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Alcuin wrote many theological and dogmatic treatises, as well as a few grammatical works and a number of poems. In 796, he was made abbot of Marmoutier Abbey, in Tours, where he remained until his death.

Alcuin was born in Northumbria, presumably sometime in the 730s. Virtually nothing is known of his parents, family background, or origin. In common hagiographical fashion, the "Vita Alcuini" asserts that Alcuin was "of noble English stock", and this statement has usually been accepted by scholars. Alcuin's own work only mentions such collateral kinsmen as Wilgils, father of the missionary saint Willibrord; and Beornrad (also spelled Beornred), abbot of Echternach and bishop of Sens. Willibrord, Alcuin and Beornrad were all related by blood.

In his "Life" of St Willibrord, Alcuin writes that Wilgils, called a "paterfamilias", had founded an oratory and church at the mouth of the Humber, which had fallen into Alcuin's possession by inheritance. Because in early Anglo-Latin writing "paterfamilias" ("head of a family, householder") usually referred to a ("churl"), Donald A. Bullough suggests that Alcuin's family was of ("churlish") status: i.e., free but subordinate to a noble lord, and that Alcuin and other members of his family rose to prominence through beneficial connections with the aristocracy. If so, Alcuin's origins may lie in the southern part of what was formerly known as Deira.

The young Alcuin came to the cathedral church of York during the golden age of Archbishop Ecgbert and his brother, the Northumbrian King Eadberht. Ecgbert had been a disciple of the Venerable Bede, who urged him to raise York to an archbishopric. King Eadberht and Archbishop Ecgbert oversaw the re-energising and reorganisation of the English church, with an emphasis on reforming the clergy and on the tradition of learning that Bede had begun. Ecgbert was devoted to Alcuin, who thrived under his tutelage.

The York school was renowned as a centre of learning in the liberal arts, literature, and science, as well as in religious matters. From here, Alcuin drew inspiration for the school he would lead at the Frankish court. He revived the school with the trivium and quadrivium disciplines, writing a codex on the trivium, while his student Hraban wrote one on the quadrivium.

Alcuin graduated to become a teacher during the 750s. His ascendancy to the headship of the York school, the ancestor of St Peter's School, began after Aelbert became Archbishop of York in 767. Around the same time, Alcuin became a deacon in the church. He was never ordained a priest. Though no real evidence shows that he took monastic vows, he lived as if he had.

In 781, King Elfwald sent Alcuin to Rome to petition the pope for official confirmation of York's status as an archbishopric and to confirm the election of the new archbishop, Eanbald I. On his way home, he met Charlemagne (whom he had met once before), this time in the Italian city of Parma.

Alcuin's intellectual curiosity allowed him to be reluctantly persuaded to join Charlemagne's court. He joined an illustrious group of scholars whom Charlemagne had gathered around him, the mainsprings of the Carolingian Renaissance: Peter of Pisa, Paulinus of Aquileia, Rado, and Abbot Fulrad. Alcuin would later write, "the Lord was calling me to the service of King Charles".

Alcuin became master of the Palace School of Charlemagne in Aachen () in 782. It had been founded by the king's ancestors as a place for the education of the royal children (mostly in manners and the ways of the court). However, Charlemagne wanted to include the liberal arts, and most importantly, the study of religion. From 782 to 790, Alcuin taught Charlemagne himself, his sons Pepin and Louis, as well as young men sent to be educated at court, and the young clerics attached to the palace chapel. Bringing with him from York his assistants Pyttel, Sigewulf, and Joseph, Alcuin revolutionised the educational standards of the Palace School, introducing Charlemagne to the liberal arts and creating a personalised atmosphere of scholarship and learning, to the extent that the institution came to be known as the 'school of Master Albinus'.

In this role as adviser, he took issue with the emperor's policy of forcing pagans to be baptised on pain of death, arguing, "Faith is a free act of the will, not a forced act. We must appeal to the conscience, not compel it by violence. You can force people to be baptised, but you cannot force them to believe." His arguments seem to have prevailed – Charlemagne abolished the death penalty for paganism in 797.

Charlemagne gathered the best men of every land in his court, and became far more than just the king at the centre. It seems that he made many of these men his closest friends and counsellors. They referred to him as 'David', a reference to the Biblical king David. Alcuin soon found himself on intimate terms with Charlemagne and the other men at court, where pupils and masters were known by affectionate and jesting nicknames. Alcuin himself was known as 'Albinus' or 'Flaccus'. While at Aachen, Alcuin bestowed pet names upon his pupils – derived mainly from Virgil's "Eclogues". According to the "Encyclopædia Britannica", "He loved Charlemagne and enjoyed the king's esteem, but his letters reveal that his fear of him was as great as his love."

After the death of Pope Adrian I, Alcuin was commissioned by Charlemagne to compose an epitaph for Adrian. The epitaph was inscribed on black stone quarried at Aachen and carried to Rome where it was set over Adrian's tomb in the south transept of St Peter's basilica just before Charlemagne's coronation in the basilica on Christmas Day 800.

Return to Northumbria and back to Francia.

In 790, Alcuin returned from the court of Charlemagne to England, to which he had remained attached. He dwelt there for some time, but Charlemagne then invited him back to help in the fight against the Adoptionist heresy, which was at that time making great progress in Toledo, the old capital of the Visigoths and still a major city for the Christians under Islamic rule in Spain. He is believed to have had contacts with Beatus of Liébana, from the Kingdom of Asturias, who fought against Adoptionism. At the Council of Frankfurt in 794, Alcuin upheld the orthodox doctrine against the views expressed by Felix of Urgel, an heresiarch according to the Catholic Encyclopaedia. Having failed during his stay in Northumbria to influence King Æthelred in the conduct of his reign, Alcuin never returned home.

He was back at Charlemagne's court by at least mid-792, writing a series of letters to Æthelred, to Hygbald, Bishop of Lindisfarne, and to Æthelhard, Archbishop of Canterbury in the succeeding months, dealing with the Viking attack on Lindisfarne in July 793. These letters and Alcuin's poem on the subject, , provide the only significant contemporary account of these events. In his description of the Viking attack, he wrote: "Never before has such terror appeared in Britain. Behold the church of St Cuthbert, splattered with the blood of God's priests, robbed of its ornaments."

In 796, Alcuin was in his 60s. He hoped to be free from court duties and upon the death of Abbot Itherius of Saint Martin at Tours, Charlemagne put Marmoutier Abbey into Alcuin's care, with the understanding that he should be available if the king ever needed his counsel. There, he encouraged the work of the monks on the beautiful Carolingian minuscule script, ancestor of modern Roman typefaces.

Alcuin died on 19 May 804, some 10 years before the emperor, and was buried at St. Martin's Church under an epitaph that partly read:

The majority of details on Alcuin's life come from his letters and poems. Also, autobiographical sections are in Alcuin's poem on York and in the "Vita Alcuini", a hagiography written for him at Ferrières in the 820s, possibly based in part on the memories of Sigwulf, one of Alcuin's pupils.

Carolingian Renaissance figure and legacy.

The collection of mathematical and logical word problems entitled "Propositiones ad acuendos juvenes" ("Problems to Sharpen Youths") is sometimes attributed to Alcuin. In a 799 letter to Charlemagne, the scholar claimed to have sent "certain figures of arithmetic for the joy of cleverness", which some scholars have identified with the "Propositiones."

The text contains about 53 mathematical word problems (with solutions), in no particular pedagogical order. Among the most famous of these problems are: four that involve river crossings, including the problem of three anxious brothers, each of whom has an unmarried sister whom he cannot leave alone with either of the other men lest she be defiled (Problem 17); the problem of the wolf, goat, and cabbage (Problem 18); and the problem of "the two adults and two children where the children weigh half as much as the adults" (Problem 19). Alcuin's sequence is the solution to one of the problems of that book.

Alcuin made the abbey school into a model of excellence and many students flocked to it. He had many manuscripts copied using outstandingly beautiful calligraphy, the Carolingian minuscule based on round and legible uncial letters. He wrote many letters to his English friends, to Arno, bishop of Salzburg and above all to Charlemagne. These letters (of which 311 are extant) are filled mainly with pious meditations, but they form an important source of information as to the literary and social conditions of the time and are the most reliable authority for the history of humanism during the Carolingian age. Alcuin trained the numerous monks of the abbey in piety, and in the midst of these pursuits, he died.

Alcuin is the most prominent figure of the Carolingian Renaissance, in which three main periods have been distinguished: in the first of these, up to the arrival of Alcuin at the court, the Italians occupy a central place; in the second, Alcuin and the English are dominant; in the third (from 804), the influence of Theodulf the Visigoth is preponderant.

Alcuin also developed manuals used in his educational work – a grammar and works on rhetoric and dialectics. These are written in the form of a dialogue, and in two of them the interlocutors are Charlemagne and Alcuin. He wrote several theological treatises: a "De fide Trinitatis", and commentaries on the Bible. Alcuin is credited with inventing the first known question mark, though it did not resemble the modern symbol.

Alcuin transmitted to the Franks the knowledge of Latin culture, which had existed in Anglo-Saxon England. A number of his works still exist. Besides some graceful epistles in the style of Venantius Fortunatus, he wrote some long poems, and notably he is the author of a history (in verse) of the church at York, "Versus de patribus, regibus et sanctis Eboracensis ecclesiae". At the same time, he is noted for making one of the only explicit comments on Old English poetry surviving from the early Middle Ages, in a letter to one Speratus, the bishop of an unnamed English see (possibly Unwona of Leicester): ("Let God's words be read at the episcopal dinner-table. It is right that a reader should be heard, not a harpist, patristic discourse, not pagan song. What has Ingeld to do with Christ?").

Use of homoerotic language in writings.

Alcuin was also a close friend of Charlemagne's sister Gisela, Abbess of Chelles, and he hailed her as "a noble sister in the bond of sweet love". He wrote to Charlemagne's daughters Rotrude and Bertha, "the devotion of my heart specially tends towards you both because of the familiarity and dedication you have shown me". He dedicated the last two books of his commentary on John's gospel to them both.

Despite inconclusive evidence of Alcuin's personal passions, he was clear in his own writings that the men of Sodom had been punished with fire for "sinning against nature with men" – a view consistent with Church teaching. Such sins, argued Alcuin, were therefore more serious than lustful acts with women, for which the earth was cleansed and revivified by the water of the Flood, and merit to be "withered by flames unto eternal barrenness".

Alcuin is honored in the Church of England and in the Episcopal Church on 20 May the first available day after the day of his death (as Dunstan is celebrated on 19 May).

Alcuin College, one of the colleges of the University of York, is named after him. In January 2020, Alcuin was the subject of the BBC Radio 4 programme "In Our Time".

For a complete census of Alcuin's works, see Marie-Hélène Jullien and Françoise Perelman, eds., "Clavis scriptorum latinorum medii aevi: Auctores Galliae 735–987. Tomus II: Alcuinus." Turnhout: Brepols, 1999.

Of Alcuin's letters, over 310 have survived.

Angilbert, Count of Ponthieu ( – 18 February 814) was a noble Frankish poet who was educated under Alcuin and served Charlemagne as a secretary, diplomat, and son-in-law. He is venerated as a pre-Congregation saint and is still honored on the day of his death, 18 February.

Angilbert seems to have been brought up at the court of Charlemagne at the palace school in (Aachen). He was educated there as the pupil and then-friend of the great English scholar Alcuin. When Charlemagne sent his young son Pepin to Italy as King of the Lombards, Angilbert went along as "primicerius palatii", a high administrator of the satellite court. As the friend and adviser of Pepin, he assisted for a while in the government of Italy. Angilbert delivered the document on Iconoclasm from the Frankish Synod of Frankfurt to Pope Adrian I, and was later sent on three important embassies to the pope, in 792, 794, and 796. At one time, he served an officer of the maritime provinces. He accompanied Charlemagne to Rome in 800 and was one of the witnesses to his will in 811.

There are various traditions concerning Angilbert's relationship with Bertha, daughter of Charlemagne. One holds that they were married, another that they were not. They had, however, at least one daughter and two sons, one of whom, Nithard, became a notable figure in the mid-9th century, while their daughter Bertha went on to marry Helgaud II, Count of Ponthieu. Control of marriage and the meanings of legitimacy were hotly contested in the Middle Ages. Bertha and Angilbert are an example of how resistance to the idea of a sacramental marriage could coincide with holding church offices. On the other hand, some historians have speculated that Charlemagne opposed formal marriages for his daughters out of concern for political rivalries from their potential husbands; none of Charlemagne's daughters were married, despite political offers of arranged marriages.

In 790, Angilbert retired to the abbey of Centulum, the "Monastery of St Richarius" () at present-day Saint-Riquier in Picardy. Elected abbot in 794, he rebuilt the monastery and endowed it with a library of 200 volumes. It was not uncommon for the Merovingian, Carolingian, or later kings to make laymen abbots of monasteries; the layman would often use the income of the monastery as his own and leave the monks a bare minimum for the necessary expenses of the foundation. Angilbert, in contrast, spent a great deal rebuilding Saint-Riquier; when he completed it, Charlemagne spent Easter of the year 800 there. In keeping with Carolingian policies, Angilbert established a school at Saint-Riquier to educate the local boys.

The poem "De conversione Saxonum" has been attributed to Angilbert.

Angilbert's poems were published by Ernst Dümmler in the "Monumenta Germaniae Historica". For criticisms of this edition, see Ludwig Traube in Max Roediger's "Schriften für germanische Philologie" (1888).

Antony van Leeuwenhoek

In chemistry, amines (, ) are compounds and functional groups that contain a basic nitrogen atom with a lone pair. Amines are formally derivatives of ammonia (), wherein one or more hydrogen atoms have been replaced by a substituent such as an alkyl or aryl group (these may respectively be called alkylamines and arylamines; amines in which both types of substituent are attached to one nitrogen atom may be called alkylarylamines). Important amines include amino acids, biogenic amines, trimethylamine, and aniline. Inorganic derivatives of ammonia are also called amines, such as monochloramine ().

The substituent is called an amino group.

Compounds with a nitrogen atom attached to a carbonyl group, thus having the structure , are called amides and have different chemical properties from amines.

Classification of amines.

Amines can be classified according to the nature and number of substituents on nitrogen. Aliphatic amines contain only H and alkyl substituents. Aromatic amines have the nitrogen atom connected to an aromatic ring.

Amines, alkyl and aryl alike, are organized into three subcategories based on the number of carbon atoms adjacent to the nitrogen (how many hydrogen atoms of the ammonia molecule are replaced by hydrocarbon groups):

A fourth subcategory is determined by the connectivity of the substituents attached to the nitrogen:

It is also possible to have four organic substituents on the nitrogen. These species are not amines but are quaternary ammonium cations and have a charged nitrogen center. Quaternary ammonium salts exist with many kinds of anions.

Amines are named in several ways. Typically, the compound is given the prefix "amino-" or the suffix "-amine". The prefix ""N"-" shows substitution on the nitrogen atom. An organic compound with multiple amino groups is called a diamine, triamine, tetraamine and so forth.

Systematic names for some common amines:

Physical properties.

Hydrogen bonding significantly influences the properties of primary and secondary amines. For example, methyl and ethyl amines are gases under standard conditions, whereas the corresponding methyl and ethyl alcohols are liquids. Amines possess a characteristic ammonia smell, liquid amines have a distinctive "fishy" and foul smell.

The nitrogen atom features a lone electron pair that can bind H+ to form an ammonium ion R3NH+. The lone electron pair is represented in this article by two dots above or next to the N. The water solubility of simple amines is enhanced by hydrogen bonding involving these lone electron pairs. Typically salts of ammonium compounds exhibit the following order of solubility in water: primary ammonium () > secondary ammonium () > tertiary ammonium (R3NH+). Small aliphatic amines display significant solubility in many solvents, whereas those with large substituents are lipophilic. Aromatic amines, such as aniline, have their lone pair electrons conjugated into the benzene ring, thus their tendency to engage in hydrogen bonding is diminished. Their boiling points are high and their solubility in water is low.

Spectroscopic identification.

Typically the presence of an amine functional group is deduced by a combination of techniques, including mass spectrometry as well as NMR and IR spectroscopies. 1H NMR signals for amines disappear upon treatment of the sample with D2O. In their infrared spectrum primary amines exhibit two N-H bands, whereas secondary amines exhibit only one. In their IR spectra, primary and secondary amines exhibit distinctive N-H stretching bands near 3300 cm-1. Somewhat less distinctive are the bands appearing below 1600 cm-1, which are weaker and overlap with C-C and C-H modes. For the case of propyl amine, the H-N-H scissor mode appears near 1600 cm-1, the C-N stretch near 1000 cm-1, and the R2N-H bend near 810 cm-1.

Alkyl amines characteristically feature tetrahedral nitrogen centers. C-N-C and C-N-H angles approach the idealized angle of 109°. C-N distances are slightly shorter than C-C distances. The energy barrier for the nitrogen inversion of the stereocenter is about 7 kcal/mol for a trialkylamine. The interconversion has been compared to the inversion of an open umbrella into a strong wind.

Amines of the type NHRR' and NRR′R′′ are chiral: the nitrogen center bears four substituents counting the lone pair. Because of the low barrier to inversion, amines of the type NHRR' cannot be obtained in optical purity. For chiral tertiary amines, NRR′R′′ can only be resolved when the R, R', and R′′ groups are constrained in cyclic structures such as "N"-substituted aziridines (quaternary ammonium salts are resolvable).

In aromatic amines ("anilines"), nitrogen is often nearly planar owing to conjugation of the lone pair with the aryl substituent. The C-N distance is correspondingly shorter. In aniline, the C-N distance is the same as the C-C distances.

Like ammonia, amines are bases. Compared to alkali metal hydroxides, amines are weaker.

The basicity of amines depends on:

Owing to inductive effects, the basicity of an amine might be expected to increase with the number of alkyl groups on the amine. Correlations are complicated owing to the effects of solvation which are opposite the trends for inductive effects. Solvation effects also dominate the basicity of aromatic amines (anilines). For anilines, the lone pair of electrons on nitrogen delocalizes into the ring, resulting in decreased basicity. Substituents on the aromatic ring, and their positions relative to the amino group, also affect basicity as seen in the table.

Solvation significantly affects the basicity of amines. N-H groups strongly interact with water, especially in ammonium ions. Consequently, the basicity of ammonia is enhanced by 1011 by solvation. The intrinsic basicity of amines, i.e. the situation where solvation is unimportant, has been evaluated in the gas phase. In the gas phase, amines exhibit the basicities predicted from the electron-releasing effects of the organic substituents. Thus tertiary amines are more basic than secondary amines, which are more basic than primary amines, and finally ammonia is least basic. The order of pKb's (basicities in water) does not follow this order. Similarly aniline is more basic than ammonia in the gas phase, but ten thousand times less so in aqueous solution.

In aprotic polar solvents such as DMSO, DMF, and acetonitrile the energy of solvation is not as high as in protic polar solvents like water and methanol. For this reason, the basicity of amines in these aprotic solvents is almost solely governed by the electronic effects.

Industrially significant alkyl amines are prepared from ammonia by alkylation with alcohols:

From alkyl and aryl halides.

Unlike the reaction of amines with alcohols the reaction of amines and ammonia with alkyl halides is used for synthesis in the laboratory:

In such reactions, which are more useful for alkyl iodides and bromides, the degree of alkylation is difficult to control such that one obtains mixtures of primary, secondary, and tertiary amines, as well as quaternary ammonium salts.

Selectivity can be improved via the Delépine reaction, although this is rarely employed on an industrial scale. Selectivity is also assured in the Gabriel synthesis, which involves organohalide reacting with potassium phthalimide.

Aryl halides are much less reactive toward amines and for that reason are more controllable. A popular way to prepare aryl amines is the Buchwald-Hartwig reaction.

Disubstituted alkenes react with HCN in the presence of strong acids to give formamides, which can be decarbonylated. This method, the Ritter reaction, is used industrially to produce tertiary amines such as "tert"-octylamine.

Hydroamination of alkenes is also widely practiced. The reaction is catalyzed by zeolite-based solid acids.

Via the process of hydrogenation, unsaturated N-containing functional groups are reduced to amines using hydrogen in the presence of a nickel catalyst. Suitable groups include nitriles, azides, imines including oximes, amides, and nitro. In the case of nitriles, reactions are sensitive to acidic or alkaline conditions, which can cause hydrolysis of the group. is more commonly employed for the reduction of these same groups on the laboratory scale.

Many amines are produced from aldehydes and ketones via reductive amination, which can either proceed catalytically or stoichiometrically.

Aniline () and its derivatives are prepared by reduction of the nitroaromatics. In industry, hydrogen is the preferred reductant, whereas, in the laboratory, tin and iron are often employed.

Specialized methods.

Many methods exist for the preparation of amines, many of these methods being rather specialized.

Alkylation, acylation, and sulfonation, etc..

Aside from their basicity, the dominant reactivity of amines is their nucleophilicity. Most primary amines are good ligands for metal ions to give coordination complexes. Amines are alkylated by alkyl halides. Acyl chlorides and acid anhydrides react with primary and secondary amines to form amides (the "Schotten–Baumann reaction").

Similarly, with sulfonyl chlorides, one obtains sulfonamides. This transformation, known as the Hinsberg reaction, is a chemical test for the presence of amines.

Because amines are basic, they neutralize acids to form the corresponding ammonium salts . When formed from carboxylic acids and primary and secondary amines, these salts thermally dehydrate to form the corresponding amides.

Amines undergo sulfamation upon treatment with sulfur trioxide or sources thereof:

Amines reacts with nitrous acid to give diazonium salts. The alkyl diazonium salts are of little importance because they are too unstable. The most important members are derivatives of aromatic amines such as aniline ("phenylamine") (A = aryl or naphthyl):

Anilines and naphthylamines form more stable diazonium salts, which can be isolated in the crystalline form. Diazonium salts undergo a variety of useful transformations involving replacement of the group with anions. For example, cuprous cyanide gives the corresponding nitriles:

Aryldiazoniums couple with electron-rich aromatic compounds such as a phenol to form azo compounds. Such reactions are widely applied to the production of dyes.

Conversion to imines.

Imine formation is an important reaction. Primary amines react with ketones and aldehydes to form imines. In the case of formaldehyde (R' H), these products typically exist as cyclic trimers.

Reduction of these imines gives secondary amines:

Similarly, secondary amines react with ketones and aldehydes to form enamines:

An overview of the reactions of amines is given below:

Biological activity.

Amines are ubiquitous in biology. The breakdown of amino acids releases amines, famously in the case of decaying fish which smell of trimethylamine. Many neurotransmitters are amines, including epinephrine, norepinephrine, dopamine, serotonin, and histamine. Protonated amino groups () are the most common positively charged moieties in proteins, specifically in the amino acid lysine. The anionic polymer DNA is typically bound to various amine-rich proteins. Additionally, the terminal charged primary ammonium on lysine forms salt bridges with carboxylate groups of other amino acids in polypeptides, which is one of the primary influences on the three-dimensional structures of proteins.

Hormones derived from the modification of amino acids are referred to as amine hormones. Typically, the original structure of the amino acid is modified such that a –COOH, or carboxyl, group is removed, whereas the , or amine, group remains. Amine hormones are synthesized from the amino acids tryptophan or tyrosine.

Application of amines.

Primary aromatic amines are used as a starting material for the manufacture of azo dyes. It reacts with nitrous acid to form diazonium salt, which can undergo coupling reaction to form an azo compound. As azo-compounds are highly coloured, they are widely used in dyeing industries, such as:

Most drugs and drug candidates contain amine functional groups:

Aqueous monoethanolamine (MEA), diglycolamine (DGA), diethanolamine (DEA), diisopropanolamine (DIPA) and methyldiethanolamine (MDEA) are widely used industrially for removing carbon dioxide (CO2) and hydrogen sulfide (H2S) from natural gas and refinery process streams. They may also be used to remove CO2 from combustion gases and flue gases and may have potential for abatement of greenhouse gases. Related processes are known as sweetening.

Epoxy resin curing agents.

Amines are often used as epoxy resin curing agents. These include dimethylethylamine, cyclohexylamine, and a variety of diamines such as 4,4-diaminodicyclohexylmethane. Multifunctional amines such as tetraethylenepentamine and triethylenetetramine are also widely used in this capacity. The reaction proceeds by the lone pair of electrons on the amine nitrogen attacking the outermost carbon on the oxirane ring of the epoxy resin. This relieves ring strain on the epoxide and is the driving force of the reaction.

Low molecular weight simple amines, such as ethylamine, are only weakly toxic with between 100 and 1000 mg/kg. They are skin irritants, especially as some are easily absorbed through the skin. Amines are a broad class of compounds, and more complex members of the class can be extremely bioactive, for example strychnine.

Absolute zero is the lowest limit of the thermodynamic temperature scale; a state at which the enthalpy and entropy of a cooled ideal gas reach their minimum value, taken as zero kelvin. The fundamental particles of nature have minimum vibrational motion, retaining only quantum mechanical, zero-point energy-induced particle motion. The theoretical temperature is determined by extrapolating the ideal gas law; by international agreement, absolute zero is taken as −273.15 degrees on the Celsius scale (International System of Units), which equals −459.67 degrees on the Fahrenheit scale (United States customary units or imperial units). The corresponding Kelvin and Rankine temperature scales set their zero points at absolute zero by definition.

It is commonly thought of as the lowest temperature possible, but it is not the lowest "enthalpy" state possible, because all real substances begin to depart from the ideal gas when cooled as they approach the change of state to liquid, and then to solid; and the sum of the enthalpy of vaporization (gas to liquid) and enthalpy of fusion (liquid to solid) exceeds the ideal gas's change in enthalpy to absolute zero. In the quantum-mechanical description, matter at absolute zero is in its ground state, the point of lowest internal energy.