqueous solution.

Approximating activities by concentrations, the chemical equilibrium constant, Keq, for this reaction is given by:

If the concentration of dissolved solutes is low, the concentration [H2O] can be taken as being constant at c. 55.5M.[1]

Expressed with activities Template:Mvar, instead of concentrations, the thermodynamic equilibrium constant for the water ionization reaction is:

which is numerically equal to the more traditional thermodynamic equilibrium constant written as:

under the assumption that the sum of the chemical potentials of H+ and H3O+ is formally equal to twice the chemical potential of H2O at the same temperature and pressure.[2]

In infinitely dilute aqueous solution, the activity of water solvent is unity.

The ionization constant, dissociation constant, self-ionization constant, or ionic product of water, symbolized by Kw may be given by:

where [H3O+] is the concentration of hydrogen or hydronium ion, and [OH−] is the concentration of hydroxide ion.

The ionic product of water can be also expressed on activity basis as:[3]

At 25 °C Kw is equal to Template:Val.

Dependence on temperature, pressure and ionic strength

Temperature dependence of the water ionization constant at 25 MPa
Pressure dependence of the water ionization constant at 25 °C
Variation of pKw with ionic strength of NaCl solutions at 25 °C

The dependence of the water ionization on temperature and pressure has been investigated thoroughly.[4] The value of pKw decreases as temperature increases from the melting point of ice to a minimum at c. 250 °C, after which it increases up to the critical point of water c. 374 °C. It decreases with increasing pressure.

pKw values for liquid water.[5]
Temperature Pressure[6] pKw
0°C 0.10 MPa 14.95
25°C 0.10 MPa 13.99
50°C 0.10 MPa 13.26
75°C 0.10 MPa 12.70
100°C 0.10 MPa 12.25
150°C 0.47 MPa 11.64
200°C 1.5 MPa 11.31
250°C 4.0 MPa 11.20
300°C 8.7 MPa 11.34
350°C 17 MPa 11.92

With electrolyte solutions, the value of pKw is dependent on ionic strength of the electrolyte. Values for sodium chloride are typical for a 1:1 electrolyte. With 1:2 electrolytes, MX2, pKw decreases with increasing ionic strength.[7]

The value of Kw is usually of interest in the liquid phase. Example values for superheated steam (gas) and supercritical water fluid are given in the table.

Comparison of pKw values for liquid water, superheated steam, and supercritical water.[2]
T/°C 350 400 450 500 600 800
0.1 MPa 11.920 (liquid)a 47.961b 47.873b 47.638 b 46.384 b 40.785 b
25 MPa 11.551 (liquid)c 16.566 18.135 18.758 19.425 20.113
100 MPa 10.600 (liquid)c 10.744 11.005 11.381 12.296 13.544
1000 MPa 8.311 (liquid)c 8.178 8.084 8.019 7.952 7.957

Notes to the Table. The values are for supercritical fluid except those marked: a at saturation pressure corresponding to 350 °C. b superheated steam. c subcooled liquid.

Isotope effects

Heavy water, D2O, self-ionizes less than normal water, H2O;

D2O + D2O Template:Eqm D3O+ + OD−

This is attributed to oxygen forming a slightly stronger bond to deuterium because the larger mass of deuterium difference results in a lower zero-point energy, a quantum mechanical effect.

The dissociation constant and the ionic product of heavy water is given by:

The following table compares the values of pKw for H2O and D2O.[8]

pKw values for pure water
T/°C 10 20 25 30 40 50
H2O 14.535 14.167 13.997 13.830 13.535 13.262
D2O 15.439 15.049 14.869 14.699 14.385 14.103

Ionization equilibria in water - heavy water mixtures

In water - heavy water mixtures equilibria several species are involved:H2O, HDO, D2O, H3O+, D3O+, H2DO+, HD2O+, HO-, DO-.

Mechanism

The rate of reaction for the dissociation

H2O → H+ + OH−

depends on the activation energy, ΔE‡. According to the Boltzmann distribution the proportion of water molecules that have sufficient energy, due to thermal population, is given by

where Template:Mvar is the Boltzmann constant. Thus some dissociation can occur because sufficient thermal energy is available. The following sequence of events has been proposed on the basis of electric field fluctuations in liquid water.[9] Random fluctuations in molecular motions occasionally (about once every 10 hours per water molecule[10]) produce an electric field strong enough to break an oxygen–hydrogen bond, resulting in a hydroxide (OH−) and hydronium ion (H3O+); the hydrogen nucleus of the hydronium ion travels along water molecules by the Grotthuss mechanism and a change in the hydrogen bond network in the solvent isolates the two ions, which are stabilized by solvation. Within 1 picosecond, however, a second reorganization of the hydrogen bond network allows rapid proton transfer down the electric potential difference and subsequent recombination of the ions. This timescale is consistent with the time it takes for hydrogen bonds to reorientate themselves in water.[11][12][13]

Relationship with the neutral point of water

Water molecules dissociate into equal amounts of H3O+ and OH−, so their concentrations are equal to Template:Val at 25 °C. A solution in which the H3O+ and OH− concentrations equal each other is considered a neutral solution. In general, the pH of the neutral point is numerically equal to pKw/2.

Pure water is neutral, but most water samples contain impurities. If an impurity is an acid or base this will affect the concentrations of hydronium ion and hydoxide ion. Water samples which are exposed to air will absorb the acid carbon dioxide and the concentration of H3O+ will increase. The concentration of OH− will decrease in such a way that the product [H3O+][OH−] remains constant for fixed temperature and pressure.

See also

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References

  1. ↑ McMurry, John. (2004) Organic Chemistry, p. 44
  2. ↑ 2.0 2.1 "Release on the Ionization Constant of H2O" The International Association for the Properties of Water and Steam, Lucerne, Switzerland, August 2007.
  3. ↑ IUPAC. Compendium of Chemical Terminology, 2nd ed. (the "Gold Book"). Compiled by A. D. McNaught and A. Wilkinson. Blackwell Scientific Publications, Oxford (1997). XML on-line corrected version: http://goldbook.iupac.org (2006–) created by M. Nic, J. Jirat, B. Kosata; updates compiled by A. Jenkins. ISBN 0-9678550-9-8. doi:10.1351/goldbook. Entry: autoprotolysis constant.
  4. ↑ International Association for the Properties of Water and Steam (IAPWS)
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    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  6. ↑ 0.1 MPa for T < 100°C. Saturation pressure for T ≥ 100°C.
  7. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  8. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  9. ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  10. ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  11. ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  12. ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  13. ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

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