Pandemonium architecture

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https://en.wikipedia.org/wiki/Pandemonium_architecture 

The original pandemonium model proposed by Oliver Selfridge in 1959

Pandemonium architecture is a theory in cognitive science that describes how visual images are processed by the brain. It has applications in artificial intelligence and pattern recognition. The theory was developed by the artificial intelligence pioneer Oliver Selfridge in 1959. It describes the process of object recognition as the exchange of signals within a hierarchical system of detection and association, the elements of which Selfridge metaphorically termed "demons". This model is now recognized as the basis of visual perception in cognitive science.

Pandemonium architecture arose in response to the inability of template matching theories to offer a biologically plausible explanation of the image constancy phenomenon. Contemporary researchers praise this architecture for its elegancy and creativity; that the idea of having multiple independent systems (e.g., feature detectors) working in parallel to address the image constancy phenomena of pattern recognition is powerful yet simple. The basic idea of the pandemonium architecture is that a pattern is first perceived in its parts before the "whole".

Pandemonium architecture was one of the first computational models in pattern recognition. Although not perfect, the pandemonium architecture influenced the development of modern connectionist, artificial intelligence, and word recognition models.

History

We perceive a dog, but how and why?

Most research in perception has been focused on the visual system, investigating the mechanisms of how we see and understand objects. A critical function of our visual system is its ability to recognize patterns, but the mechanism by which this is achieved is unclear.

The earliest theory that attempted to explain how we recognize patterns is the template matching model. According to this model, we compare all external stimuli against an internal mental representation. If there is "sufficient" overlap between the perceived stimulus and the internal representation, we will "recognize" the stimulus. Although some machines follow a template matching model (e.g., bank machines verifying signatures and accounting numbers), the theory is critically flawed in explaining the phenomena of image constancy: we can easily recognize a stimulus regardless of the changes in its form of presentation (e.g., T and T are both easily recognized as the letter T). It is highly unlikely that we have a stored template for all of the variations of every single pattern.

As a result of the biological plausibility criticism of the template matching model, feature detection models began to rise. In a feature detection model, the image is first perceived in its basic individual elements before it is recognized as a whole object. For example, when we are presented with the letter A, we would first see a short horizontal line and two slanted long diagonal lines. Then we would combine the features to complete the perception of A. Each unique pattern consists of different combination of features, which means those that are formed with the same features will generate the same recognition. That is, regardless of how we rotate the letter A, is still perceived as the letter A. It is easy for this sort of architecture to account for the image constancy phenomena because you only need to "match" at the basic featural level, which is presumed to be limited and finite, thus biologically plausible. The best known feature detection model is called the pandemonium architecture.

Pandemonium architecture

The pandemonium architecture was originally developed by Oliver Selfridge in the late 1950s. The architecture is composed of different groups of "demons" working independently to process the visual stimulus. Each group of demons is assigned to a specific stage in recognition, and within each group, the demons work in parallel. There are four major groups of demons in the original architecture.

The four major processing stages of pandemonium architecture

Stage	Demon name	Function
1	Image demon	Records the image that is received in the retina.
2	Feature demons	There are many feature demons, each representing a specific feature. For example, there is a feature demon for short straight lines, another for curved lines, and so forth. Each feature demon's job is to "yell" if they detect a feature that they correspond to. Note that, feature demons are not meant to represent any specific neurons, but to represent a group of neurons that have similar functions. For example, the vertical line feature demon is used to represent the neurons that respond to the vertical lines in the retina image.
3	Cognitive demons	Watch the "yelling" from the feature demons. Each cognitive demon is responsible for a specific pattern (e.g., a letter in the alphabet). The "yelling" of the cognitive demons is based on how much of their pattern was detected by the feature demons. The more features the cognitive demons find that correspond to their pattern, the louder they "yell". For example, if the curved, long straight and short angled line feature demons are yelling really loud, the R letter cognitive demon might get really excited, and the P letter cognitive demon might be somewhat excited as well; but the Z letter cognitive demon is very likely to be quiet.
4	Decision demon	Represents the final stage of processing. It listens to the "yelling" produced by the cognitive demons. It selects the loudest cognitive demon. The demon that gets selected becomes our conscious perception. Continuing with our previous example, the R cognitive demon would be the loudest, seconded by P; therefore we will perceive R, but if we were to make a mistake because of poor displaying conditions (e.g., letters are quickly flashed or have parts occluded), it is likely to be P. Note that, the "pandemonium" simply represents the cumulative "yelling" produced by the system.

The concept of feature demons, that there are specific neurons dedicated to perform specialized processing is supported by research in neuroscience. Hubel and Wiesel found there were specific cells in a cat's brain that responded to specific lengths and orientations of a line. Similar findings were discovered in frogs, octopuses and a variety of other animals. Octopuses were discovered to be only sensitive to verticality of lines, whereas frogs demonstrated a wider range of sensitivity. These animal experiments demonstrate that feature detectors seem to be a very primitive development. That is, it did not result from the higher cognitive development of humans. Not surprisingly, there is also evidence that the human brain possesses these elementary feature detectors as well.

Moreover, this architecture is capable of learning, similar to a back-propagation styled neural network. The weight between the cognitive and feature demons can be adjusted in proportion to the difference between the correct pattern and the activation from the cognitive demons. To continue with our previous example, when we first learned the letter R, we know is composed of a curved, long straight, and a short angled line. Thus when we perceive those features, we perceive R. However, the letter P consists of very similar features, so during the beginning stages of learning, it is likely for this architecture to mistakenly identify R as P. But through constant exposure of confirming R's features to be identified as R, the weights of R's features to P are adjusted so the P response becomes inhibited (e.g., learning to inhibit the P response when a short angled line is detected). In principle, a pandemonium architecture can recognize any pattern.

As mentioned earlier, this architecture makes error predictions based on the amount of overlapping features. Such as, the most likely error for R should be P. Thus, in order to show this architecture represents the human pattern recognition system we must put these predictions into test. Researchers have constructed scenarios where various letters are presented in situations that make them difficult to identify; then types of errors were observed, which was used to generate confusion matrices: where all of the errors for each letter are recorded. Generally, the results from these experiments matched the error predictions from the pandemonium architecture. Also as a result of these experiments, some researchers have proposed models that attempted to list all of the basic features in the Roman alphabet.

Criticism

A major criticism of the pandemonium architecture is that it adopts a completely bottom-up processing: recognition is entirely driven by the physical characteristics of the targeted stimulus. This means that it is unable to account for any top-down processing effects, such as context effects (e.g., pareidolia), where contextual cues can facilitate (e.g., word superiority effect: it is relatively easier to identify a letter when it is part of a word than in isolation) processing. However, this is not a fatal criticism to the overall architecture, because is relatively easy to add a group of contextual demons to work along with the cognitive demons to account for these context effects.

A demonstration of the global-to-local hypothesis: people see the A before the Hs

Although the pandemonium architecture is built on the fact that it can account for the image constancy phenomena, some researchers have argued otherwise; and pointed out that the pandemonium architecture might share the same flaws from the template matching models. For example, the letter H is composed of 2 long vertical lines and a short horizontal line; but if we rotate the H 90 degrees in either direction, it is now composed of 2 long horizontal lines and a short vertical line. In order to recognize the rotated H as H, we would need a rotated H cognitive demon. Thus we might end up with a system that requires a large number of cognitive demons in order to produce accurate recognition, which would lead to the same biological plausibility criticism of the template matching models. However, it is rather difficult to judge the validity of this criticism because the pandemonium architecture does not specify how and what features are extracted from incoming sensory information, it simply outlines the possible stages of pattern recognition. But of course that raises its own questions, to which it is almost impossible to criticize such a model if it does not include specific parameters. Also, the theory appears to be rather incomplete without defining how and what features are extracted, which proves to be especially problematic with complex patterns (e.g., extracting the weight and features of a dog).

Some researchers have also pointed out that the evidence supporting the pandemonium architecture has been very narrow in its methodology. Majority of the research that supports this architecture has often referred to its ability to recognize simple schematic drawings that are selected from a small finite set (e.g., letters in the Roman alphabet). Evidence from these types of experiments can lead to overgeneralized and misleading conclusions, because the recognition process of complex, three-dimensional patterns could be very different from simple schematics. Furthermore, some have criticized the methodology used in generating the confusion matrix, because it confounds perceptual confusion (error in identification caused by overlapping features between the error and the correct answer) with post-perceptual guessing (people randomly guessing because they cannot be sure what they saw). However, these criticisms were somewhat addressed when similar results were replicated with other paradigms (e.g., go/no go and same-different tasks), supporting the claim that humans do have elementary feature detectors. These new paradigms relied on reaction time as the dependent variable, which also avoided the problem of empty cells that is inherent with the confusion matrix (statistical analyses are difficult to conduct and interpret when the data have empty cells).

Additionally, some researchers have pointed out that feature accumulation theories like the pandemonium architecture have the processing stages of pattern recognition almost backwards. This criticism was mainly used by advocates of the global-to-local theory, who argued and provided evidence that perception begins with a blurry view of the whole that refines overtime, implying feature extraction does not happen in the early stages of recognition. However, there is nothing to prevent a demon from recognizing a global pattern in parallel with other demons recognizing local patterns within the global pattern.

Applications and influences

A pandemonium-based system was able to correctly identify all these stimuli as the letter A, without being exposed to these exact stimuli beforehand.

The pandemonium architecture has been applied to solve several real-world problems, such as translating hand-sent Morse codes and identifying hand-printed letters. The overall accuracy of pandemonium-based models are impressive, even when the system was given a short learning period. For example, Doyle constructed a pandemonium-based system with over 30 complex feature-analyzers. He then fed his system several hundred letters for learning. During this phase, the system analyzed the inputted letter and generated its own output (what the system identifies the letter as). The output from the system was compared against the correct identification, which sends an error signal back to the system to adjust the weights between the features analyzers accordingly. In the testing phase, unfamiliar letters were presented (different style and size of the letters than those that were presented in the learning phase), and the system was able to achieve a near 90% accuracy. Because of its impressive capability to recognize words, all modern theories on how humans read and recognize words follow this hierarchal structure: word recognition begins with feature extractions of the letters, which then activates the letter detectors (e.g., SOLAR, SERIOL, IA, DRC).

Based on the original pandemonium architecture, John Jackson has extended the theory to explain phenomena beyond perception. Jackson offered the analogy of an arena to account for "consciousness". His arena consisted of a stand, a playing field, and a sub-arena. The arena was populated by a multitude of demons. The demons that were designated in the playing fields were the active demons, as they represent the active elements of human consciousness. The demons in the stands are to watch those in the playing field until something excites them; each demon is excited by different things. The more excited the demons get, the louder they yell. If a demon yells pass a set threshold, it gets to join the other demons in the playing field and perform its function, which may then excite other demons, and this cycle continues. The sub-arena in the analogy functions as the learning and feedback mechanism of the system. The learning system here is similar to any other neural styled networks, which is through modifying the connection strength between the demons; in other words, how the demons respond to each other's yelling. This multiple agent approach to human information processing became the assumption for many modern artificial intelligence systems.

Comparisons

Comparison with template matching theories

Although the pandemonium architecture arose as a response to address a major criticism of the template matching theories, the two are actually rather similar in some sense: there is a process where a specific set of features for items is matched against some sort of mental representation. The critical difference between the two is that the image is directly compared against an internal representation in the template matching theories, whereas with the pandemonium architecture, the image is first diffused and processed at the featural level. This granted pandemonium architectures tremendous power because it is capable of recognizing a stimulus despite its changes in size, style and other transformations; without the presumption of an unlimited pattern memory. It is also unlikely that the template matching theories will function properly when faced with realistic visual inputs, where objects are presented in three dimensions and often occluded by other objects (e.g., half of a book is covered by a piece of paper, but we can still recognize it as a book with relative ease). Nonetheless, some researchers have conducted experiments comparing the two theories. Not surprisingly, the results often favored a hierarchal feature building model like the pandemonium architecture.

Comparison with Hebbian pattern recognition

The Hebbian model resembles feature-oriented theories like the pandemonium architecture in many aspects. The first level of processing in the Hebbian model is called the cell assemblies, which have very similar functions to feature demons. However, cell assemblies are more limited than the feature demons, because it can only extracts lines, angles and contours. The cell assemblies are combined to form phase sequences, which is very similar to the function of the cognitive demons. In a sense, many consider the Hebbian model to be a crossover between the template and feature matching theories, as the features extracted from the Hebbian models can be considered as basic templates.

Fetal rights

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Fetal_rights

 

Fetal rights or foetal rights (alternatively prenatal rights) are the moral rights or legal rights of the human fetus under natural and civil law. The term fetal rights came into wide usage after Roe v. Wade, the 1973 landmark case that legalized abortion in the United States and was essentially overturned in 2022. The concept of fetal rights has evolved to include the issues of maternal substance use disorders, including alcohol use disorder and opioid use disorder. Most international human rights charters "clearly reject claims that human rights should attach from conception or any time before birth." While most international human rights instruments lack a universal inclusion of the fetus as a person for the purposes of human rights, the fetus is granted various rights in the constitutions and civil codes of some countries.

Prenatal or fetal?

There are jurisdictions where children in the prenatal stage are assigned rights even before they reach the fetal stage. For example, in Italy, embryos are considered subjects of law already after fertilization and even before implantation (if fertilization happens in vitro). The term fetus itself is not exactly suitable for legal purposes, as it denotes only a part of the prenatal period with a starting point that remains unclear. Furthermore, its archetypal meaning is associated with plants, contributing to the dehumanization of the child in the prenatal stage.

History

See also: History of abortion law debate

In antiquity, the fetus was sometimes protected by restrictions on abortion. Some versions of the Hippocratic Oath indirectly protected the fetus by prohibiting abortifacients. Until approximately the mid-19th century, philosophical views on the fetus were influenced in part by Aristotelian concept of delayed hominization. According to it, human fetuses only gradually acquire their souls, and in the early stages of pregnancy the fetus is not fully human. Relying on examinations of miscarried fetuses, Aristotle believed that male fetuses acquire their basic form at around day 40, and female ones at day 90. For Pythagoreans, however, fetal life was co-equal in moral worth with adult human life from the moment of conception; similar views were held by Stoics. Ancient Athenian law did not recognise fetal right to life before the ritual acknowledgement of the child. The law, however, allowed for the postponement of the execution of sentenced pregnant women until a baby was delivered.

Several Hindu texts on ethics and righteousness, such as Dharmaśāstra, give fetus a right to life from conception, although in practice such texts are not always followed.

The property law of the Roman Empire granted fetus inheritance rights. As long as the fetus was conceived before the testator's death (usually, the father) and then born alive, their inheritance rights were equal to those born before the testator's death. Even though under Roman law the fetus was not a legal subject, it was a potential person whose property rights were protected after birth. Roman jurist Ulpian noted that "in the Law of the Twelve Tables he who was in the womb is admitted to the legitimate succession, if he has been born". Another jurist Julius Paulus similarly noted, that "the ancients provided for the free unborn child in such a way that they preserved for it all legal rights intact until the time of birth". The inheritance rights of the fetus were means of fulfilling the testator's will. The interests of the fetus could be protected by a custodian, usually a male relative, but in some cases a woman herself could be appointed the custodian. The Digest granted the fetus consanguinity rights, vesting the protection of fetal interests in the praetor. The Digest also prohibited the execution of pregnant women until delivery. The Roman law also envisaged that if a slave mother had been free for any period between the time of the conception and childbirth, the child would be regarded as born free. Although the mother might have become slave again before the childbirth, it was considered that the unborn should not be prejudiced by the mother's misfortune. At the same time, Greek and Roman sources do not mention issues of alcohol consumption by pregnant women. On that basis it is believed that Greeks and Romans were not aware of the fetal alcohol syndrome.

A baby holding the "Petition of the Unborn Babes", 18th century illustration.

After the spread of Christianity, an issue emerged: whether it was permissible for a pregnant woman to be baptised before childbirth, due to uncertainty as to whether the fetus would be cobaptised with its mother. The Synod of Neo-Caesarea decided that the baptism of a pregnant woman in any stage of gestation did not include the fetus. In the Middle Ages, fetal rights were closely associated with the concept of ensoulment. In some cases the fetus could also inherit or be in the order of succession. In the Byzantine Empire, a fetus was regarded as a natural person and could inherit alongside blood descendants and slaves. Byzantine Emperor Michael VIII Palaiologos allowed soldiers to transfer their pronoiai to their unborn children. The unborn royals were increasingly granted the right to succession. In 1284, King of Scotland Alexander III designated his future unborn children as heirs presumptive by the act of parliament to avoid potential squabbles among loyal descendants of his lineage. The 1315 entail of Scottish king Robert the Bruce allowed the unborn collateral individuals to be in line for the throne beyond his brother Edward and daughter Marjorie Bruce. After the death of Albert II of Germany in 1439, his then-unborn son Ladislaus the Posthumous inherited his father's sovereign rights. In 1536, the British Parliament gave the unborn children of Henry VIII and Jane Seymour precedence in the line of royal succession. The medieval distinction between the ensouled and the unensouled fetus was removed after Pope Pius IX decreed in 1854 that the ensoulment of Virgin Mary occurred at conception.

In 1751, a pamphlet "The Petition of the Unborn Babes to the Censors of the Royal College of Physicians of London" by physician Frank Nicholls was published, advocating fetal right to life and protection. The pamphlet anticipated many of the arguments of the 21st century's pro-life movement. In 1762, English jurist and judge William Blackstone wrote that an "infant in its mother's womb" could benefit from a legacy and receive an estate as if it were actually bom. The fetus was thus considered a person for purposes of inheritance. Similarly to the Roman law, the Napoleonic Code envisaged that if a woman becomes a widow, a male guardian should be appointed for her unborn child.

In the 20th century and particularly after World War II fetal rights issues continued to develop. In 1948, the Declaration of Geneva was adopted which prior to amendments in 1983 and 2005, advised physicians to "maintain the utmost respect for human life from the time of its conception". In 1967, American Bar Association Journal noted "the modern trend of legal decisions that grant every property and personal right to the unborn child, including the right to life itself, from conception on". In 1975, while interpreting the right to life under the Basic Law of Germany, the Federal Constitutional Court opined that "life in the sense of historical existence of a human individual" exists "at least from the 14th day after conception (nidation, individuation)" and thus everyone's right to life under the Basic Law of Germany includes the unborn as human beings. The 1980s witnessed the reappearance of fetal protection in the workplace, aimed at guarding fetal health in potentially hazardous working conditions. In 1983, Ireland was one of the first countries in the world to constitutionalize a fetal right to life by passing the Eighth Amendment to the Constitution, later repealed in September 2018.

Modern regulations

The only modern international treaty specifically tackling the fetal rights is the American Convention on Human Rights which envisages the fetal right to life from the moment of conception. The convention was ratified by twenty five countries of the Americas (two countries later denounced the convention leading the current number of ratifiers to be twenty three) in 1973–1993. Mexico ratified the convention with the reservation that the expression "in general" concerning the fetal right to life does not constitute an obligation and that this matter falls within the domain of the states. While the convention may be interpreted to permit domestic abortion laws in exceptional circumstances, it effectively declares the fetus a person. However, only a minority of state ratifiers completely prohibit abortion without allowing for an exception when the pregnant woman's life is in danger (Dominican Republic, El Salvador and Nicaragua).

Based on the 1959 Declaration of the Rights of the Child, preambular paragraph 9 of the Convention on the Rights of the Child (CRC) states that "the child... needs... appropriate legal protection before as well as after birth", but due to ambiguity the legal protection of the fetus conflicts with the rights of a pregnant girl under the same Convention. Such conflict is sometimes called maternal-fetal conflict. Under CRC, the rights of a pregnant girl are interpreted as superseding those of her fetus. The states retain the power to decide for themselves what prenatal legal protection they would adopt under CRC. A proposal to grant fetus the right to life from conception was put forward by Belgium, Brazil, El Salvador, Mexico and Morocco during drafting of the International Covenant on Civil and Political Rights (ICCPR), but it was rejected in favor of less stringent wording. At the same time, ICCPR prohibits the execution of pregnant women.

The World Medical Association Declaration on Therapeutic Abortion notes that "circumstances bringing the interests of a mother into conflict with the interests of her unborn child create a dilemma and raise the question as to whether or not the pregnancy should be deliberately terminated". The Dublin Declaration on Maternal Healthcare, signed in 2012, prioritizes fetal right to life by noting that "there is a fundamental difference between abortion, and necessary medical treatments that are carried out to save the life of the mother, even if such treatment results in the loss of life of her unborn child". Several organizations, such as World Health Organization (WHO) and Human Rights Watch prioritize women's reproductive rights over fetal rights.

Under European law, a fetus is generally regarded as an in utero part of the mother and thus its rights are held by the mother. The European Court of Human Rights opined that the right to life does not extend to fetuses under Article 2 of the European Convention on Human Rights (ECHR). In H. v. Norway, the European Commission did not exclude that "in certain circumstances" the fetus may enjoy "a certain protection under Article 2, first sentence". Two European Union member states (Hungary and Slovakia) grant the fetus the constitutional right to life. The Constitution of Norway grants the unborn royal children the right of succession to the throne. In English common law, fetus is granted inheritance rights under the born alive rule.

Islamic law grants the fetus the right to life particularly after ensoulment, which according to various Islamic jurists happens after 40–42 days or four months after conception (some Shiite jurists believe the ensoulment occurs after 11 to 14 days, during the implantation of the fertilized egg in the uterine wall). Both the Sunni and Shiite jurists accord the fetus inheritance rights under two conditions: if a man dies and a pregnant wife survives him, the fetal right to inherit is secure and the inheritance cannot be disposed of before the fetus' share is set aside. Under the second condition, if a woman aborts the fetus at any stage and ignores any vital signs, the fetus is entitled to the inheritance of any legitimate legator who dies after its conception.

The legal debate on fetal rights sometimes invokes the notion of fetal viability. Its primary determinant is fetal lung capacity which typically develops at twenty-three to twenty-four weeks. The twenty-three weeks is usually regarded as the lower bound of fetal viability because technology has been unable to surpass the limit set by lung development. It was nonetheless stated that technology has made it possible to regard the fetus as a patient independent of the mother. In Winnipeg Child and Family Services v. G., the judges argued that "technologies like real-time ultrasound, fetal heart monitors and foetoscopy can clearly show us that the fetus is alive" and thus the born alive rule is "outdated and indefensible".

The creation of human embryos for all research purposes is prohibited by the Convention for the Protection of Human Rights and Dignity of the Human Being with regard to the Application of Biology and Medicine. However, similarly to the abortion debate, in the normative debate on embryo research two views can be distinguished: a "fetalist" view focusing on the moral value of the embryo, and a "feminist" view advocating the interests of women, particularly candidate oocyte donors.

Prenatal personhood

In most jurisdictions, children in the prenatal stage are not considered subjects of law, or bearers of subjective rights. According to Aude Bertrand-Mirkovic, children in the prenatal stage are human persons but do not need legal personhood, as their rights can be protected by objective law. In civil law, legislature often resort to infans conceptus rule, which is rooted in Roman law and means that the pre-born child is treated as born whenever it is considered in their interest. Enforcement of this rule is subjected to a condition of live (and sometimes viable) birth. In Common law, there exists the Born alive rule, — the principle applied in criminal matters which says that ‘a person cannot be held responsible for injuries inflicted on a foetus in utero unless and until it is born alive’.

In some countries, such as the Republic of El Salvador, the Italian Republic, and the Republic of Peru, the child in the prenatal stage is endowed with the status of a subject of law. Recognition of a child in the prenatal stage as a subject of law in practice does not imply the prohibition of abortion and does generally imply the emergence of subjective non-property rights. In contrast, most subjective property rights may arise before birth but can be exercised already after birth.

According to Oleksandra Steshenko, prenatal personhood should be seen as legal personhood of a particular type, as its scope and content differ from the personhood of born people: a) prenatal personhood always lacks the capacity to act; b) it does not necessarily require the civil registration of prenatal existence; c) the beginning of prenatal existence can be determined, when necessary, using legal presumptions; d) every prenatal right should be weighed against the rights of a pregnant mother and, where relevant, against rights of a father; e) as the duration of prenatal existence increases, the child gradually acquires a broader range of rights with increasing scope.

Fetal (prenatal) rights by country

Country	Constitutional protection of fetal rights		Recognition of personhood
Canada	No		No	223. When child becomes human being A child becomes a human being within the meaning of this Act when it has completely proceeded, in a living state, from the body of its mother, whether or not: (a) it has breathed; (b) it has an independent circulation; or (c) the navel string is severed.
Chile	Yes	Article 19 The Constitution guarantees all persons: 1.The right to life and to the physical and mental integrity of the person. The law protects the life of the unborn.	Yes
Dominican Republic	Yes	Article 37 The right to life is inviolable from conception until death. The death penalty may not be established, pronounced, nor applied in any case.	Yes
Ecuador	Yes	Article 45 Children and adolescents shall enjoy the rights that are common to all human beings, in addition to those that are specific to their age. The State shall recognize and guarantee life, including care and protection from the time of conception.	Yes
El Salvador	Yes	Article 1 El Salvador recognizes the human person as the origin and the end of the activity of the State, which is organized to attain justice, judicial security, and the common good. In that same manner, it recognizes as a human person every human being since the moment of conception.	Yes
Guatemala	Yes	Article 3 The State guarantees and protects the human life from its conception, as well as the integrity and security of the person.	Yes
Hungary	Yes	Article 2 Human dignity shall be inviolable. Every human being shall have the right to life and human dignity; the life of the foetus shall be protected from the moment of conception.	Yes
Honduras	Yes	Article 67 The unborn shall be considered as born for all rights accorded within the limits established by law.	Yes
Madagascar	Yes	Article 19 The State recognizes and organizes for all individuals the right to the protection of health from their conception through the organization of free public health care, which gratuitousness results from the capacity of the national solidarity.	Yes
Peru	Yes	Article 2. To life, his identity, his moral, psychical, and physical integrity, and his free development and well-being. The unborn child is a rights-bearing subject in all cases that benefit him.	Yes
Brazil	No		Yes	Article 2.  The civil personality of the person starts in the birth with the life, but the law safeguard, since the conception, the rights of the unborn.
Philippines	Yes	Section 12 The State recognizes the sanctity of family life and shall protect and strengthen the family as a basic autonomous social institution. It shall equally protect the life of the mother and the life of the unborn from conception.	Yes
Slovakia	Yes	Article 15 1. Everyone has the right to life. Human life is worthy of protection already before birth.	Yes
Solomon Islands	No		Yes[85]

1. 
   

1. Argentina, Barbados, Bolivia, Brazil, Chile, Colombia, Costa Rica, Dominica, Ecuador, El Salvador, Grenada, Guatemala, Haiti, Honduras, Jamaica, Mexico, Nicaragua, Panama, Paraguay, Peru, Dominican Republic, Suriname, Trinidad and Tobago (denounced), Uruguay and Venezuela (denounced).

United States President George W. Bush signs the Unborn Victims of Violence Act of 2004

The Eighth Amendment of the Constitution of Ireland gave "the unborn" a right to life equal to that of "the mother". In 2018, the Supreme Court ruled that the fetus' only inherent constitutionally protected right is the right to be born, overturning a High Court ruling that a fetus additionally possessed the children's rights guaranteed by Article 42A of the Constitution. On 25 May 2018, a referendum was passed which amended the Constitution by the substitution of the former provision recognising the right to life of the unborn, with one permitting the Oireachtas, the Irish Parliament, to legislate for the termination of pregnancies. This amendment took effect when it was signed into law by the President of Ireland on 18 September 2018, and abortion was governed by the Protection of Life During Pregnancy Act 2013 until it was replaced and repealed by the Health (Regulation of Termination of Pregnancy) Act 2018, which took effect on 1 January 2019.

In the United States, as of 2014, thirty-eight states provide certain level of criminal protection for the unborn, and twenty-three of these states have laws that protect the fetus from conception until birth. All US states–by statute, court rule or case law–permit a guardian ad litem to represent the interests of the unborn. In 1999, the Unborn Victims of Violence Act was introduced into United States Congress which defines violent assault committed against pregnant women as being a crime against two victims: the woman and the fetus she carries. This law was passed in 2004 after the murder of Laci Peterson and the fetus she was carrying. In 2002, U.S. President George W. Bush announced a plan to ensure health care coverage for fetuses under the State Children's Health Insurance Program (SCHIP).

The civil codes of several countries, such as China (including Hong Kong and Macau) and Russia, as well as some US states, grant fetus inheritance rights, usually under the born alive rule. In the civil code of Iran, fetus can inherit in case of abortion that took place due to a crime, as long as the fetus was alive even for a second after birth. Under the civil code of Japan, for the purposes of inheritance the fetus is deemed to have already been born. The civil codes of the Philippines and Spain envisage that donations to the unborn children can be made and accepted by "persons who would legally represent them if they were already born". The same is allowed by the Malikis.

Alongside Norway, the Constitution of Bhutan grants the unborn royal children the right to succession, but only if there is no male heir.

Behavioral intervention

Various initiatives, prompted by concern for the ill effects which might be posed to the health or development of a fetus, seek to restrict or discourage women from engaging in certain behaviors while pregnant. Also, in some countries, laws have been passed to restrict the practice of abortion based upon the gender of the fetus.

• Many jurisdictions actively warn against the consumption of alcoholic beverages by pregnant women, recommending a maximum intake or total abstinence, due to its association with fetal alcohol syndrome. Countries that encourage those who are pregnant to avoid alcohol either entirely or partially include Australia, Canada, France, Iceland, Israel, the Netherlands, Norway, New Zealand, Spain, the United Kingdom, and the United States. In Poland in 2017 a bill was proposed that would limit the liberty of a woman (coercive and preventive measures such as isolation) who was known to drink during pregnancy. But the draw of the bill was not brought up because it was considered controversial and an erosion of women's autonomy and liberty.
• Many national and international agencies recommend dietary guidelines for pregnant women due to the health risks posed by the consumption of fish contaminated with methylmercury through industrial pollution. Studies have linked exposure to various levels of methylmercury in utero to neurological disorders in children.
• The use of tobacco products or exposure to secondhand smoke during pregnancy has been linked to low birth weight. Governor Mike Huckabee of Arkansas, citing studies which attribute 10% of infant deaths to tobacco-smoking mothers, considered adopting a smoking ban for pregnant women in 2006 with the aim of reducing infant mortality.
• No U.S. state has enacted a law which criminalizes specific behavior during pregnancy, but, nonetheless, it has been estimated that at least 200 American women have been criminally prosecuted or arrested under existing child abuse statutes for allegedly bringing about harm in-utero through their conduct during pregnancy. Reasons for pressing charges included use of illicit substances, consumption of alcohol, and failure to comply with a doctor's order of bedrest or caesarean section. Drug addicts have been accused of "supplying drugs to a minor" through unintentional chemical subjection via the umbilical cord. Others have been charged with assault with a deadly weapon with the "deadly weapon" in question being an illegal drug. Minnesota, Wisconsin and South Dakota allow women who continue to use substances while pregnant to be civilly committed. Some states require that medical providers report any infant who is born with a physical dependency, or who tests positive for residual traces of alcohol or drugs, to child welfare authorities.
• Cultural preferences for male children in some parts of Asia, such as Mainland China, India, South Korea, and Taiwan, have sometimes led to sex-selective abortion of female fetuses, leading to the disparity between male-to-female birth rates which is observed in some places. It is a crime in all these jurisdictions to procure an abortion for the purposes of sex selection.

Solubility equilibrium

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Solubility_equilibrium 

Solubility equilibrium is a type of dynamic equilibrium that exists when a chemical compound in the solid state is in chemical equilibrium with a solution of that compound. The solid may dissolve unchanged, with dissociation, or with chemical reaction with another constituent of the solution, such as acid or alkali. Each solubility equilibrium is characterized by a temperature-dependent solubility product which functions like an equilibrium constant. Solubility equilibria are important in pharmaceutical, environmental and many other scenarios.

Definitions

A solubility equilibrium exists when a chemical compound in the solid state is in chemical equilibrium with a solution containing the compound. This type of equilibrium is an example of dynamic equilibrium in that some individual molecules migrate between the solid and solution phases such that the rates of dissolution and precipitation are equal to one another. When equilibrium is established and the solid has not all dissolved, the solution is said to be saturated. The concentration of the solute in a saturated solution is known as the solubility. Units of solubility may be molar (mol dm⁻³) or expressed as mass per unit volume, such as μg mL⁻¹. Solubility is temperature dependent. A solution containing a higher concentration of solute than the solubility is said to be supersaturated. A supersaturated solution may be induced to come to equilibrium by the addition of a "seed" which may be a tiny crystal of the solute, or a tiny solid particle, which initiates precipitation.

There are three main types of solubility equilibria.

1. Simple dissolution.
2. Dissolution with dissociation reaction. This is characteristic of salts. The equilibrium constant is known in this case as a solubility product.
3. Dissolution with ionization reaction. This is characteristic of the dissolution of weak acids or weak bases in aqueous media of varying pH.

In each case an equilibrium constant can be specified as a quotient of activities. This equilibrium constant is dimensionless as activity is a dimensionless quantity. However, use of activities is very inconvenient, so the equilibrium constant is usually divided by the quotient of activity coefficients, to become a quotient of concentrations. See Equilibrium chemistry § Equilibrium constant for details. Moreover, the activity of a solid is, by definition, equal to 1 so it is omitted from the defining expression.

For a chemical equilibrium $${\displaystyle {\mathrm{A}}_p{\mathrm{B}}_q\leftrightharpoons p\mathrm{A}+q\mathrm{B}}$$ the solubility product, K_sp for the compound A_pB_q is defined as follows $${\displaystyle K_{\mathrm{s}\mathrm{p}}=[\mathrm{A}{]}^p[\mathrm{B}{]}^q}$$ where [A] and [B] are the concentrations of A and B in a saturated solution. A solubility product has a similar functionality to an equilibrium constant though formally K_sp has the dimension of (concentration)^p+q.

Effects of conditions

Temperature effect

Solubility is sensitive to changes in temperature. For example, sugar is more soluble in hot water than cool water. It occurs because solubility products, like other types of equilibrium constants, are functions of temperature. In accordance with Le Chatelier's Principle, when the dissolution process is endothermic (heat is absorbed), solubility increases with rising temperature. This effect is the basis for the process of recrystallization, which can be used to purify a chemical compound. When dissolution is exothermic (heat is released) solubility decreases with rising temperature. Sodium sulfate shows increasing solubility with temperature below about 32.4 °C, but a decreasing solubility at higher temperature. This is because the solid phase is the decahydrate (Na
₂SO
₄·10H
₂O) below the transition temperature, but a different hydrate above that temperature.

The dependence on temperature of solubility for an ideal solution (achieved for low solubility substances) is given by the following expression containing the enthalpy of melting, Δ_mH, and the mole fraction ${\displaystyle x_i}$ of the solute at saturation: $${\displaystyle {\left(\frac{\mathrm{\partial }\ln x_i}{\mathrm{\partial }T}\right)}_P=\frac{{\overline{H}}_{i,\mathrm{a}\mathrm{q}}-H_{i,\mathrm{c}\mathrm{r}}}{R{T}^2}}$$ where ${\displaystyle {\overline{H}}_{i,\mathrm{a}\mathrm{q}}}$ is the partial molar enthalpy of the solute at infinite dilution and ${\displaystyle H_{i,\mathrm{c}\mathrm{r}}}$ the enthalpy per mole of the pure crystal.

This differential expression for a non-electrolyte can be integrated on a temperature interval to give: $${\displaystyle \ln x_i=\frac{{\mathrm{\Delta }}_m{H}_i}{R}\left(\frac{1}{T_f}-\frac{1}{T}\right)}$$

For nonideal solutions activity of the solute at saturation appears instead of mole fraction solubility in the derivative with respect to temperature: $${\displaystyle {\left(\frac{\mathrm{\partial }\ln a_i}{\mathrm{\partial }T}\right)}_P=\frac{H_{i,\mathrm{a}\mathrm{q}}-H_{i,\mathrm{c}\mathrm{r}}}{R{T}^2}}$$

Common-ion effect

The common-ion effect is the effect of decreased solubility of one salt when another salt that has an ion in common with it is also present. For example, the solubility of silver chloride, AgCl, is lowered when sodium chloride, a source of the common ion chloride, is added to a suspension of AgCl in water. $${\displaystyle \mathrm{A}\mathrm{g}\mathrm{C}\mathrm{l}(\mathrm{s})\leftrightharpoons \mathrm{A}{\mathrm{g}}^{+}(\mathrm{a}\mathrm{q})+\mathrm{C}{\mathrm{l}}^{-}(\mathrm{a}\mathrm{q})}$$ The solubility, S, in the absence of a common ion can be calculated as follows. The concentrations [Ag⁺] and [Cl⁻] are equal because one mole of AgCl would dissociate into one mole of Ag⁺ and one mole of Cl⁻. Let the concentration of [Ag⁺(aq)] be denoted by x. Then $${\displaystyle K_{\mathrm{s}\mathrm{p}}=[\mathrm{A}{\mathrm{g}}^{+}][\mathrm{C}{\mathrm{l}}^{-}]=x^2}$$ $${\displaystyle \text{Solubility}=[\mathrm{A}{\mathrm{g}}^{+}]=[\mathrm{C}{\mathrm{l}}^{-}]=x=\sqrt{K_{\mathrm{s}\mathrm{p}}}}$$ K_sp for AgCl is equal to 1.77×10⁻¹⁰ mol² dm⁻⁶ at 25 °C, so the solubility is 1.33×10⁻⁵ mol dm⁻³.

Now suppose that sodium chloride is also present, at a concentration of 0.01 mol dm⁻³ = 0.01 M. The solubility, ignoring any possible effect of the sodium ions, is now calculated by $${\displaystyle K_{\mathrm{s}\mathrm{p}}=[\mathrm{A}{\mathrm{g}}^{+}][\mathrm{C}{\mathrm{l}}^{-}]=x(0.01\text{M}+x)}$$ This is a quadratic equation in x, which is also equal to the solubility. $${\displaystyle x^2+0.01\text{M}x-K_{sp}=0}$$ In the case of silver chloride, x² is very much smaller than 0.01 M x, so the first term can be ignored. Therefore $${\displaystyle \text{Solubility}=[\mathrm{A}{\mathrm{g}}^{+}]=x=\frac{K_{\mathrm{s}\mathrm{p}}}{0.01\text{M}}=1.77\times {10}^{-8}\mathrm{m}\mathrm{o}\mathrm{l}\mathrm{d}{\mathrm{m}}^{-3}}$$ a considerable reduction from 1.33×10⁻⁵ mol dm⁻³. In gravimetric analysis for silver, the reduction in solubility due to the common ion effect is used to ensure "complete" precipitation of AgCl.

Particle size effect

The thermodynamic solubility constant is defined for large monocrystals. Solubility will increase with decreasing size of solute particle (or droplet) because of the additional surface energy. This effect is generally small unless particles become very small, typically smaller than 1 μm. The effect of the particle size on solubility constant can be quantified as follows: $${\displaystyle \log {(}^{\ast }{K}_A)=\log {(}^{\ast }{K}_{A\to 0})+\frac{\gamma A_{\mathrm{m}}}{3.454RT}}$$ where *K_A is the solubility constant for the solute particles with the molar surface area A, *K_A→0 is the solubility constant for substance with molar surface area tending to zero (i.e., when the particles are large), γ is the surface tension of the solute particle in the solvent, A_m is the molar surface area of the solute (in m²/mol), R is the universal gas constant, and T is the absolute temperature.

Salt effects

The salt effects (salting in and salting-out) refers to the fact that the presence of a salt which has no ion in common with the solute, has an effect on the ionic strength of the solution and hence on activity coefficients, so that the equilibrium constant, expressed as a concentration quotient, changes.

Phase effect

Equilibria are defined for specific crystal phases. Therefore, the solubility product is expected to be different depending on the phase of the solid. For example, aragonite and calcite will have different solubility products even though they have both the same chemical identity (calcium carbonate). Under any given conditions one phase will be thermodynamically more stable than the other; therefore, this phase will form when thermodynamic equilibrium is established. However, kinetic factors may favor the formation the unfavorable precipitate (e.g. aragonite), which is then said to be in a metastable state.

In pharmacology, the metastable state is sometimes referred to as amorphous state. Amorphous drugs have higher solubility than their crystalline counterparts due to the absence of long-distance interactions inherent in crystal lattice. Thus, it takes less energy to solvate the molecules in amorphous phase. The effect of amorphous phase on solubility is widely used to make drugs more soluble.

Pressure effect

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: $${\displaystyle {\left(\frac{\mathrm{\partial }\ln x_i}{\mathrm{\partial }P}\right)}_T=-\frac{{\overline{V}}_{i,\mathrm{a}\mathrm{q}}-V_{i,\mathrm{c}\mathrm{r}}}{RT}}$$ where ${\displaystyle x_i}$ is the mole fraction of the ${\displaystyle i}$-th component in the solution, ${\displaystyle P}$ is the pressure, ${\displaystyle T}$ is the absolute temperature, ${\displaystyle {\overline{V}}_{i,\text{aq}}}$ is the partial molar volume of the ${\displaystyle i}$th component in the solution, ${\displaystyle V_{i,\text{cr}}}$ is the partial molar volume of the ${\displaystyle i}$th component in the dissolving solid, and ${\displaystyle R}$ is the universal gas constant.

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

Quantitative aspects

Simple dissolution

Dissolution of an organic solid can be described as an equilibrium between the substance in its solid and dissolved forms. For example, when sucrose (table sugar) forms a saturated solution $${\displaystyle {\mathrm{C}}_{12}{\mathrm{H}}_{22}{\mathrm{O}}_{11}(\mathrm{s})\leftrightharpoons {\mathrm{C}}_{12}{\mathrm{H}}_{22}{\mathrm{O}}_{11}(\mathrm{a}\mathrm{q})}$$ An equilibrium expression for this reaction can be written, as for any chemical reaction (products over reactants): $${\displaystyle K^{\ominus }=\frac{\left\{{\mathrm{C}}_{12}{\mathrm{H}}_{22}{\mathrm{O}}_{11}(\mathrm{a}\mathrm{q})\right\}}{\left\{{\mathrm{C}}_{12}{\mathrm{H}}_{22}{\mathrm{O}}_{11}(\mathrm{s})\right\}}}$$ where K^o is called the thermodynamic solubility constant. The braces indicate activity. The activity of a pure solid is, by definition, unity. Therefore $${\displaystyle K^{\ominus }=\left\{{\mathrm{C}}_{12}{\mathrm{H}}_{22}{\mathrm{O}}_{11}(\mathrm{a}\mathrm{q})\right\}}$$ The activity of a substance, A, in solution can be expressed as the product of the concentration, [A], and an activity coefficient, γ. When K^o is divided by γ, the solubility constant, K_s, $${\displaystyle K_{\mathrm{s}}=\left[{\mathrm{C}}_{12}{\mathrm{H}}_{22}{\mathrm{O}}_{11}(\mathrm{a}\mathrm{q})\right]}$$ is obtained. This is equivalent to defining the standard state as the saturated solution so that the activity coefficient is equal to one. The solubility constant is a true constant only if the activity coefficient is not affected by the presence of any other solutes that may be present. The unit of the solubility constant is the same as the unit of the concentration of the solute. For sucrose K_s = 1.971 mol dm⁻³ at 25 °C. This shows that the solubility of sucrose at 25 °C is nearly 2 mol dm⁻³ (540 g/L). Sucrose is unusual in that it does not easily form a supersaturated solution at higher concentrations, as do most other carbohydrates.

Dissolution with dissociation

Ionic compounds normally dissociate into their constituent ions when they dissolve in water. For example, for silver chloride: $${\displaystyle {\text{AgCl}}_{(\text{s})}^{}\stackrel{-\rightharpoonup }{\leftharpoondown -}{\text{Ag}}_{(\text{aq})}^{+}+\text{Cl}{-}_{(\text{aq})}^{}}$$ The expression for the equilibrium constant for this reaction is: $${\displaystyle K^{\ominus }=\frac{\left\{{{\text{Ag}}^{+}}_{(\text{aq})}\right\}\left\{{{\text{Cl}}^{-}}_{(\text{aq})}\right\}}{\left\{{\text{AgCl}}_{(\text{s})}^{}\right\}}=\left\{{{\text{Ag}}^{+}}_{(\text{aq})}\right\}\left\{{{\text{Cl}}^{-}}_{(\text{aq})}\right\}}$$ where ${\displaystyle K^{\ominus }}$ is the thermodynamic equilibrium constant and braces indicate activity. The activity of a pure solid is, by definition, equal to one.

When the solubility of the salt is very low the activity coefficients of the ions in solution are nearly equal to one. By setting them to be actually equal to one this expression reduces to the solubility product expression: $${\displaystyle K_{\text{sp}}=[{\text{Ag}}^{+}][{\text{Cl}}^{-}]=[{\text{Ag}}^{+}{]}^2=[{\text{Cl}}^{-}{]}^2.}$$

For 2:2 and 3:3 salts, such as CaSO₄ and FePO₄, the general expression for the solubility product is the same as for a 1:1 electrolyte $${\displaystyle \mathrm{A}\mathrm{B}\leftrightharpoons {\mathrm{A}}^{p+}+{\mathrm{B}}^{p-}}$$

${\displaystyle K_{sp}=[\mathrm{A}][\mathrm{B}]=[\mathrm{A}{]}^2=[\mathrm{B}{]}^2}$ (electrical charges are omitted in general expressions, for simplicity of notation)

With an unsymmetrical salt like Ca(OH)₂ the solubility expression is given by $${\displaystyle \text{Ca}{(\text{OH})}_2^{}\stackrel{-\rightharpoonup }{\leftharpoondown -}{\text{Ca}}^{2+}+2{\text{OH}}^{-}}$$ $${\displaystyle K_{sp}=[\text{Ca}]{[\text{OH}]}^2}$$ Since the concentration of hydroxide ions is twice the concentration of calcium ions this reduces to ${\displaystyle {\mathrm{K}}_{\mathrm{s}\mathrm{p}}=4[\mathrm{C}\mathrm{a}{]}^3}$

In general, with the chemical equilibrium $${\displaystyle {\text{A}}_p{\text{B}}_q\leftrightharpoons p{\text{A}}^{n+}+q{\text{B}}^{m-}}$$ $${\displaystyle [\text{B}]=\frac{q}{p}[\text{A}]}$$ and the following table, showing the relationship between the solubility of a compound and the value of its solubility product, can be derived.

Salt	p	q	Solubility, S
AgCl Ca(SO4) Fe(PO4)	1	1	√Ksp
Na2(SO4) Ca(OH)2	2 1	1 2	${\displaystyle \sqrt[3]{\frac{K_{\text{sp}}}{4}}}$
Na3(PO4) FeCl3	3 1	1 3	${\displaystyle \sqrt[4]{\frac{K_{\text{sp}}}{27}}}$
Al2(SO4)3 Ca3(PO4)2	2 3	3 2	${\displaystyle \sqrt[5]{\frac{K_{\text{sp}}}{108}}}$
Mp(An)q	p	q	${\displaystyle \sqrt[p+q]{\frac{K_{\text{sp}}}{p^p{q}^q}}}$

Solubility products are often expressed in logarithmic form. Thus, for calcium sulfate, with K_sp = 4.93×10⁻⁵ mol² dm⁻⁶, log K_sp = −4.32. The smaller the value of K_sp, or the more negative the log value, the lower the solubility.

Some salts are not fully dissociated in solution. Examples include MgSO₄, famously discovered by Manfred Eigen to be present in seawater as both an inner sphere complex and an outer sphere complex. The solubility of such salts is calculated by the method outlined in dissolution with reaction.

Hydroxides

The solubility product for the hydroxide of a metal ion, Mⁿ⁺, is usually defined, as follows: $${\displaystyle \mathrm{M}(\mathrm{O}\mathrm{H}{)}_{\mathrm{n}}\leftrightharpoons {\mathrm{M}}^{\mathrm{n}+}+\mathrm{n}\mathrm{O}{\mathrm{H}}^{-}}$$ $${\displaystyle K_{sp}=[{\mathrm{M}}^{\mathrm{n}+}][\mathrm{O}{\mathrm{H}}^{-}{]}^{\mathrm{n}}}$$ However, general-purpose computer programs are designed to use hydrogen ion concentrations with the alternative definitions. $${\displaystyle \mathrm{M}(\mathrm{O}\mathrm{H}{)}_{\mathrm{n}}+\mathrm{n}{\mathrm{H}}^{+}\leftrightharpoons {\mathrm{M}}^{\mathrm{n}+}+\mathrm{n}{\mathrm{H}}_2\mathrm{O}}$$ $${\displaystyle K_{\text{sp}}^{\ast }=[{\mathrm{M}}^{\mathrm{n}+}][{\mathrm{H}}^{+}{]}^{-\mathrm{n}}}$$

For hydroxides, solubility products are often given in a modified form, K*_sp, using hydrogen ion concentration in place of hydroxide ion concentration. The two values are related by the self-ionization constant for water, K_w. $${\displaystyle K_{\mathrm{w}}=[{\mathrm{H}}^{+}][\mathrm{O}{\mathrm{H}}^{-}]}$$ $${\displaystyle K_{\text{sp}}^{\ast }=\frac{K_{\text{sp}}}{(K_{\text{w}}{)}^n}}$$ $${\displaystyle \log K_{\text{sp}}^{\ast }=\log K_{\text{sp}}-n\log K_{\text{w}}}$$ For example, at ambient temperature, for calcium hydroxide, Ca(OH)₂, lg K_sp is ca. −5 and lg K*_sp ≈ −5 + 2 × 14 ≈ 23.

Dissolution with reaction

When a concentrated solution of ammonia is added to a suspension of silver chloride dissolution occurs because a complex of Ag⁺ is formed

A typical reaction with dissolution involves a weak base, B, dissolving in an acidic aqueous solution. $${\displaystyle \mathrm{B}(\mathrm{s})+{\mathrm{H}}^{+}(\mathrm{a}\mathrm{q})\leftrightharpoons {\mathrm{B}\mathrm{H}}^{+}(\mathrm{a}\mathrm{q})}$$ This reaction is very important for pharmaceutical products. Dissolution of weak acids in alkaline media is similarly important. $${\displaystyle \mathrm{H}\mathrm{A}(\mathrm{s})+\mathrm{O}{\mathrm{H}}^{-}(\mathrm{a}\mathrm{q})\leftrightharpoons {\mathrm{A}}^{-}(\mathrm{a}\mathrm{q})+{\mathrm{H}}_2\mathrm{O}}$$ The uncharged molecule usually has lower solubility than the ionic form, so solubility depends on pH and the acid dissociation constant of the solute. The term "intrinsic solubility" is used to describe the solubility of the un-ionized form in the absence of acid or alkali.

Leaching of aluminium salts from rocks and soil by acid rain is another example of dissolution with reaction: alumino-silicates are bases which react with the acid to form soluble species, such as Al³⁺(aq).

Formation of a chemical complex may also change solubility. A well-known example is the addition of a concentrated solution of ammonia to a suspension of silver chloride, in which dissolution is favoured by the formation of an ammine complex. $${\displaystyle \mathrm{A}\mathrm{g}\mathrm{C}\mathrm{l}(\mathrm{s})+2\mathrm{N}{\mathrm{H}}_3(\mathrm{a}\mathrm{q})\leftrightharpoons [\mathrm{A}\mathrm{g}(\mathrm{N}{\mathrm{H}}_3{)}_2{]}^{+}(\mathrm{a}\mathrm{q})+\mathrm{C}{\mathrm{l}}^{-}(\mathrm{a}\mathrm{q})}$$ When sufficient ammonia is added to a suspension of silver chloride, the solid dissolves. The addition of water softeners to washing powders to inhibit the formation of soap scum provides an example of practical importance.

Experimental determination

The determination of solubility is fraught with difficulties. First and foremost is the difficulty in establishing that the system is in equilibrium at the chosen temperature. This is because both precipitation and dissolution reactions may be extremely slow. If the process is very slow solvent evaporation may be an issue. Supersaturation may occur. With very insoluble substances, the concentrations in solution are very low and difficult to determine. The methods used fall broadly into two categories, static and dynamic.

Static methods

In static methods a mixture is brought to equilibrium and the concentration of a species in the solution phase is determined by chemical analysis. This usually requires separation of the solid and solution phases. In order to do this the equilibration and separation should be performed in a thermostatted room. Very low concentrations can be measured if a radioactive tracer is incorporated in the solid phase.

A variation of the static method is to add a solution of the substance in a non-aqueous solvent, such as dimethyl sulfoxide, to an aqueous buffer mixture. Immediate precipitation may occur giving a cloudy mixture. The solubility measured for such a mixture is known as "kinetic solubility". The cloudiness is due to the fact that the precipitate particles are very small resulting in Tyndall scattering. In fact the particles are so small that the particle size effect comes into play and kinetic solubility is often greater than equilibrium solubility. Over time the cloudiness will disappear as the size of the crystallites increases, and eventually equilibrium will be reached in a process known as precipitate ageing.

Dynamic methods

Solubility values of organic acids, bases, and ampholytes of pharmaceutical interest may be obtained by a process called "Chasing equilibrium solubility". In this procedure, a quantity of substance is first dissolved at a pH where it exists predominantly in its ionized form and then a precipitate of the neutral (un-ionized) species is formed by changing the pH. Subsequently, the rate of change of pH due to precipitation or dissolution is monitored and strong acid and base titrant are added to adjust the pH to discover the equilibrium conditions when the two rates are equal. The advantage of this method is that it is relatively fast as the quantity of precipitate formed is quite small. However, the performance of the method may be affected by the formation supersaturated solutions.