formulasearchengine - User contributions [en]

Method of matched asymptotic expansions

2014-08-19T03:31:38Z

131.181.251.130: /* Accuracy */ that function isn't y(1) anyway, not sure why it was there

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2014-05-22T02:16:19Z

131.181.251.130:

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Lucas primality test

2014-05-06T05:06:18Z

131.181.251.130: /* Concepts */ - "equality" changed to "equivalence" each time it occurs, as the expressions given are equivalences, not equalities.

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ElGamal encryption

2014-01-13T00:57:32Z

131.181.251.130: /* Security */

{{Redirect|Soluble|the algebraic object called a "soluble group"|Solvable group}}
'''Solubility''' is the property of a [[solid]], [[liquid]], or [[gaseous]] [[chemical substance]] called ''[[solution|solute]]'' to [[dissolution (chemistry)|dissolve]] in a [[solid]], [[liquid]], or [[gaseous]] [[solvent]] to form a homogeneous [[solution]] of the solute in the solvent. The solubility of a substance fundamentally depends on the physical and chemical properties of the solute and solvent as well as on temperature, pressure and the pH of the solution. The extent of the solubility of a substance in a specific solvent is measured as the [[saturation (chemistry)|saturation]] concentration, where adding more solute does not increase the concentration of the solution and begin to precipitate the excess amount of solute.

Most often, the solvent is a liquid, which can be a pure substance or a [[mixture]]. One may also speak of [[solid solution]], but rarely of solution in a gas (see [[vapor-liquid equilibrium]] instead).

The extent of solubility ranges widely, from infinitely soluble (without limit) (fully [[miscible]]<ref>{{cite book|author=Clugston M. and Fleming R. |year=2000|page=108| title=Advanced Chemistry| edition=1st| publisher = Oxford Publishing| location = Oxford}}</ref>) such as [[ethanol]] in [[water]], to poorly soluble, such as [[silver chloride]] in water. The term ''insoluble'' is often applied to poorly or very poorly soluble compounds.

Under certain conditions, the [[solubility equilibrium|equilibrium solubility]] can be exceeded to give a so-called [[supersaturation|supersaturated]] solution, which is [[Metastability in molecules|metastable]].<ref>{{cite web| url=http://cancerweb.ncl.ac.uk/cgi-bin/omd?metastable |title = Cancerweb.ncl.ac.uk|work= Online Medical Dictionary|publisher=[[University of Newcastle Upon Tyne]]}}</ref>

Solubility is not to be confused with the ability to dissolve or liquefy a substance, because the solution might occur not only because of dissolution but also because of a chemical reaction. For example zinc, which is insoluble in hydrochloric acid, does dissolve in hydrochloric acid but by chemical reaction into hydrogen gas and zinc chloride, which in turn is soluble in the acid. Solubility does not also depend on particle size or other [[Chemical kinetics|kinetic]] factors; given enough time, even large particles will eventually dissolve.

== IUPAC definition ==
According to an [[IUPAC]] definition,<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/S05740.html Entry: Solubility].</ref> solubility is the analytical composition of a saturated solution expressed as a proportion of a designated solute in a designated solvent. Solubility may be stated in units of concentration, molality, mole fraction, mole ratio, and other units.

==Molecular view==
Solubility occurs under dynamic equilibrium, which means that solubility results from the simultaneous and opposing processes of [[solvation|dissolution]] and phase joining (e.g., [[precipitation (chemistry)|precipitation]] of [[solids]]). The solubility equilibrium occurs when the two processes proceed at a constant rate.

The term ''solubility'' is also used in some fields where the solute is altered by [[solvolysis]]. For example, many metals and their [[oxide]]s are said to be "soluble in hydrochloric acid," whereas the aqueous acid degrades the solid to irreversibly give soluble products. It is also true that most ionic solids are degraded by polar solvents, but such processes are reversible. In those cases where the solute is not recovered upon evaporation of the solvent, the process is referred to as solvolysis. The thermodynamic concept of solubility does not apply straightforwardly to solvolysis.

When a solute dissolves, it may form several species in the solution. For example, an [[aqueous]] [[Suspension (chemistry)|suspension]] of [[Iron(II) hydroxide|ferrous hydroxide]], {{chem|Fe(OH)|2}}, will contain the series [{{chem|Fe(H|2|O)}}6 − ''x''(OH)''x''](2 − x)+ as well as other [[oligomer]]ic species. Furthermore, the solubility of ferrous hydroxide and the composition of its soluble components depend on [[pH]]. In general, solubility in the solvent phase can be given only for a specific solute that is thermodynamically stable, and the value of the solubility will include all the species in the solution (in the example above, all the iron-containing complexes).{{Citation needed|date=June 2008}}

==Factors affecting solubility==
Solubility is defined for specific [[phase (matter)|phases]]. For example, the solubility of [[aragonite]] and [[calcite]] in water are expected to differ, even though they are both [[Polymorphism (materials science)|polymorphs]] of [[calcium carbonate]] and have the same [[chemical formula]].

The solubility of one substance in another is determined by the balance of [[intermolecular force]]s between the solvent and solute, and the [[entropy]] change that accompanies the solvation. Factors such as temperature and pressure will alter this balance, thus changing the solubility.

Solubility may also strongly depend on the presence of other species dissolved in the solvent, for example, [[complex (chemistry)|complex-]]forming anions ([[ligand]]s) in liquids. Solubility will also depend on the excess or deficiency of a common ion in the solution, a phenomenon known as the [[common-ion effect]]. To a lesser extent, solubility will depend on the [[ionic strength]] of solutions. The last two effects can be quantified using the equation for [[solubility equilibrium]].

For a solid that dissolves in a redox reaction, solubility is expected to depend on the potential (within the range of potentials under which the solid remains the thermodynamically stable phase). For example, solubility of gold in high-temperature water is observed to be almost an order of magnitude higher when the redox potential is controlled using a highly oxidizing Fe3O4-Fe2O3 [[redox buffer]] than with a moderately oxidizing Ni-NiO buffer.<ref>{{cite book|author=I.Y. Nekrasov| title=Geochemistry, Mineralogy and Genesis of Gold Deposits|publisher=Taylor & Francis| year= 1996|pages=135–136 |url=http://books.google.ca/books?id=HUWRZecignoC&pg=PA135#PPA135,M1|isbn=978-90-5410-723-1}}</ref>

[[Image:SolubilityVsTemperature.png|right|400px]]
Solubility (metastable) also depends on the physical size of the crystal or droplet of solute (or, strictly speaking, on the [[specific surface area]] or molar surface area of the solute). For quantification, see the equation in the article on [[Solubility_equilibrium#Particle_size_effect|solubility equilibrium]]. For highly defective crystals, solubility may increase with the increasing degree of disorder. Both of these effects occur because of the dependence of solubility constant on the Gibbs energy of the crystal. The last two effects, although often difficult to measure, are of practical importance.{{Citation needed|date=July 2008}} For example, they provide the driving force for [[Ostwald ripening|precipitate aging]] (the crystal size spontaneously increasing with time).

===Temperature===
The solubility of a given solute in a given solvent typically depends on temperature. For many solids dissolved in liquid water, the solubility increases with temperature up to 100 °C.<ref name = hill>John W. Hill, Ralph H. Petrucci, ''General Chemistry'', 2nd edition, Prentice Hall, 1999.</ref> In liquid water at high temperatures, (e.g., that approaching the [[critical temperature]]), the solubility of ionic solutes tends to decrease due to the change of properties and structure of liquid water; the lower [[dielectric constant]] results in a less [[polar solvent]].

[[Gas]]eous solutes exhibit more complex behavior with temperature. As the temperature is raised, gases usually become less soluble in water (to minimum, which is below 120 °C for most permanent gases<ref>{{cite book|editor=P. Cohen|title=The ASME handbook on Water Technology for Thermal Power Systems|publisher=The American Society of Mechanical Engineers|year=1989| page =442}}</ref>), but more soluble in organic solvents.<ref name=hill/>

The chart shows solubility curves for some typical solid inorganic [[salt]]s (temperature is in degrees [[Celsius]]).<ref>{{cite book|title=Handbook of Chemistry and Physics| edition= 27th|location= Cleveland, Ohio|year=1943 |publisher= Chemical Rubber Publishing Co.}}</ref> Many salts behave like [[barium nitrate]] and [[disodium hydrogen arsenate]], and show a large increase in solubility with temperature. Some solutes (e.g., [[sodium chloride]] in water) exhibit solubility that is fairly independent of temperature. A few, such as [[cerium(III) sulfate]], become less soluble in water as temperature increases. This temperature dependence is sometimes referred to as "retrograde" or "inverse" solubility. Occasionally, a more complex pattern is observed, as with [[sodium sulfate]], where the less soluble deca[[hydrate]] crystal loses [[water of crystallization]] at 32 °C to form a more soluble [[anhydrous]] phase.{{Citation needed|date=July 2008}}

[[Image:Temperature dependence solublity of solid in liquid water high temperature.svg|right|400px]]
The solubility of [[organic compounds]] nearly always increases with temperature. The technique of [[Recrystallization (chemistry)|recrystallization]], used for purification of solids, depends on a solute's different solubilities in hot and cold solvent. A few exceptions exist, such as certain [[cyclodextrin]]s.<ref>{{cite journal|title=A highly water-soluble 2+1 b-cyclodextrin–fullerene conjugate|author=Salvatore Filippone, Frank Heimanna and André Rassat|journal=[[Chem. Commun.]]|volume=2002|pages=1508–1509|doi=10.1039/b202410a|year=2002|issue=14}}</ref>

===Pressure===
For condensed phases (solids and liquids), the pressure dependence of solubility is typically weak and usually neglected in practice. Assuming an ideal solution, the dependence can be quantified as:

:<math> \left(\frac{\partial \ln N_i}{\partial P} \right)_T = -\frac{V_{i,aq}-V_{i,cr}} {RT} </math>

where the index i iterates the components, Ni is the mole fraction of the ith component in the solution, P is the pressure, the index T refers to constant temperature, Vi,aq is the [[partial molar volume]] of the ith component in the solution, Vi,cr is the partial molar volume of the ith component in the dissolving solid, and R is the [[universal gas constant]].<ref>{{cite book|author=E.M. Gutman| title=Mechanochemistry of Solid Surfaces|publisher= World Scientific Publishing Co.|year=1994}}</ref>

The pressure dependence of solubility does occasionally have practical significance. For example, [[Fouling#Precipitation_fouling|precipitation fouling]] of oil fields and wells by [[calcium sulfate]] (which decreases its solubility with decreasing pressure) can result in decreased productivity with time.

==Solubility of gases==
[[Henry's law]] is used to quantify the solubility of gases in solvents. The solubility of a gas in a solvent is directly proportional to the [[partial pressure]] of that gas above the solvent. This relationship is written as:
:<math> p = k_{\rm H}\, c </math>
where ''k''H is a temperature-dependent constant (for example, 769.2 [[litre|L]]·[[Atmosphere (unit)|atm]]/[[Mole (unit)|mol]] for [[Oxygen#Allotropes|dioxygen]] (O2) in water at 298 K), ''p'' is the partial pressure (atm), and ''c'' is the [[concentration]] of the dissolved gas in the liquid (mol/L).

The solubility of gases is sometimes also quantified using [[Bunsen solubility coefficient]].

In the presence of small [[Liquid bubble|bubble]]s, the solubility of the gas does not depend on the bubble radius in any other way than through the effect of the radius on pressure (i.e., the solubility of gas in the liquid in contact with small bubbles is increased due to pressure increase by Δp = 2γ/r; see [[Young–Laplace equation]]).<ref>{{cite journal| doi=10.1007/BF00550401| author=G.W. Greenwood|title=The Solubility of Gas Bubbles|journal=Journal of Materials Science|volume=4|pages= 320–322|year= 1969|bibcode = 1969JMatS...4..320G| issue=4 }}</ref>

Henry's law is valid for gases that do not undergo speciation on dissolution. [[Sieverts' law]] shows a case when this assumption does not hold.

==Polarity==
A popular [[aphorism]] used for predicting solubility is "''like dissolves like''".<ref>{{cite book| author=Kenneth J. Williamson| title=Macroscale and Microscale Organic Experiments| page=40|edition= 2nd | publisher=D. C, Heath| location=Lexington, Mass.| year=1994| isbn=0-669-19429-8}}</ref> This statement indicates that a solute will dissolve best in a solvent that has a similar [[chemical structure]] to itself. This view is simplistic, but it is a useful rule of thumb. The overall solvation capacity of a solvent depends primarily on its [[Chemical polarity|polarity]].<ref>The solvent polarity is ''defined'' as its solvation power according to Reichardt</ref> For example, a very polar ([[hydrophile|hydrophilic]]) solute such as [[urea]] is very soluble in highly polar water, less soluble in fairly polar [[methanol]], and practically insoluble in non-polar solvents such as [[benzene]]. In contrast, a non-polar or [[lipophilicity|lipophilic]] solute such as [[naphthalene]] is insoluble in water, fairly soluble in methanol, and highly soluble in non-polar benzene.<ref>{{cite book| title = Merck Index| edition=7th| publisher= Merck & Co.|year=1960}}</ref>

The solubility is favored by [[entropy of mixing]] and depends on [[enthalpy of dissolution]] and the [[hydrophobic effect]].

Synthetic chemists often exploit differences in solubilities to separate and purify compounds from reaction mixtures, using the technique of [[liquid-liquid extraction]].

==Rate of dissolution==
{{main|Dissolution (chemistry)}}
{{main|Solubilization}}
Dissolution is not always an instantaneous process. It is fast when salt and sugar dissolve in water but much slower for a tablet of [[aspirin]] or a large crystal of hydrated [[copper(II) sulfate]]. These observations are the consequence of two factors: the rate of solubilization (in kg/s) is related to the solubility product and the surface area of the material. The speed at which a solid dissolves may depend on its crystallinity or lack thereof in the case of [[amorphous]] solids and the surface area (crystallite size) and the presence of [[Polymorphism (materials science)|polymorphism]]. Many practical systems illustrate this effect, for example in designing methods for controlled [[drug delivery]]. Critically, the dissolution rate may depend on the presence of mixing and other factors that determine the degree of undersaturation in the liquid solvent film immediately adjacent to the solid solute crystal. In some cases, solubility equilibria can take a long time to establish (hours, days, months, or many years; depending on the nature of the solute and other factors). In practice, it means that the amount of solute in a solution is not always determined by its thermodynamic solubility, but may depend on kinetics of dissolution (or precipitation).

The rate of dissolution and solubility should not be confused as they are different concepts, kinetic and thermodynamic, respectively. The solubilization kinetics, as well as apparent solubility can be improved after complexation of an active ingredient with cyclodextrin. This can be used in the case of drug with poor solubility.<ref>{{cite journal| doi=10.1016/j.ejps.2004.06.002| author=A. Gil ''et al.''| year=2004| title = Evolution of the interaction of a new chemical entity, eflucimibe, with gamma-cyclodextrin during kneading process| journal=Eur. J. Pharm. Sciences|volume=23|pages= 123–129| issue=2| pmid=15451000}}</ref>

==Quantification of solubility==
Solubility is commonly expressed as a concentration; for example, as g of solute per kg of solvent, [[Mass concentration (chemistry)|g per dL (100mL) of solvent]], [[molarity]], [[molality]], [[mole fraction]], etc. The maximum equilibrium amount of solute that can dissolve per amount of solvent is the solubility of that solute in that solvent under the specified conditions. The advantage of expressing solubility in this manner is its simplicity, while the disadvantage is that it can strongly depend on the presence of other species in the solvent (for example, the common ion effect).

[[Solubility constant]]s are used to describe saturated solutions of ionic compounds  of relatively low solubility (see [[solubility equilibrium]]). The solubility constant is a special case of an [[equilibrium constant]]. It describes the balance between dissolved ions from the salt and undissolved salt. The solubility constant is also "applicable" (i.e., useful) to [[precipitation (chemistry)|precipitation]], the reverse of the dissolving reaction. As with other equilibrium constants, [[temperature]] can affect the numerical value of solubility constant. The solubility constant is not as simple as solubility, however the value of this constant is generally independent of the presence of other species in the solvent.

The [[Flory-Huggins solution theory]] is a theoretical model describing the solubility of polymers. The [[Hansen Solubility Parameters]] and the [[Hildebrand solubility parameter]]s are empirical methods for the prediction of solubility. It is also possible to predict solubility from other physical constants such as the [[enthalpy of fusion]].

The [[partition coefficient]] ([[Log P]]) is a measure of differential solubility of a compound in a [[hydrophobe|hydrophobic]] solvent ([[1-octanol]]) and a [[hydrophile|hydrophilic]] solvent ([[water]]). The logarithm of these two values enables compounds to be ranked in terms of hydrophilicity (or hydrophobicity).

The energy change associated with dissolving is usually given per mole of solute as the [[enthalpy of solution]].

==Applications==
Solubility is of fundamental importance in a large number of scientific disciplines and practical applications, ranging from ore processing, to the use of medicines, and the transport of pollutants.

Solubility is often said to be one of the "characteristic properties of a substance," which means that solubility is commonly used to describe the substance, to indicate a substance's polarity, to help to distinguish it from other substances, and as a guide to applications of the substance. For example, [[Indigo_dye#Chemical_properties|indigo]] is described as "insoluble in water, alcohol, or ether but soluble in chloroform, nitrobenzene, or concentrated sulfuric acid".{{Citation needed|date=July 2008}}

Solubility of a substance is useful when separating mixtures. For example, a mixture of salt ([[sodium chloride]]) and silica may be separated by dissolving the salt in water, and filtering off the undissolved silica. The synthesis of chemical compounds, by the milligram in a laboratory, or by the ton in industry, both make use of the relative solubilities of the desired product, as well as unreacted starting materials, byproducts, and side products to achieve separation.

Another example of this is the synthesis of [[benzoic acid]] from [[phenylmagnesium bromide]] and [[dry ice]]. Benzoic acid is more soluble in an organic solvent such as [[dichloromethane]] or [[diethyl ether]], and when shaken with this organic solvent in a [[separatory funnel]], will preferentially dissolve in the organic layer. The other reaction products, including the magnesium bromide, will remain in the aqueous layer, clearly showing that separation based on solubility is achieved. This process, known as [[liquid-liquid extraction]], is an important technique in [[synthetic chemistry]].

==Solubility of ionic compounds in water==
{{main|Solubility chart}}
{{main|Solubility table}}

Some ionic compounds ([[salts]]) dissolve in water, which arises because of the attraction between positive and negative charges (see: [[solvation]]). For example, the salt's positive ions (e.g. Ag+) attract the partially negative oxygens in H2O. Likewise, the salt's negative ions (e.g. Cl−) attract the partially positive hydrogens in H2O. Note: oxygen is partially negative because it is more [[electronegativity|electronegative]] than hydrogen, and vice-versa (see: [[chemical polarity]]).

:AgCl(s) {{eqm}} Ag+(aq) + Cl−(aq)

However, there is a limit to how much salt can be dissolved in a given volume of water. This amount is given by the [[solubility product]], Ksp. This value depends on the type of salt (AgCl vs. NaCl, for example), temperature, and the common ion effect.

One can calculate the amount of AgCl that will dissolve in 1 liter of water, some algebra is required.

:Ksp = [Ag+] × [Cl−] (definition of solubility product)
:Ksp = 1.8 × 10−10 (from a table of solubility products)
[Ag+] = [Cl−], in the absence of other silver or chloride salts,
:[Ag+]2 = 1.8 × 10−10
:[Ag+] = 1.34 × 10−5

The result: 1 liter of water can dissolve 1.34 × 10−5 [[mole (unit)|moles]] of AgCl(s) at room temperature. Compared with other types of salts, AgCl is poorly soluble in water. In contrast, table salt (NaCl) has a higher Ksp and is, therefore, more soluble.

<center><table class="wikitable">
<tr><th>Soluble</th><th>Insoluble</th></tr>
<tr><td>[[Alkali metal|Group I]] and [[Ammonium|NH4+]] compounds</td><td>[[Carbonate]]s (Except [[Alkali metal|Group I]], [[Ammonium|NH4+]] and [[uranyl]] compounds)</td></tr>
<tr><td>[[Nitrate]]s</td><td>[[Sulfite]]s (Except [[Alkali metal|Group I]] and [[Ammonium|NH4+]] compounds)</td></tr>
<tr><td>[[Acetate]]s (Ethanoates) (Except [[Silver|Ag+]] compounds)</td><td>[[Phosphate]]s (Except [[Alkali metal|Group I]] and [[Ammonium|NH4+]] compounds)</td></tr>
<tr><td>[[Chloride]]s (Chlorates and Perchlorates), [[bromide]]s and [[iodide]]s (Except [[Silver|Ag+]], [[Lead|Pb2+]], [[Copper|Cu+]] and [[Mercury (element)|Hg22+]])</td><td>[[Hydroxide]]s and [[oxide]]s (Except [[Alkali metal|Group I]], [[Ammonium|NH4+]], [[Barium|Ba2+]], [[Strontium|Sr2+]] and [[Thallium|Tl+]])</td></tr>
<tr><td>[[Sulfate]]s (Except [[Silver|Ag+]], [[Lead|Pb2+]], [[Barium|Ba2+]], [[Strontium|Sr2+]] and [[Calcium|Ca2+]])</td><td>[[Sulfide]]s (Except [[Alkali metal|Group I]], [[Alkaline earth metal|Group II]] and [[Ammonium|NH4+]] compounds)</td></tr>
</table>
</center>
<ref>{{cite book| editors=C. Houk, R. Post |title=Chemistry, Concept and Problems|publisher=John Wiley & Sons|year=1997| page=121| isbn=0-471-12120-7}}</ref>

==Solubility of organic compounds==
The principle outlined above under [[#Polarity|polarity]], that ''like dissolves like,'' is the usual guide to solubility with organic systems. For example, [[petroleum jelly]] will dissolve in [[gasoline]] because both petroleum jelly and gasoline are non-polar hydrocarbons. It will not, on the other hand, dissolve in [[ethyl alcohol]] or water, since the polarity of these solvents is too high. Sugar will not dissolve in gasoline, since sugar is too polar in comparison with gasoline. A mixture of gasoline and sugar can therefore be separated by [[filtration]], or [[solvent extraction|extraction]] with water.

==Solubility in non-aqueous solvents==
Most publicly available solubility values are those for solubility in water.<ref>{{cite web|url=http://srdata.nist.gov/solubility/casNO.aspx| title= NIST solubility database}}</ref> The reference also lists some for non-aqueous solvents. Solubility data for non-aqueous solvents is currently being collected via an [[open notebook science]] [[crowdsourcing]] project.<ref>{{cite web| url=http://onschallenge.wikispaces.com/ |title=ONS Solubility challenge}}</ref><ref>{{cite web|url=http://lxsrv7.oru.edu/~alang/onsc/solubility/allsolvents.php?solute=vanillin| title= Solubility of Vanillin in various non-aqueous solvents}}</ref>

==Solid solution==
This term is often used in the field of [[metallurgy]] to refer to the extent that an [[alloy]]ing element will dissolve into the [[base metal]] without forming a separate phase. The [[solvus]] or solubility line (or curve) is the line (or lines) on a [[phase diagram]] that give the limits of solute addition. That is, the lines show the maximum amount of a component that can be added to another component and still be in [[solid solution]]. In the solid's crystalline structure, the 'solute' element can either take the place of the matrix within the lattice (a substitutional position; for example, chromium in iron) or take a place in a space between the lattice points (an interstitial position; for example, carbon in iron).

In microelectronic fabrication, solid solubility refers to the maximum concentration of impurities one can place into the substrate.

==Incongruent dissolution==
Many substances dissolve congruently; i.e., the composition of the solid and the dissolved solute stoichiometrically match. However, some substances may dissolve [[Incongruent transition|incongruently]], whereby the composition of the solute in solution does not match that of the solid. This solubilization is accompanied by alteration of the "primary solid" and possibly formation of a secondary solid phase. However, in general, some primary solid also remains and a complex solubility equilibrium establishes. For example, dissolution of [[albite]] may result in formation of [[gibbsite]].<ref>{{cite book| editors=O.M. Saether & P. de Caritat |title=Geochemical processes, weathering and groundwater recharge in catchments|publisher=Taylor & Francis| location=Rotterdam|year=1997| page=6| isbn=90-5410-641-7}}</ref>

:NaAlSi3O8(s) + H+ + 7H2O = Na+ + Al(OH)3(s) + 3H4SiO4.

In this case, the solubility of albite is expected to depend on the solid-to-solvent ratio. This kind of solubility is of great importance in geology, where it results in formation of [[metamorphic rock]]s.

==See also==

*[[Biopharmaceutics Classification System]]
*[[Dühring's rule]]
*[[Fajans-Paneth-Hahn Law]]
*[[Flexible SPC water model]]
*[[Hot water extraction]]
*[[Hydrotrope]]
*[[Raoult's law]]
*[[Henry's law]]
*[[Solubility equilibrium]]
*[[Solubilization]]
*[[Apparent molar property]]

==References==
{{reflist|35em}}

==External links==
{{Wiktionary|soluble|solubility}}
*[http://www.vcclab.org/lab/alogps VCClab.org], "ALOGPS" free interactive calculation of aqueous solubility of compounds at Virtual Computational Chemistry Laboratory using several algorithms.
*[http://www.acdlabs.com/products/phys_chem_lab/aqsol/ ACDlabs.com]? ACD/Solubility DB aqueous solubility prediction
*[http://www.simulations-plus.com/Definitions.aspx?lID=58&pID=13 Simulations-plus.com], S+Sw, an aqueous solubility prediction model.

{{Chemical solutions}}
{{Diving medicine, physiology and physics}}

[[Category:Solutions]]
[[Category:Physical quantities]]
[[Category:Underwater diving physics]]

Shape factor (image analysis and microscopy)

2013-09-10T03:06:50Z

131.181.251.131: /* An application of shape factors */ Removing subjective language

{{introduction|Systolic geometry}}

'''[[Systolic geometry]]''' is a branch of [[differential geometry]], a field within mathematics, studying problems such as the relationship between the [[area]] inside a [[closed curve]] ''C'', and the [[length]] or perimeter of ''C''. Since the area ''A'' may be small while the length ''l'' is large, when ''C'' looks elongated, the relationship can only take the form of an [[inequality (mathematics)|inequality]]. What is more, such an inequality would be an [[upper bound]] for ''A'': there is no interesting lower bound just in terms of the length.

[[Mikhail Gromov (mathematician)|Mikhail Gromov]] once voiced the opinion that the [[isoperimetric inequality]] was known already to the Ancient Greeks. The mythological tale of [[Dido (Queen of Carthage)|Dido, Queen of Carthage]] shows that problems about making a maximum area for a given perimeter were posed in a natural way, in past eras.

The relation between length and area is closely related to the physical phenomenon known as [[surface tension]], which gives a visible form to the comparable relation between [[surface area]] and [[volume]]. The familiar shapes of drops of water express minima of surface area.

The purpose of this article is to explain another such relation between length and area. A space is called [[simply connected]] if every loop in the space can be contracted to a point in a continuous fashion. For example, a room with a pillar in the middle, connecting floor to ceiling, is not simply connected. In [[geometry]], a ''systole'' is a distance which is characteristic of a [[compact set|compact]] [[metric space]] which is not simply connected. It is the length of a shortest loop in the space that cannot be contracted to a point in the space. '''Systolic geometry''' gives lower bounds for various attributes of the space in terms of its systole.

It is known that the [[Fubini–Study metric]] is the natural metric for the geometrisation of quantum mechanics. In an intriguing connection to global geometric phenomena, it turns out that the Fubini–Study metric can be characterized as the boundary case of equality in [[Gromov's inequality for complex projective space]], involving an [[area]] quantity called the 2-systole, pointing to a possible connection to quantum mechanical phenomena.

In the following, these systolic inequalities will be compared to the classical isoperimetric inequalities, which can in turn be motivated by physical phenomena observed in the behavior of a water drop.

==Surface tension and shape of a water drop==
[[Image:Dew 2.jpg|thumb|Water beading on a leaf]]
Perhaps the most familiar physical manifestation of the 3-dimensional isoperimetric inequality is the shape of a drop of water. Namely, a drop will typically assume a symmetric round shape. Since the amount of water in a drop is fixed, surface tension forces the drop into a shape which minimizes the surface area of the drop, namely a round sphere. Thus the round shape of the drop is a consequence of the phenomenon of surface tension. Mathematically, this phenomenon is expressed by the isoperimetric inequality.

==Isoperimetric inequality in the plane==

The solution to the isoperimetric problem in the plane is usually expressed in the form of an inequality that relates the length <math>L</math> of a closed curve and the area <math>A</math> of the planar region that it encloses. The isoperimetric inequality states that

:<math> 4\pi A \le L^2,\, </math>

and that the equality holds if and only if the curve is a round circle. The inequality is an upper bound for area in terms of length. It can be rewritten as follows:

:<math> L^2 -4\pi A \geq 0. \, </math>

==Central symmetry==

Recall the notion of central symmetry: a Euclidean polyhedron is called centrally symmetric if it is invariant under the [[antipodal map]]

:<math> x \mapsto -x. \, </math>

Thus, in the plane central symmetry is the rotation by 180 degrees. For example, an ellipse is centrally symmetric, as is any ellipsoid in 3-space.

==Property of a centrally symmetric polyhedron in 3-space==

There is a geometric inequality that is in a sense dual to the isoperimetric inequality in the following sense. Both involve a length and an area. The isoperimetric inequality is an upper bound for area in terms of length. There is a geometric inequality which provides an upper bound for a certain length in terms of area. More precisely it can be described as follows.

Any centrally symmetric convex body of surface area <math>A</math> can be squeezed through a noose of length <math>\sqrt{\pi A}</math>, with the tightest fit achieved by a sphere. This property is equivalent to a special case of [[Introduction to systolic geometry#Pu's inequality|Pu's inequality]], one of the earliest systolic inequalities.

For example, an ellipsoid is an example of a convex centrally symmetric body in 3-space. It may be helpful to the reader to develop an intuition for the property mentioned above in the context of thinking about ellipsoidal examples.

An alternative formulation is as follows. Every convex centrally symmetric body <math>P</math> in <math>{\mathbb R}^3</math> admits a pair of opposite (antipodal) points and a path of length
<math>L</math> joining them and lying on the boundary <math>\partial P</math> of <math>P</math>, satisfying

:<math> L^2 \leq \frac{\pi}{4} \mathrm{area}(\partial P). </math>

==Notion of systole==
[[Image:TorusSystoleLoop.png|right|thumb|200px|Shortest loop on a torus]]

The ''systole'' of a compact metric space <math>X</math> is a metric
invariant of <math>X</math>, defined to be the least length of a
noncontractible loop in <math>X</math>. We will denote it as follows:

:<math> \mathrm{sys}(X). \, </math>

Note that a loop minimizing length is necessarily a [[closed geodesic]]. When <math>X</math> is a [[graph (mathematics)|graph]], the invariant is usually referred to as the [[girth (graph theory)|girth]], ever since the 1947 article by [[William Tutte]]. Possibly inspired by Tutte's article, [[Charles Loewner]] started thinking about systolic questions on surfaces in the late 1940s, resulting in a 1950 thesis by his student P. M. Pu. The actual term ''systole'' itself was not coined until a quarter century later, by [[Marcel Berger]].

This line of research was, apparently, given further impetus by a remark of [[René Thom]], in a conversation with Berger in the library of Strasbourg University during the 1961-62 academic year, shortly after the publication of the papers of R. Accola and C. Blatter. Referring to these systolic inequalities, Thom reportedly exclaimed: ''Mais c'est fondamental!'' [These results are of fundamental importance!]

Subsequently, Berger popularized the subject in a series of articles and books, most recently in the march '08 issue of the [[Notices of the American Mathematical Society]]. A bibliography at the ''Website for systolic geometry and topology'' currently contains over 170 articles. Systolic geometry is a rapidly developing field, featuring a number of recent publications in leading journals. Recently, an intriguing link has emerged with the [[Lusternik-Schnirelmann category]]. The existence of such a link can be thought of as a theorem in [[systolic topology]].

==The real projective plane==
[[Image:Steiner's Roman Surface.gif|thumb|An animation of the [[Roman Surface]] representing RP2 in R3]]
In [[projective geometry]], the [[real projective plane]] <math>\mathbb {RP}^2</math> is defined as the collection of lines through the origin in <math>\mathbb {R}^3</math>. The distance function on <math>\mathbb {RP}^2</math> is most readily understood from this point of view. Namely, the distance between two lines through the origin is by definition the angle between them (measured in radians), or more precisely the lesser of the two angles. This distance function corresponds to the metric of constant [[Gaussian curvature]] +1.

Alternatively, <math>\mathbb {RP}^2</math> can be defined as the surface obtained by identifying each pair of antipodal points on the 2-sphere.

Other metrics on <math>\mathbb {RP}^2</math> can be obtained by quotienting metrics on <math>S^2</math> imbedded in 3-space in a centrally symmetric way.

Topologically, <math>\mathbb {RP}^2</math> can be obtained from the Möbius strip by attaching a disk along the boundary.

Among [[closed surface]]s, the real projective plane is the simplest non-orientable such surface.

==Pu's inequality==

[[Pu's inequality|Pu's inequality for the real projective plane]] applies to general [[Riemannian metric]]s on <math>\mathbb {RP}^2</math>.

A student of [[Charles Loewner]]'s, [[Pao Ming Pu]] proved in a 1950 thesis (published in 1952) that every metric <math>g</math> on the real projective plane <math>\mathbb {RP}^2 </math> satisfies the optimal inequality

:<math> \mathrm{sys}(g)^2 \leq \frac{\pi}{2} \mathrm{area}(g), </math>

where <math>\mathrm{sys}</math> is the systole. The boundary case of equality is attained precisely when the metric is of constant Gaussian curvature. Alternatively, the inequality can be presented as follows:

:<math> \mathrm{area}(g) - \frac{2}{\pi} \mathrm{sys}(g)^2 \geq 0. </math>

There is a vast generalisation of Pu's inequality, due to [[Mikhail Gromov (mathematician)|Mikhail Gromov]], called [[Gromov's systolic inequality for essential manifolds]]. To state his result, one requires a topological notion of an [[essential manifold]].

==Loewner's torus inequality==
[[Image:TorusSystoleLoop.png|right|thumb|200px|Shortest loop on a torus]]
Similarly to Pu's inequality, [[Loewner's torus inequality]] relates
the total area, to the systole, i.e. least length of a noncontractible
loop on the torus <math>(T^2, g)</math>:

:<math> \mathrm{area}(g) - \tfrac{\sqrt{3}}{2} \mathrm{sys}(g)^2 \geq 0. </math>

The boundary case of equality is attained if and only if the metric is
homothetic to the flat metric obtained as the quotient of
<math>{\mathbb R}^2</math> by the lattice formed by the
[[Eisenstein integers]].

==Bonnesen's inequality==

The classical [[Bonnesen's inequality]] is the strengthened
isoperimetric inequality

:<math> L^2 - 4\pi A \geq \pi^2(R-r)^2. \, </math>

Here <math>A</math> is the area of the region bounded by a closed Jordan curve of length (perimeter) <math>L</math> in the plane, <math>R</math> is the circumradius of the bounded region, and <math>r</math> is its inradius. The error term <math>\pi^2(R-r)^2</math> on the right hand side is traditionally called the ''isoperimetric defect''. There exists a similar strengthening of Loewner's inequality.

==Loewner's inequality with a defect term==
The explanation of the strengthened version of Loewner's inequality is somewhat more technical than the rest of this article. It seems worth including it here for the sake of completeness. The strengthened version is the inequality

:<math> \mathrm{area}(g) - \tfrac{\sqrt{3}}{2} \mathrm{sys}(g)^2 \geq \mathrm{Var}(f),</math>

where Var is the probabilistic [[variance]] while ''f'' is the conformal factor expressing the metric ''g'' in terms of the flat metric of unit area in the conformal class of ''g''. The proof results from a combination of the [[computational formula for the variance]] and [[Fubini's theorem]] (see Horowitz ''et al'', 2009).

==See also==
*[[systoles of surfaces]]

==References==
* [[Victor Bangert|Bangert, V.]]; Croke, C.; Ivanov, S.; [[Mikhail Katz|Katz, M.]]: Filling area conjecture and ovalless real hyperelliptic surfaces. Geometric and Functional Analysis (GAFA) 15 (2005), no. 3, 577-597.
* Berger, M.: Systoles et applications selon Gromov. (French. French summary) [Systoles and their applications according to Gromov] Séminaire Bourbaki, Vol. 1992/93. Astérisque No. 216 (1993), Exp. No. 771, 5, 279—310.
* Berger, M.: A panoramic view of Riemannian geometry. Springer-Verlag, Berlin, 2003.
* Berger, M.: What is... a Systole? Notices of the AMS 55 (2008), no. 3, 374-376.
* Buser, P.; Sarnak, P.: On the period matrix of a Riemann surface of large genus. With an appendix by J. H. Conway and N. J. A. Sloane. Invent. Math. 117 (1994), no. 1, 27—56.
* Gromov, M. Systoles and intersystolic inequalities. (English, French summary) Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992), 291—362, Sémin. Congr., 1, Soc. Math. France, Paris, 1996.
* Gromov, M. Metric structures for Riemannian and non-Riemannian spaces. Based on the 1981 French original. With appendices by M. Katz, P. Pansu and S. Semmes. Translated from the French by Sean Michael Bates. Progress in Mathematics, 152. Birkhäuser Boston, Inc., Boston, MA, 1999.
* Charles Horowitz, Karin Usadi Katz and [[Mikhail Katz|Mikhail G. Katz]] (2008), Loewner's torus inequality with isosystolic defect, Journal of Geometric Analysis 19 (2009), no. 4, 796-808. See [http://arxiv.org/abs/0803.0690 arXiv:0803.0690]
* Katz, M. Systolic geometry and topology. With an appendix by J. Solomon. Mathematical Surveys and Monographs, volume 137. [[American Mathematical Society]], 2007.
* Katz, M.; Rudyak, Y.: Systolic category and Lusternik-Schnirelman category of low-dimensional manifolds. [[Communications on Pure and Applied Mathematics]] 59 ('06), 1433-1456.
* Katz, M.; Sabourau, S.: Entropy of systolically extremal surfaces and asymptotic bounds. Ergo. Th. Dynam. Sys. 25 (2005), 1209-1220.
* Katz, M.; Schaps, M.; Vishne, U.: Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups. J. Differential Geom. 76 (2007), no. 3, 399-422. Available at {{arxiv|math.DG/0505007}}
* Pu, P. M.: Some inequalities in certain nonorientable Riemannian manifolds. Pacific J. Math. 2 (1952), 55—71.

==External links==
*[http://people.hofstra.edu/Stefan_Waner/RealWorld/pdfs/DiffGeom.pdf Introduction to Differential Geometry & General Relativity]

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