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giving a contradiction. More generally, this shows that there is no smooth retraction from any non-empty smooth oriented compact manifold onto its boundary. The proof using Stokes' theorem is closely related to the proof using homology, because the form generates the de Rham cohomology group (∂) which is isomorphic to the homology group (∂) by de Rham's theorem. A combinatorial proof. The BFPT can be proved using Sperner's lemma. We now give an outline of the proof for the special case in which "f" is a function from the standard "n"-simplex, formula_56 to itself, where For every point formula_58 also formula_59 Hence the sum of their coordinates is equal: Hence, by the pigeonhole principle, for every formula_58 there must be an index formula_62 such that the formula_63th coordinate of formula_64 is greater than or equal to the formula_63th coordinate of its image under "f": Moreover, if formula_64 lies on a "k"-dimensional sub-face of formula_56 then by the same argument, the index formula_63 can be selected from among the coordinates which are not zero on this sub-face.
We now use this fact to construct a Sperner coloring. For every triangulation of formula_56 the color of every vertex formula_64 is an index formula_63 such that formula_73 By construction, this is a Sperner coloring. Hence, by Sperner's lemma, there is an "n"-dimensional simplex whose vertices are colored with the entire set of available colors. Because "f" is continuous, this simplex can be made arbitrarily small by choosing an arbitrarily fine triangulation. Hence, there must be a point formula_64 which satisfies the labeling condition in all coordinates: formula_75 for all formula_76 Because the sum of the coordinates of formula_64 and formula_78 must be equal, all these inequalities must actually be equalities. But this means that: That is, formula_64 is a fixed point of formula_81 A proof by Hirsch. There is also a quick proof, by Morris Hirsch, based on the impossibility of a differentiable retraction. The indirect proof starts by noting that the map "f" can be approximated by a smooth map retaining the property of not fixing a point; this can be done by using the Weierstrass approximation theorem or by convolving with smooth bump functions. One then defines a retraction as above which must now be differentiable. Such a retraction must have a non-singular value, by Sard's theorem, which is also non-singular for the restriction to the boundary (which is just the identity). Thus the inverse image would be a 1-manifold with boundary. The boundary would have to contain at least two end points, both of which would have to lie on the boundary of the original ball—which is impossible in a retraction.
R. Bruce Kellogg, Tien-Yien Li, and James A. Yorke turned Hirsch's proof into a computable proof by observing that the retract is in fact defined everywhere except at the fixed points. For almost any point, "q", on the boundary, (assuming it is not a fixed point) the one manifold with boundary mentioned above does exist and the only possibility is that it leads from "q" to a fixed point. It is an easy numerical task to follow such a path from "q" to the fixed point so the method is essentially computable. gave a conceptually similar path-following version of the homotopy proof which extends to a wide variety of related problems. A proof using oriented area. A variation of the preceding proof does not employ the Sard's theorem, and goes as follows. If formula_82 is a smooth retraction, one considers the smooth deformation formula_83 and the smooth function Differentiating under the sign of integral it is not difficult to check that ""("t") = 0 for all "t", so "φ" is a constant function, which is a contradiction because "φ"(0) is the "n"-dimensional volume of the ball, while "φ"(1) is zero. The geometric idea is that "φ"("t") is the oriented area of "g""t"("B") (that is, the Lebesgue measure of the image of the ball via "g""t", taking into account multiplicity and orientation), and should remain constant (as it is very clear in the one-dimensional case). On the other hand, as the parameter "t" passes from 0 to 1 the map "g""t" transforms continuously from the identity map of the ball, to the retraction "r", which is a contradiction since the oriented area of the identity coincides with the volume of the ball, while the oriented area of "r" is necessarily 0, as its image is the boundary of the ball, a set of null measure.
A proof using the game Hex. A quite different proof given by David Gale is based on the game of Hex. The basic theorem regarding Hex, first proven by John Nash, is that no game of Hex can end in a draw; the first player always has a winning strategy (although this theorem is nonconstructive, and explicit strategies have not been fully developed for board sizes of dimensions 10 x 10 or greater). This turns out to be equivalent to the Brouwer fixed-point theorem for dimension 2. By considering "n"-dimensional versions of Hex, one can prove in general that Brouwer's theorem is equivalent to the determinacy theorem for Hex. A proof using the Lefschetz fixed-point theorem. The Lefschetz fixed-point theorem says that if a continuous map "f" from a finite simplicial complex "B" to itself has only isolated fixed points, then the number of fixed points counted with multiplicities (which may be negative) is equal to the Lefschetz number and in particular if the Lefschetz number is nonzero then "f" must have a fixed point. If "B" is a ball (or more generally is contractible) then the Lefschetz number is one because the only non-zero simplicial homology group is: formula_86 and "f" acts as the identity on this group, so "f" has a fixed point.
A proof in a weak logical system. In reverse mathematics, Brouwer's theorem can be proved in the system WKL0, and conversely over the base system RCA0 Brouwer's theorem for a square implies the weak Kőnig's lemma, so this gives a precise description of the strength of Brouwer's theorem. Generalizations. The Brouwer fixed-point theorem forms the starting point of a number of more general fixed-point theorems. The straightforward generalization to infinite dimensions, i.e. using the unit ball of an arbitrary Hilbert space instead of Euclidean space, is not true. The main problem here is that the unit balls of infinite-dimensional Hilbert spaces are not compact. For example, in the Hilbert space ℓ2 of square-summable real (or complex) sequences, consider the map "f" : ℓ2 → ℓ2 which sends a sequence ("x""n") from the closed unit ball of ℓ2 to the sequence ("y""n") defined by It is not difficult to check that this map is continuous, has its image in the unit sphere of ℓ2, but does not have a fixed point. The generalizations of the Brouwer fixed-point theorem to infinite dimensional spaces therefore all include a compactness assumption of some sort, and also often an assumption of convexity. See fixed-point theorems in infinite-dimensional spaces for a discussion of these theorems.
There is also finite-dimensional generalization to a larger class of spaces: If formula_88 is a product of finitely many chainable continua, then every continuous function formula_89 has a fixed point, where a chainable continuum is a (usually but in this case not necessarily metric) compact Hausdorff space of which every open cover has a finite open refinement formula_90, such that formula_91 if and only if formula_92. Examples of chainable continua include compact connected linearly ordered spaces and in particular closed intervals of real numbers. The Kakutani fixed point theorem generalizes the Brouwer fixed-point theorem in a different direction: it stays in R"n", but considers upper hemi-continuous set-valued functions (functions that assign to each point of the set a subset of the set). It also requires compactness and convexity of the set. The Lefschetz fixed-point theorem applies to (almost) arbitrary compact topological spaces, and gives a condition in terms of singular homology that guarantees the existence of fixed points; this condition is trivially satisfied for any map in the case of "D""n".
Benzoic acid Benzoic acid () is a white (or colorless) solid organic compound with the formula , whose structure consists of a benzene ring () with a carboxyl () substituent. The benzoyl group is often abbreviated "Bz" (not to be confused with "Bn," which is used for benzyl), thus benzoic acid is also denoted as BzOH, since the benzoyl group has the formula –. It is the simplest aromatic carboxylic acid. The name is derived from gum benzoin, which was for a long time its only source. Benzoic acid occurs naturally in many plants and serves as an intermediate in the biosynthesis of many secondary metabolites. Salts of benzoic acid are used as food preservatives. Benzoic acid is an important precursor for the industrial synthesis of many other organic substances. The salts and esters of benzoic acid are known as benzoates (). History. Benzoic acid was discovered in the sixteenth century. The dry distillation of gum benzoin was first described by Nostradamus (1556), and then by Alexius Pedemontanus (1560) and Blaise de Vigenère (1596).
Justus von Liebig and Friedrich Wöhler determined the composition of benzoic acid. These latter also investigated how hippuric acid is related to benzoic acid. In 1875 Salkowski discovered the antifungal properties of benzoic acid, which explains the preservation of benzoate-containing cloudberry fruits. Production. Industrial preparations. Benzoic acid is produced commercially by partial oxidation of toluene with oxygen. The process is catalyzed by cobalt or manganese naphthenates. The process uses abundant materials, and proceeds in high yield. The first industrial process involved the reaction of benzotrichloride (trichloromethyl benzene) with calcium hydroxide in water, using iron or iron salts as catalyst. The resulting calcium benzoate is converted to benzoic acid with hydrochloric acid. The product contains significant amounts of chlorinated benzoic acid derivatives. For this reason, benzoic acid for human consumption was obtained by dry distillation of gum benzoin. Food-grade benzoic acid is now produced synthetically.
Laboratory synthesis. Benzoic acid is cheap and readily available, so the laboratory synthesis of benzoic acid is mainly practiced for its pedagogical value. It is a common undergraduate preparation. Benzoic acid can be purified by recrystallization from water because of its high solubility in hot water and poor solubility in cold water. The avoidance of organic solvents for the recrystallization makes this experiment particularly safe. This process usually gives a yield of around 65%. By hydrolysis. Like other nitriles and amides, benzonitrile and benzamide can be hydrolyzed to benzoic acid or its conjugate base in acid or basic conditions. From Grignard reagent. Bromobenzene can be converted to benzoic acid by "carboxylation" of the intermediate phenylmagnesium bromide. This synthesis offers a convenient exercise for students to carry out a Grignard reaction, an important class of carbon–carbon bond forming reaction in organic chemistry. Oxidation of benzyl compounds. Benzyl alcohol and benzyl chloride and virtually all benzyl derivatives are readily oxidized to benzoic acid.
Uses. Benzoic acid is mainly consumed in the production of phenol by oxidative decarboxylation at 300−400 °C: The temperature required can be lowered to 200 °C by the addition of catalytic amounts of copper(II) salts. The phenol can be converted to cyclohexanol, which is a starting material for nylon synthesis. Precursor to plasticizers. Benzoate plasticizers, such as the glycol-, diethyleneglycol-, and triethyleneglycol esters, are obtained by transesterification of methyl benzoate with the corresponding diol. These plasticizers, which are used similarly to those derived from terephthalic acid ester, represent alternatives to phthalates. Precursor to sodium benzoate and related preservatives. Benzoic acid and its salts are used as food preservatives, represented by the E numbers E210, E211, E212, and E213. Benzoic acid inhibits the growth of mold, yeast and some bacteria. It is either added directly or created from reactions with its sodium, potassium, or calcium salt. The mechanism starts with the absorption of benzoic acid into the cell. If the intracellular pH changes to 5 or lower, the anaerobic fermentation of glucose through phosphofructokinase is decreased by 95%. The efficacy of benzoic acid and benzoate is thus dependent on the pH of the food. Benzoic acid, benzoates
and their derivatives are used as preservatives for acidic foods and beverages such as citrus fruit juices (citric acid), sparkling drinks (carbon dioxide), soft drinks (phosphoric acid), pickles (vinegar) and other acidified foods. Typical concentrations of benzoic acid as a preservative in food are between 0.05 and 0.1%. Foods in which benzoic acid may be used and maximum levels for its application are controlled by local food laws. Concern has been expressed that benzoic acid and its salts may react with ascorbic acid (vitamin C) in some soft drinks, forming small quantities of carcinogenic benzene. Medicinal. Benzoic acid is a constituent of Whitfield's ointment which is used for the treatment of fungal skin diseases such as ringworm and athlete's foot. As the principal component of gum benzoin, benzoic acid is also a major ingredient in both tincture of benzoin and Friar's balsam. Such products have a long history of use as topical antiseptics and inhalant decongestants. Benzoic acid was used as an expectorant, analgesic, and antiseptic in the early 20th century.
Niche and laboratory uses. In teaching laboratories, benzoic acid is a common standard for calibrating a bomb calorimeter. Biology and health effects. Benzoic acid occurs naturally as do its esters in many plant and animal species. Appreciable amounts are found in most berries (around 0.05%). Ripe fruits of several "Vaccinium" species (e.g., cranberry, "V. vitis macrocarpon"; bilberry, "V. myrtillus") contain as much as 0.03–0.13% free benzoic acid. Benzoic acid is also formed in apples after infection with the fungus "Nectria galligena". Among animals, benzoic acid has been identified primarily in omnivorous or phytophageous species, e.g., in viscera and muscles of the rock ptarmigan ("Lagopus muta") as well as in gland secretions of male muskoxen ("Ovibos moschatus") or Asian bull elephants ("Elephas maximus"). Gum benzoin contains up to 20% of benzoic acid and 40% benzoic acid esters. In terms of its biosynthesis, benzoate is produced in plants from cinnamic acid. A pathway has been identified from phenol via 4-hydroxybenzoate.
Reactions. Reactions of benzoic acid can occur at either the aromatic ring or at the carboxyl group. Aromatic ring. Electrophilic aromatic substitution reaction will take place mainly in 3-position due to the electron-withdrawing carboxylic group; i.e. benzoic acid is "meta" directing. Carboxyl group. Reactions typical for carboxylic acids apply also to benzoic acid. Safety and mammalian metabolism. It is excreted as hippuric acid. Benzoic acid is metabolized by butyrate-CoA ligase into an intermediate product, benzoyl-CoA, which is then metabolized by glycine "N"-acyltransferase into hippuric acid. Humans metabolize toluene which is also excreted as hippuric acid. For humans, the World Health Organization's International Programme on Chemical Safety (IPCS) suggests a provisional tolerable intake would be 5 mg/kg body weight per day. Cats have a significantly lower tolerance against benzoic acid and its salts than rats and mice. Lethal dose for cats can be as low as 300 mg/kg body weight. The oral for rats is 3040 mg/kg, for mice it is 1940–2263 mg/kg. In Taipei, Taiwan, a city health survey in 2010 found that 30% of dried and pickled food products had benzoic acid.
Boltzmann distribution In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution) is a probability distribution or probability measure that gives the probability that a system will be in a certain state as a function of that state's energy and the temperature of the system. The distribution is expressed in the form: where is the probability of the system being in state , is the exponential function, is the energy of that state, and a constant of the distribution is the product of the Boltzmann constant and thermodynamic temperature . The symbol formula_2 denotes proportionality (see for the proportionality constant). The term "system" here has a wide meaning; it can range from a collection of 'sufficient number' of atoms or a single atom to a macroscopic system such as a natural gas storage tank. Therefore, the Boltzmann distribution can be used to solve a wide variety of problems. The distribution shows that states with lower energy will always have a higher probability of being occupied.
The "ratio" of probabilities of two states is known as the Boltzmann factor and characteristically only depends on the states' energy difference: The Boltzmann distribution is named after Ludwig Boltzmann who first formulated it in 1868 during his studies of the statistical mechanics of gases in thermal equilibrium. Boltzmann's statistical work is borne out in his paper “On the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability Calculations Regarding the Conditions for Thermal Equilibrium" The distribution was later investigated extensively, in its modern generic form, by Josiah Willard Gibbs in 1902. The Boltzmann distribution should not be confused with the Maxwell–Boltzmann distribution or Maxwell-Boltzmann statistics. The Boltzmann distribution gives the probability that a system will be in a certain "state" as a function of that state's energy, while the Maxwell-Boltzmann distributions give the probabilities of particle "speeds" or "energies" in ideal gases. The distribution of energies in a one-dimensional gas however, does follow the Boltzmann distribution.
The distribution. The Boltzmann distribution is a probability distribution that gives the probability of a certain state as a function of that state's energy and temperature of the system to which the distribution is applied. It is given as formula_4 where: Using Lagrange multipliers, one can prove that the Boltzmann distribution is the distribution that maximizes the entropy formula_6 subject to the normalization constraint that formula_7 and the constraint that formula_8 equals a particular mean energy value, except for two special cases. (These special cases occur when the mean value is either the minimum or maximum of the energies . In these cases, the entropy maximizing distribution is a limit of Boltzmann distributions where approaches zero from above or below, respectively.) The partition function can be calculated if we know the energies of the states accessible to the system of interest. For atoms the partition function values can be found in the NIST Atomic Spectra Database. The distribution shows that states with lower energy will always have a higher probability of being occupied than the states with higher energy. It can also give us the quantitative relationship between the probabilities of the two states being occupied. The ratio of probabilities for states and is given as
formula_9 where: The corresponding ratio of populations of energy levels must also take their degeneracies into account. The Boltzmann distribution is often used to describe the distribution of particles, such as atoms or molecules, over bound states accessible to them. If we have a system consisting of many particles, the probability of a particle being in state is practically the probability that, if we pick a random particle from that system and check what state it is in, we will find it is in state . This probability is equal to the number of particles in state divided by the total number of particles in the system, that is the fraction of particles that occupy state . where is the number of particles in state and is the total number of particles in the system. We may use the Boltzmann distribution to find this probability that is, as we have seen, equal to the fraction of particles that are in state i. So the equation that gives the fraction of particles in state as a function of the energy of that state is
formula_11 This equation is of great importance to spectroscopy. In spectroscopy we observe a spectral line of atoms or molecules undergoing transitions from one state to another. In order for this to be possible, there must be some particles in the first state to undergo the transition. We may find that this condition is fulfilled by finding the fraction of particles in the first state. If it is negligible, the transition is very likely not observed at the temperature for which the calculation was done. In general, a larger fraction of molecules in the first state means a higher number of transitions to the second state. This gives a stronger spectral line. However, there are other factors that influence the intensity of a spectral line, such as whether it is caused by an allowed or a forbidden transition. The softmax function commonly used in machine learning is related to the Boltzmann distribution: Generalized Boltzmann distribution. Distribution of the form is called generalized Boltzmann distribution by some authors.
The Boltzmann distribution is a special case of the generalized Boltzmann distribution. The generalized Boltzmann distribution is used in statistical mechanics to describe canonical ensemble, grand canonical ensemble and isothermal–isobaric ensemble. The generalized Boltzmann distribution is usually derived from the principle of maximum entropy, but there are other derivations. The generalized Boltzmann distribution has the following properties: In statistical mechanics. The Boltzmann distribution appears in statistical mechanics when considering closed systems of fixed composition that are in thermal equilibrium (equilibrium with respect to energy exchange). The most general case is the probability distribution for the canonical ensemble. Some special cases (derivable from the canonical ensemble) show the Boltzmann distribution in different aspects: Although these cases have strong similarities, it is helpful to distinguish them as they generalize in different ways when the crucial assumptions are changed: In economics.
The Boltzmann distribution can be introduced to allocate permits in emissions trading. The new allocation method using the Boltzmann distribution can describe the most probable, natural, and unbiased distribution of emissions permits among multiple countries. The Boltzmann distribution has the same form as the multinomial logit model. As a discrete choice model, this is very well known in economics since Daniel McFadden made the connection to random utility maximization.
Leg theory Leg theory is a bowling tactic in the sport of cricket. The term "leg theory" is somewhat archaic, but the basic tactic remains a play in modern cricket. Simply put, leg theory involves concentrating the bowling attack at or near the line of leg stump. This may or may not be accompanied by a concentration of fielders on the leg side. The line of attack aims to cramp the batsman, making him play the ball with the bat close to the body. This makes it difficult to hit the ball freely and score runs, especially on the off side. Since a leg theory attack means the batsman is more likely to hit the ball on the leg side, additional fielders on that side of the field can be effective in preventing runs and taking catches. Stifling the batsman in this manner can lead to impatience and frustration, resulting in rash play by the batsman which in turn can lead to a quick dismissal. Concentrating attack on the leg stump is considered by many cricket fans and commentators to lead to boring play, as it stifles run scoring and encourages batsmen to play conservatively.
Leg theory can be a moderately successful tactic when used with both fast bowling and spin bowling, particularly leg spin to right-handed batsmen or off spin to left-handed batsmen. However, because it relies on lack of concentration or discipline by the batsman, it can be risky against patient and skilled players, especially batsmen who are strong on the leg side. The English opening bowlers Sydney Barnes and Frank Foster used leg theory with some success in Australia in 1911–12. In England, at around the same time, Fred Root was one of the main proponents of the same tactic. Fast leg theory. In 1930, England captain Douglas Jardine, together with Nottinghamshire's captain Arthur Carr and his bowlers Harold Larwood and Bill Voce, developed a variant of leg theory in which the bowlers bowled fast, short-pitched balls that would rise into the batsman's body, together with a heavily stacked ring of close fielders on the leg side. The idea was that when the batsman defended against the ball, he would be likely to deflect the ball into the air for a catch.
Jardine called this modified form of the tactic "fast leg theory". On the 1932–33 English tour of Australia, Larwood and Voce bowled fast leg theory at the Australian batsmen. It turned out to be extremely dangerous, and most Australian players sustained injuries from being hit by the ball. Wicket-keeper Bert Oldfield's skull was fractured by a ball hitting his head (although the ball had first glanced off the bat and Larwood had an orthodox field), almost precipitating a riot by the Australian crowd. The Australian press dubbed the tactic "Bodyline", and claimed it was a deliberate attempt by the English team to intimidate and injure the Australian players. Reports of the controversy reaching England at the time described the bowling as "fast leg theory", which sounded to many people to be a harmless and well-established tactic. This led to a serious misunderstanding amongst the English public and the Marylebone Cricket Club – the administrators of English cricket – of the dangers posed by Bodyline. The English press and cricket authorities declared the Australian protests to be a case of sore losing and "squealing". It was only with the return of the English team and the subsequent use of Bodyline against English players in England by the touring West Indian cricket team in 1933 that demonstrated to the country the dangers it posed. The MCC subsequently revised the Laws of Cricket to prevent the use of "fast leg theory" tactics in future, also limiting the traditional tactic.
Blythe Danner Blythe Katherine Danner (born February 3, 1943) is an American actress. Accolades she has received include two Primetime Emmy Awards for Best Supporting Actress in a Drama Series for her role as Izzy Huffstodt on "Huff" (2004–2006), and a Tony Award for Best Featured Actress for her performance in "Butterflies Are Free" on Broadway (1969–1972). Danner was twice nominated for the Primetime Emmy for Outstanding Guest Actress in a Comedy Series for portraying Marilyn Truman on "Will & Grace" (2001–06; 2018–20), and the Primetime Emmy for Outstanding Lead Actress in a Miniseries or Movie for her roles in "We Were the Mulvaneys" (2002) and "Back When We Were Grownups" (2004). For the latter, she also received a Golden Globe Award nomination. Danner played Dina Byrnes in "Meet the Parents" (2000) and its sequels "Meet the Fockers" (2004) and "Little Fockers" (2010). She has collaborated on several occasions with Woody Allen, appearing in three of his films: "Another Woman" (1988), "Alice" (1990), and "Husbands and Wives" (1992). Her other notable film credits include "1776" (1972), "Hearts of the West" (1975), "The Great Santini" (1979), "Mr. & Mrs. Bridge" (1990), "The Prince of Tides" (1991), "To Wong Foo, Thanks for Everything! Julie Newmar" (1995), "The Myth of Fingerprints" (1997), "The X-Files" (1998), "Forces of Nature" (1999), "The Love Letter" (1999), "The Last Kiss" (2006), "Paul" (2011), "Hello I Must Be Going" (2012), "I'll See You in My Dreams" (2015), and "What They Had" (2018).
Danner is the sister of Harry Danner and the widow of Bruce Paltrow. She is the mother of actress Gwyneth Paltrow and director Jake Paltrow. She is the grandmother of media personality Apple Martin. Early life. Danner was born in Philadelphia, Pennsylvania, the daughter of Katharine (née Kile) and Harry Earl Danner, a bank executive. She has a brother, opera singer and actor Harry Danner, a sister and a maternal half-brother. Danner has Pennsylvania Dutch, some English and Irish ancestry; her maternal grandmother was a German immigrant, and one of her paternal great-grandmothers was born in Barbados to a family of European descent. Danner graduated from George School, a Quaker high school located near Newtown, Bucks County, Pennsylvania, in 1960. Career. A graduate of Bard College, Danner's first roles included the 1967 musical "Mata Hari" and the 1968 Off-Broadway production of "Summertree". Her early Broadway appearances included "Cyrano de Bergerac" (1968) and her Theatre World Award-winning performance in "The Miser" (1969). She won the Tony Award for Best Featured Actress in a Play for portraying a free-spirited divorcée in "Butterflies Are Free" (1970).
In 1972, Danner portrayed Martha Jefferson in the film version of "1776". That same year, she played the unknowing wife of a husband who committed murder, opposite Peter Falk and John Cassavetes, in the "Columbo" episode "Étude in Black". Her earliest starring film role was opposite Alan Alda in "To Kill a Clown" (1972). Danner appeared in the episode of "MAS*H" entitled "The More I See You", playing the love interest of Alda's character Hawkeye Pierce. She played lawyer Amanda Bonner in television's "Adam's Rib", opposite Ken Howard as Adam Bonner. She played Zelda Fitzgerald in "F. Scott Fitzgerald and 'The Last of the Belles"' (1974). She was the eponymous heroine in the film "Lovin' Molly" (1974) (directed by Sidney Lumet). She appeared in "Futureworld", playing Tracy Ballard with co-star Peter Fonda (1976). In the 1982 TV movie "Inside the Third Reich", she played the wife of Albert Speer. In the film version of Neil Simon's semi-autobiographical play "Brighton Beach Memoirs" (1986), she portrayed a middle-aged Jewish mother. She has appeared in two films based on the novels of Pat Conroy, "The Great Santini" (1979) and "The Prince of Tides" (1991), as well as two television movies adapted from books by Anne Tyler, "Saint Maybe" and "Back When We Were Grownups", both for the Hallmark Hall of Fame.
Danner appeared opposite Robert De Niro in the 2000 comedy hit "Meet the Parents", and its sequels, "Meet the Fockers" (2004) and "Little Fockers" (2010). From 2001 to 2006, she regularly appeared on NBC's sitcom "Will & Grace" as Will Truman's mother Marilyn. From 2004 to 2006, she starred in the main cast of the comedy-drama series "Huff". In 2005, she was nominated for three Primetime Emmy Awards for her work on "Will & Grace", "Huff", and the television film "Back When We Were Grownups", winning for her role in "Huff". The following year, she won a second consecutive Emmy Award for "Huff". For 25 years, she has been a regular performer at the Williamstown Summer Theater Festival, where she also serves on the board of directors. In 2006, Danner was awarded an inaugural Katharine Hepburn Medal by Bryn Mawr College's Katharine Houghton Hepburn Center. In 2015, Danner was inducted into the American Theater Hall of Fame. Environmental activism. Danner has been involved in environmental issues such as recycling and conservation for over 30 years. She has been active with INFORM, Inc., is on the Board of Environmental Advocates of New York and the board of directors of the Environmental Media Association, and won the 2002 EMA Board of Directors Ongoing Commitment Award. In 2011, Danner joined Moms Clean Air Force, to help call on parents to join in the fight against toxic air pollution.
Health care activism. After the death of her husband Bruce Paltrow from oral cancer, she became involved with the nonprofit Oral Cancer Foundation. In 2005, she filmed a public service announcement to raise public awareness of the disease and the need for early detection. She has since appeared on morning talk shows and given interviews in such magazines as "People". The Bruce Paltrow Oral Cancer Fund, administered by the Oral Cancer Foundation, raises funding for oral cancer research and treatment, with a particular focus on those communities in which healthcare disparities exist. She has also appeared in commercials for Prolia, a brand of denosumab used in the treatment of osteoporosis. Personal life. Danner was married to producer and director Bruce Paltrow, who died of oral cancer in 2002. She and Paltrow had two children together, actress Gwyneth Paltrow and director Jake Paltrow. Danner's niece is the actress Katherine Moennig, the daughter of her maternal half-brother William. Danner co-starred with her daughter in the 1992 television film "Cruel Doubt" and again in the 2003 film "Sylvia", in which she portrayed Aurelia Plath, mother to Gwyneth's title role of Sylvia Plath. Danner is a practitioner of transcendental meditation, which she has described as "very helpful and comforting".
Bioleaching Bioleaching is the extraction or liberation of metals from their ores through the use of living organisms. Bioleaching is one of several applications within biohydrometallurgy and several methods are used to treat ores or concentrates containing copper, zinc, lead, arsenic, antimony, nickel, molybdenum, gold, silver, and cobalt. Bioleaching falls into two broad categories. The first, is the use of microorganisms to oxidize refractory minerals to release valuable metals such and gold and silver. Most commonly the minerals that are the target of oxidization are pyrite and arsenopyrite. The second category is leaching of sulphide minerals to release the associated metal, for example, leaching of pentlandite to release nickel, or the leaching of chalcocite, covellite or chalcopyrite to release copper. Process. Bioleaching can involve numerous ferrous iron and sulfur oxidizing bacteria, including "Acidithiobacillus ferrooxidans" (formerly known as "Thiobacillus ferrooxidans") and "Acidithiobacillus thiooxidans " (formerly known as "Thiobacillus thiooxidans"). As a general principle, in one proposed method of bacterial leaching known as Indirect Leaching, Fe3+ ions are used to oxidize the ore. This step is entirely independent of microbes. The role of the bacteria is further oxidation of the ore, but also the regeneration of the chemical oxidant Fe3+ from Fe2+. For example, bacteria catalyse the breakdown of the mineral pyrite (FeS2) by oxidising the sulfur and metal (in this case ferrous iron, (Fe2+)) using oxygen. This yields soluble products that can be further purified and refined to yield the desired metal.
Pyrite leaching (FeS2): In the first step, disulfide is spontaneously oxidized to thiosulfate by ferric ion (Fe3+), which in turn is reduced to give ferrous ion (Fe2+): The ferrous ion is then oxidized by bacteria using oxygen: Thiosulfate is also oxidized by bacteria to give sulfate: The ferric ion produced in reaction (2) oxidized more sulfide as in reaction (1), closing the cycle and given the net reaction: The net products of the reaction are soluble ferrous sulfate and sulfuric acid. The microbial oxidation process occurs at the cell membrane of the bacteria. The electrons pass into the cells and are used in biochemical processes to produce energy for the bacteria while reducing oxygen to water. The critical reaction is the oxidation of sulfide by ferric iron. The main role of the bacterial step is the regeneration of this reactant. The process for copper is very similar, but the efficiency and kinetics depend on the copper mineralogy. The most efficient minerals are supergene minerals such as chalcocite, Cu2S and covellite, CuS. The main copper mineral chalcopyrite (CuFeS2) is not leached very efficiently, which is why the dominant copper-producing technology remains flotation, followed by smelting and refining. The leaching of CuFeS2 follows the two stages of being dissolved and then further oxidised, with Cu2+ ions being left in solution.
Chalcopyrite leaching: net reaction: In general, sulfides are first oxidized to elemental sulfur, whereas disulfides are oxidized to give thiosulfate, and the processes above can be applied to other sulfidic ores. Bioleaching of non-sulfidic ores such as pitchblende also uses ferric iron as an oxidant (e.g., UO2 + 2 Fe3+ ==> UO22+ + 2 Fe2+). In this case, the sole purpose of the bacterial step is the regeneration of Fe3+. Sulfidic iron ores can be added to speed up the process and provide a source of iron. Bioleaching of non-sulfidic ores by layering of waste sulfides and elemental sulfur, colonized by "Acidithiobacillus" spp., has been accomplished, which provides a strategy for accelerated leaching of materials that do not contain sulfide minerals. Further processing. The dissolved copper (Cu2+) ions are removed from the solution by ligand exchange solvent extraction, which leaves other ions in the solution. The copper is removed by bonding to a ligand, which is a large molecule consisting of a number of smaller groups, each possessing a lone electron pair. The ligand-copper complex is extracted from the solution using an organic solvent such as kerosene:
The ligand donates electrons to the copper, producing a complex - a central metal atom (copper) bonded to the ligand. Because this complex has no charge, it is no longer attracted to polar water molecules and dissolves in the kerosene, which is then easily separated from the solution. Because the initial reaction is reversible, it is determined by pH. Adding concentrated acid reverses the equation, and the copper ions go back into an aqueous solution. Then the copper is passed through an electro-winning process to increase its purity: An electric current is passed through the resulting solution of copper ions. Because copper ions have a 2+ charge, they are attracted to the negative cathodes and collect there. The copper can also be concentrated and separated by displacing the copper with Fe from scrap iron: The electrons lost by the iron are taken up by the copper. Copper is the oxidising agent (it accepts electrons), and iron is the reducing agent (it loses electrons). Traces of precious metals such as gold may be left in the original solution. Treating the mixture with sodium cyanide in the presence of free oxygen dissolves the gold. The gold is removed from the solution by adsorbing (taking it up on the surface) to charcoal.
With fungi. Several species of fungi can be used for bioleaching. Fungi can be grown on many different substrates, such as electronic scrap, catalytic converters, and fly ash from municipal waste incineration. Experiments have shown that two fungal strains ("Aspergillus niger, Penicillium simplicissimum") were able to mobilize Cu and Sn by 65%, and Al, Ni, Pb, and Zn by more than 95%. "Aspergillus niger" can produce some organic acids such as citric acid. This form of leaching does not rely on microbial oxidation of metal but rather uses microbial metabolism as source of acids that directly dissolve the metal. Feasibility. Economic feasibility. Bioleaching is in general simpler and, therefore, cheaper to operate and maintain than traditional processes, since fewer specialists are needed to operate complex chemical plants. And low concentrations are not a problem for bacteria because they simply ignore the waste that surrounds the metals, attaining extraction yields of over 90% in some cases. These microorganisms actually gain energy by breaking down minerals into their constituent elements. The company simply collects the ions out of the solution after the bacteria have finished.
Bioleaching can be used to extract metals from low concentration ores such as gold that are too poor for other technologies. It can be used to partially replace the extensive crushing and grinding that translates to prohibitive cost and energy consumption in a conventional process. Because the lower cost of bacterial leaching outweighs the time it takes to extract the metal. High concentration ores, such as copper, are more economical to smelt rather bioleach due to the slow speed of the bacterial leaching process compared to smelting. The slow speed of bioleaching introduces a significant delay in cash flow for new mines. Nonetheless, at the largest copper mine of the world, Escondida in Chile the process seems to be favorable. Economically it is also very expensive and many companies once started can not keep up with the demand and end up in debt. In space. In 2020 scientists showed, with an experiment with different gravity environments on the ISS, that microorganisms could be employed to mine useful elements from basaltic rocks via bioleaching in space.
Environmental impact. The process is more environmentally friendly than traditional extraction methods. For the company this can translate into profit, since the necessary limiting of sulfur dioxide emissions during smelting is expensive. Less landscape damage occurs, since the bacteria involved grow naturally, and the mine and surrounding area can be left relatively untouched. As the bacteria breed in the conditions of the mine, they are easily cultivated and recycled. Toxic chemicals are sometimes produced in the process. Sulfuric acid and H+ ions that have been formed can leak into the ground and surface water turning it acidic, causing environmental damage. Heavy ions such as iron, zinc, and arsenic leak during acid mine drainage. When the pH of this solution rises, as a result of dilution by fresh water, these ions precipitate, forming "Yellow Boy" pollution. For these reasons, a setup of bioleaching must be carefully planned, since the process can lead to a biosafety failure. Unlike other methods, once started, bioheap leaching cannot be quickly stopped, because leaching would still continue with rainwater and natural bacteria. Projects like Finnish Talvivaara proved to be environmentally and economically disastrous.
Bouldering Bouldering is a form of rock climbing that is performed on small rock formations or artificial rock walls without the use of ropes or harnesses. While bouldering can be done without any equipment, most climbers use climbing shoes to help secure footholds, chalk to keep their hands dry and to provide a firmer grip, and bouldering mats to prevent injuries from falls. Unlike free solo climbing, which is also performed without ropes, bouldering problems (the sequence of moves that a climber performs to complete the climb) are usually less than tall. Traverses, which are a form of boulder problem, require the climber to climb horizontally from one end to another. Artificial climbing walls allow boulderers to climb indoors in areas without natural boulders. In addition, bouldering competitions take place in both indoor and outdoor settings. The sport was originally a method of training for roped climbs and mountaineering, so climbers could practice specific moves at a safe distance from the ground. Additionally, the sport served to build stamina and increase finger strength. Throughout the 20th century, bouldering evolved into a separate discipline. Individual problems are assigned ratings based on difficulty. Although there have been various rating systems used throughout the history of bouldering, modern problems usually use either the V-scale or the Fontainebleau scale.
Outdoor bouldering. The characteristics of boulder problems depend largely on the type of rock being climbed. For example, granite often features long cracks and slabs while sandstone rocks are known for their steep overhangs and frequent horizontal breaks. Limestone and volcanic rock are also used for bouldering. There are many prominent bouldering areas throughout the United States, including Hueco Tanks in Texas, Mount Blue Sky in Colorado, The Appalachian Mountains in The Eastern United States, and The Buttermilks in Bishop, California. Squamish, British Columbia is one of the most popular bouldering areas in Canada. Europe is also home to a number of bouldering sites, such as Fontainebleau in France, Meschia in Italy, Albarracín in Spain, and various mountains throughout Switzerland. Indoor bouldering. Artificial climbing walls are used to simulate boulder problems in an indoor environment, usually at climbing gyms. These walls are constructed with wooden panels, polymer cement panels, concrete shells, or precast molds of actual rock walls. Holds, usually made of plastic, are then bolted onto the wall to create problems. Some problems use steep overhanging surfaces which force the climber to support much of their weight using their upper body strength.
Climbing gyms often feature multiple problems within the same section of wall. Historically, the most common method route-setters used to designate the intended problem was by placing colored tape next to each hold. For example, red tape would indicate one bouldering problem while green tape would be used to set a different problem in the same area. Indoor bouldering requires very little in terms of equipment: at minimum, climbing shoes; at maximum, a chalk bag, chalk, a brush, and climbing shoes. Grading. Bouldering problems are assigned numerical difficulty ratings by route-setters and climbers. The two most widely used rating systems are the V-scale and the Fontainebleau system. The V-scale, which originated in the United States, is an open-ended rating system with higher numbers indicating a higher degree of difficulty. The V1 rating indicates that a problem can be completed by a novice climber in good physical condition after several attempts. The scale begins at V0, and as of 2024, the highest V rating that has been assigned to a bouldering problem is V17. Some climbing gyms also use a VB grade to indicate beginner problems.
The Fontainebleau scale follows a similar system, with each numerical grade divided into three ratings with the letters "a", "b", and "c". For example, Fontainebleau 7A roughly corresponds with V6, while Fontainebleau 7C+ is equivalent to V10. In both systems, grades are further differentiated by appending "+" to indicate a small increase in difficulty. Despite this level of specificity, ratings of individual problems are often controversial, as ability level is not the only factor that affects how difficult a problem may be for a particular climber. Height, arm length, flexibility, and other body characteristics can also affect difficulty. Highball bouldering. Highball bouldering is "a sub-discipline of bouldering in which climbers seek out tall, imposing lines to climb ropeless above crash pads." It may have begun in 1961 when John Gill, without top-rope rehearsal or bouldering pads (which did not exist), bouldered a steep face on a granite spire called "The Thimble". In 2002 Jason Kehl completed the first highball at double-digit V-difficulty, called Evilution, a boulder in the Buttermilks of California, earning the grade of V12.
Important milestone ascents in this style include: Competition bouldering. The International Federation of Sport Climbing (IFSC) employs an indoor format (although competitions can also take place in an outdoor setting) that breaks the competition into three rounds: qualifications, semi-finals, and finals. The rounds feature different sets of four to six boulder problems, and each competitor has a fixed amount of time to attempt each problem. At the end of each round, competitors are ranked by the number of completed problems with ties settled by the total number of attempts taken to solve the problems. Some competitions only permit climbers a fixed number of attempts at each problem with a timed rest period in between. In an open-format competition, all climbers compete simultaneously, and are given a fixed amount of time to complete as many problems as possible. More points are awarded for more difficult problems, while points are deducted for multiple attempts on the same problem. In 2012, the IFSC submitted a proposal to the International Olympic Committee (IOC) to include lead climbing in the 2020 Summer Olympics. The proposal was later revised to an "overall" competition, which would feature bouldering, lead climbing, and speed climbing. In 2016, the International Olympic Committee (IOC) officially approved climbing, along with four other sports, as an Olympic sport, based on their "impact on gender equality, the youth appeal of the sports and the legacy value of adding them to the Tokyo Games".
History. Modern bouldering. Modern recreational climbing began in the late 19th century in England, southeastern Germany, northern Italy, and France. Bouldering on the rocks of Fontainbleau outside of Paris began in the late 1800s, with the first guidebook written by Maurice Martin in 1945. Bouldering as training or a recreational past-time began also in the late 1800s in England and perhaps elsewhere. Oscar Eckenstein was an early proponent. In the late 1950s, John Gill, who is frequently called "the father of modern bouldering", combined gymnastics with rock climbing, and felt that the best place to do that was on boulders or small outcrops. He developed a rating system that was closed-ended: B1 problems were as difficult as the most challenging roped routes of the time, B2 problems were more difficult, and B3 problems had been completed once. He also introduced chalk as a method of keeping the climber's hands dry, promoted a dynamic climbing style, and emphasized the importance of strength training to complement skill. His 1969 article in the Journal of the American Alpine Club entitled "The Art of Bouldering" defines modern bouldering. As Gill improved in ability and influence, his ideas became the norm.
In the 1980s, two important training tools emerged. One important training tool was bouldering mats, also referred to as "crash pads", which protected against injuries from falling and enabled boulderers to climb in areas that would have been too dangerous otherwise. The second important tool was indoor climbing walls, which helped spread the sport to areas without outdoor climbing and allowed serious climbers to train year-round. As the sport grew in popularity, new bouldering areas were developed throughout Europe and the United States, and more athletes began participating in bouldering competitions. The visibility of the sport greatly increased in the early 2000s, as YouTube videos and climbing blogs helped boulderers around the world to quickly learn techniques, find hard problems, and announce newly completed projects. Notable ascents. Notable boulder climbs are chronicled by the climbing media to track progress in boulder climbing standards and levels of technical difficulty; in contrast, the hardest traditional climbing routes tend to be of lower technical difficulty due to the additional burden of having to place protection during the course of the climb, and due to the lack of any possibility of using natural protection on the most extreme climbs.
As of November 2022, the world's hardest bouldering routes are "Burden of Dreams" by Nalle Hukkataival and "Return of the Sleepwalker" by Daniel Woods, both at proposed grades of . There are a number of routes with a confirmed climbing grade of , the first of which was "Gioia" by Christian Core in 2008 (and confirmed by Adam Ondra in 2011). As of December 2021, female climbers Josune Bereziartu, Ashima Shiraishi, and Kaddi Lehmann have repeated boulder problems at the boulder grade. On 28 July 2023, Katie Lamb became the first female climber to climb an -rated boulder by repeating "Box Therapy" at Rocky Mountain National Park. However, after Brooke Raboutou repeated the climb In October 2023, the boulder was ultimately downgraded to . Equipment. Unlike other climbing sports, bouldering can be performed safely and effectively with very little equipment, an aspect which makes the discipline highly appealing, but opinions differ. While bouldering pioneer John Sherman asserted that "The only gear really needed to go bouldering is boulders," others suggest the use of climbing shoes and a chalkbag – a small pouch where ground-up chalk is kept – as the bare minimum, and more experienced boulderers typically bring multiple pairs of climbing shoes, chalk, brushes, crash pads, and a skincare kit.
Climbing shoes have the most direct impact on performance. Besides protecting the climber's feet from rough surfaces, climbing shoes are designed to help the climber secure footholds. Climbing shoes typically fit much tighter than other athletic footwear and often curl the toes downwards to enable precise footwork. They are manufactured in a variety of different styles to perform in different situations. Stiffer shoes excel at securing small edges, whereas softer shoes provide greater sensitivity. The front of the shoe, called the "toe box", can be asymmetric, which performs well on overhanging rocks, or symmetric, which is better suited for vertical problems and slabs. To absorb sweat, most boulderers use gymnastics chalk on their hands, stored in a chalk bag, which can be tied around the waist (also called sport climbing chalk bags), allowing the climber to reapply chalk during the climb. There are also versions of floor chalk bags (also called bouldering chalk bags), which are usually bigger than sport climbing chalk bags and are meant to be kept on the floor while climbing; this is because boulders do not usually have so many movements as to require chalking up more than once. Different sizes of brushes are used to remove excess chalk and debris from boulders in between climbs; they are often attached to the end of a long straight object in order to reach higher holds. Crash pads, also referred to as bouldering mats, are foam cushions placed on the ground to protect climbers from injury after falling.
Boulder problems are generally shorter than from ground to top. This makes the sport significantly safer than free solo climbing, which is also performed without ropes, but with no upper limit on the height of the climb. However, minor injuries are common in bouldering, particularly sprained ankles and wrists. To prevent injuries, boulderers position crash pads near the boulder to provide a softer landing, as well as one or more spotters to help redirect the climber towards the pads. Upon landing, boulderers employ falling techniques similar to those used in gymnastics: spreading the impact across the entire body to avoid bone fractures and positioning limbs to allow joints to move freely throughout the impact. Techniques. Although every type of rock climbing requires a high level of strength and technique, bouldering is the most dynamic form of the sport, requiring the highest level of power and placing considerable strain on the body. Training routines that strengthen fingers and forearms are useful in preventing injuries such as tendonitis and ruptured ligaments. However, as with other forms of climbing, bouldering technique begins with proper footwork. Leg muscles are significantly stronger than arm muscles; thus, proficient boulderers use their arms to maintain balance and body positioning as much as possible, relying on their legs to push them up the rock. Boulderers also keep their arms straight with their shoulders engaged whenever feasible, allowing their bones to support their body weight rather than their muscles.
Bouldering movements are described as either "static" or "dynamic". Static movements are those that are performed slowly, with the climber's position controlled by maintaining contact on the boulder with the other three limbs. Dynamic movements use the climber's momentum to reach holds that would be difficult or impossible to secure statically, with an increased risk of falling if the movement is not performed accurately. Environmental impact. Bouldering can damage vegetation that grows on rocks, such as moss and lichens. This can occur as a result of the climber intentionally cleaning the boulder, or unintentionally from repeated use of handholds and footholds. Vegetation on the ground surrounding the boulder can also be damaged from overuse, particularly by climbers laying down crash pads. Soil erosion can occur when boulderers trample vegetation while hiking off of established trails, or when they unearth small rocks near the boulder in an effort to make the landing zone safer in case of a fall. Other environmental concerns include littering, improperly disposed feces, and graffiti. These issues have caused some land managers to prohibit bouldering, as was the case in Tea Garden, a popular bouldering area in Rocklands, South Africa.
Boiling point The boiling point of a substance is the temperature at which the vapor pressure of a liquid equals the pressure surrounding the liquid and the liquid changes into a vapor. The boiling point of a liquid varies depending upon the surrounding environmental pressure. A liquid in a partial vacuum, i.e., under a lower pressure, has a lower boiling point than when that liquid is at atmospheric pressure. Because of this, water boils at 100°C (or with scientific precision: ) under standard pressure at sea level, but at at altitude. For a given pressure, different liquids will boil at different temperatures. The normal boiling point (also called the atmospheric boiling point or the atmospheric pressure boiling point) of a liquid is the special case in which the vapor pressure of the liquid equals the defined atmospheric pressure at sea level, one atmosphere. At that temperature, the vapor pressure of the liquid becomes sufficient to overcome atmospheric pressure and allow bubbles of vapor to form inside the bulk of the liquid. The standard boiling point has been defined by IUPAC since 1982 as the temperature at which boiling occurs under a pressure of one bar.
The heat of vaporization is the energy required to transform a given quantity (a mol, kg, pound, etc.) of a substance from a liquid into a gas at a given pressure (often atmospheric pressure). Liquids may change to a vapor at temperatures below their boiling points through the process of evaporation. Evaporation is a surface phenomenon in which molecules located near the liquid's edge, not contained by enough liquid pressure on that side, escape into the surroundings as vapor. On the other hand, boiling is a process in which molecules anywhere in the liquid escape, resulting in the formation of vapor bubbles within the liquid. Saturation temperature and pressure. A "saturated liquid" contains as much thermal energy as it can without boiling (or conversely a "saturated vapor" contains as little thermal energy as it can without condensing). Saturation temperature means "boiling point". The saturation temperature is the temperature for a corresponding saturation pressure at which a liquid boils into its vapor phase. The liquid can be said to be saturated with thermal energy. Any addition of thermal energy results in a phase transition.
If the pressure in a system remains constant (isobaric), a vapor at saturation temperature will begin to condense into its liquid phase as thermal energy (heat) is removed. Similarly, a liquid at saturation temperature and pressure will boil into its vapor phase as additional thermal energy is applied. The boiling point corresponds to the temperature at which the vapor pressure of the liquid equals the surrounding environmental pressure. Thus, the boiling point is dependent on the pressure. Boiling points may be published with respect to the NIST, USA standard pressure of 101.325 kPa (1 atm), or the IUPAC standard pressure of 100.000 kPa (1 bar). At higher elevations, where the atmospheric pressure is much lower, the boiling point is also lower. The boiling point increases with increased pressure up to the critical point, where the gas and liquid properties become identical. The boiling point cannot be increased beyond the critical point. Likewise, the boiling point decreases with decreasing pressure until the triple point is reached. The boiling point cannot be reduced below the triple point.
If the heat of vaporization and the vapor pressure of a liquid at a certain temperature are known, the boiling point can be calculated by using the Clausius–Clapeyron equation, thus: where: Saturation pressure is the pressure for a corresponding saturation temperature at which a liquid boils into its vapor phase. Saturation pressure and saturation temperature have a direct relationship: as saturation pressure is increased, so is saturation temperature. If the temperature in a system remains constant (an "isothermal" system), vapor at saturation pressure and temperature will begin to condense into its liquid phase as the system pressure is increased. Similarly, a liquid at saturation pressure and temperature will tend to flash into its vapor phase as system pressure is decreased. There are two conventions regarding the "standard boiling point of water": The "normal boiling point" is commonly given as (actually following the thermodynamic definition of the Celsius scale based on the kelvin) at a pressure of 1 atm (101.325 kPa). The IUPAC-recommended "standard boiling point of water" at a standard pressure of 100 kPa (1 bar) is . For comparison, on top of Mount Everest, at elevation, the pressure is about and the boiling point of water is . The Celsius temperature scale was defined until 1954 by two points: 0 °C being defined by the water freezing point and 100 °C being defined by the water boiling point at standard atmospheric pressure.
Relation between the normal boiling point and the vapor pressure of liquids. The higher the vapor pressure of a liquid at a given temperature, the lower the normal boiling point (i.e., the boiling point at atmospheric pressure) of the liquid. The vapor pressure chart to the right has graphs of the vapor pressures versus temperatures for a variety of liquids. As can be seen in the chart, the liquids with the highest vapor pressures have the lowest normal boiling points. For example, at any given temperature, methyl chloride has the highest vapor pressure of any of the liquids in the chart. It also has the lowest normal boiling point (−24.2 °C), which is where the vapor pressure curve of methyl chloride (the blue line) intersects the horizontal pressure line of one atmosphere (atm) of absolute vapor pressure. The critical point of a liquid is the highest temperature (and pressure) it will actually boil at. See also Vapour pressure of water. Boiling point of chemical elements. The element with the lowest boiling point is helium. Both the boiling points of rhenium and tungsten exceed 5000 K at standard pressure; because it is difficult to measure extreme temperatures precisely without bias, both have been cited in the literature as having the higher boiling point.
Boiling point as a reference property of a pure compound. As can be seen from the above plot of the logarithm of the vapor pressure vs. the temperature for any given pure chemical compound, its normal boiling point can serve as an indication of that compound's overall volatility. A given pure compound has only one normal boiling point, if any, and a compound's normal boiling point and melting point can serve as characteristic physical properties for that compound, listed in reference books. The higher a compound's normal boiling point, the less volatile that compound is overall, and conversely, the lower a compound's normal boiling point, the more volatile that compound is overall. Some compounds decompose at higher temperatures before reaching their normal boiling point, or sometimes even their melting point. For a stable compound, the boiling point ranges from its triple point to its critical point, depending on the external pressure. Beyond its triple point, a compound's normal boiling point, if any, is higher than its melting point. Beyond the critical point, a compound's liquid and vapor phases merge into one phase, which may be called a superheated gas. At any given temperature, if a compound's normal boiling point is lower, then that compound will generally exist as a gas at atmospheric external pressure. If the compound's normal boiling point is higher, then that compound can exist as a liquid or solid at that given temperature at atmospheric external pressure, and will so exist in equilibrium with its vapor (if volatile) if its vapors are contained. If a compound's vapors are not contained, then some volatile compounds can eventually evaporate away in spite of their higher boiling points.
In general, compounds with ionic bonds have high normal boiling points, if they do not decompose before reaching such high temperatures. Many metals have high boiling points, but not all. Very generally—with other factors being equal—in compounds with covalently bonded molecules, as the size of the molecule (or molecular mass) increases, the normal boiling point increases. When the molecular size becomes that of a macromolecule, polymer, or otherwise very large, the compound often decomposes at high temperature before the boiling point is reached. Another factor that affects the normal boiling point of a compound is the polarity of its molecules. As the polarity of a compound's molecules increases, its normal boiling point increases, other factors being equal. Closely related is the ability of a molecule to form hydrogen bonds (in the liquid state), which makes it harder for molecules to leave the liquid state and thus increases the normal boiling point of the compound. Simple carboxylic acids dimerize by forming hydrogen bonds between molecules. A minor factor affecting boiling points is the shape of a molecule. Making the shape of a molecule more compact tends to lower the normal boiling point slightly compared to an equivalent molecule with more surface area.
Most volatile compounds (anywhere near ambient temperatures) go through an intermediate liquid phase while warming up from a solid phase to eventually transform to a vapor phase. By comparison to boiling, a sublimation is a physical transformation in which a solid turns directly into vapor, which happens in a few select cases such as with carbon dioxide at atmospheric pressure. For such compounds, a sublimation point is a temperature at which a solid turning directly into vapor has a vapor pressure equal to the external pressure. Impurities and mixtures. In the preceding section, boiling points of pure compounds were covered. Vapor pressures and boiling points of substances can be affected by the presence of dissolved impurities (solutes) or other miscible compounds, the degree of effect depending on the concentration of the impurities or other compounds. The presence of non-volatile impurities such as salts or compounds of a volatility far lower than the main component compound decreases its mole fraction and the solution's volatility, and thus raises the normal boiling point in proportion to the concentration of the solutes. This effect is called boiling point elevation. As a common example, salt water boils at a higher temperature than pure water.
In other mixtures of miscible compounds (components), there may be two or more components of varying volatility, each having its own pure component boiling point at any given pressure. The presence of other volatile components in a mixture affects the vapor pressures and thus boiling points and dew points of all the components in the mixture. The dew point is a temperature at which a vapor condenses into a liquid. Furthermore, at any given temperature, the composition of the vapor is different from the composition of the liquid in most such cases. In order to illustrate these effects between the volatile components in a mixture, a boiling point diagram is commonly used. Distillation is a process of boiling and [usually] condensation which takes advantage of these differences in composition between liquid and vapor phases. Boiling point of water with elevation. Following is a table of the change in the boiling point of water with elevation, at intervals of 500 meters over the range of human habitation [the Dead Sea at to La Rinconada, Peru at ], then of 1,000 meters over the additional range of uninhabited surface elevation [up to Mount Everest at ], along with a similar range in Imperial.
Big Bang The Big Bang is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models based on the Big Bang concept explain a broad range of phenomena, including the abundance of light elements, the cosmic microwave background (CMB) radiation, and large-scale structure. The uniformity of the universe, known as the horizon and flatness problems, is explained through cosmic inflation: a phase of accelerated expansion during the earliest stages. A wide range of empirical evidence strongly favors the Big Bang event, which is now essentially universally accepted. Detailed measurements of the expansion rate of the universe place the Big Bang singularity at an estimated billion years ago, which is considered the age of the universe. Extrapolating this cosmic expansion backward in time using the known laws of physics, the models describe an extraordinarily hot and dense primordial universe. Physics lacks a widely accepted theory that can model the earliest conditions of the Big Bang. As the universe expanded, it cooled sufficiently to allow the formation of subatomic particles, and later atoms. These primordial elements—mostly hydrogen, with some helium and lithium—then coalesced under the force of gravity aided by dark matter, forming early stars and galaxies. Measurements of the redshifts of supernovae indicate that the expansion of the universe is accelerating, an observation attributed to a concept called dark energy.
The concept of an expanding universe was scientifically originated by the physicist Alexander Friedmann in 1922 with the mathematical derivation of the Friedmann equations. The earliest empirical observation of an expanding universe is known as Hubble's law, published in work by physicist Edwin Hubble in 1929, which discerned that galaxies are moving away from Earth at a rate that accelerates proportionally with distance. Independent of Friedmann's work, and independent of Hubble's observations, physicist Georges Lemaître proposed that the universe emerged from a "primeval atom" in 1931, introducing the modern notion of the Big Bang. In 1964, the CMB was discovered, which convinced many cosmologists that the competing steady-state model of cosmic evolution was falsified, since the Big Bang models predict a uniform background radiation caused by high temperatures and densities in the distant past. There remain aspects of the observed universe that are not yet adequately explained by the Big Bang models. These include the unequal abundances of matter and antimatter known as baryon asymmetry, the detailed nature of dark matter surrounding galaxies, and the origin of dark energy.
Features of the models. Assumptions. Big Bang cosmology models depend on three major assumptions: the universality of physical laws, the cosmological principle, and that the matter content can be modeled as a perfect fluid. The universality of physical laws is one of the underlying principles of the theory of relativity. The cosmological principle states that on large scales the universe is homogeneous and isotropic—appearing the same in all directions regardless of location. A perfect fluid has no viscosity; the pressure of a perfect fluid is proportional to its density. These ideas were initially taken as postulates, but later efforts were made to test each of them. For example, the first assumption has been tested by observations showing that the largest possible deviation of the fine-structure constant over much of the age of the universe is of order 10−5. The key physical law behind these models, general relativity has passed stringent tests on the scale of the Solar System and binary stars. The cosmological principle has been confirmed to a level of 10−5 via observations of the temperature of the CMB. At the scale of the CMB horizon, the universe has been measured to be homogeneous with an upper bound on the order of 10% inhomogeneity, as of 1995.
Expansion prediction. The cosmological principle dramatically simplifies the equations of general relativity, giving the Friedmann–Lemaître–Robertson–Walker metric to describe the geometry of the universe and, with the assumption of a perfect fluid, the Friedmann equations giving the time dependence of that geometry. The only parameter at this level of description is the mass-energy density: the geometry of the universe and its expansion is a direct consequence of its density. All of the major features of Big Bang cosmology are related to these results. Mass-energy density. In Big Bang cosmology, the mass–energy density controls the shape and evolution of the universe. By combining astronomical observations with known laws of thermodynamics and particle physics, cosmologists have worked out the components of the density over the lifespan of the universe. In the current universe, luminous matter, the stars, planets, and so on makes up less than 5% of the density. Dark matter accounts for 27% and dark energy the remaining 68%.
Horizons. An important feature of the Big Bang spacetime is the presence of particle horizons. Since the universe has a finite age, and light travels at a finite speed, there may be events in the past whose light has not yet had time to reach earth. This places a limit or a "past horizon" on the most distant objects that can be observed. Conversely, because space is expanding, and more distant objects are receding ever more quickly, light emitted by us today may never "catch up" to very distant objects. This defines a "future horizon", which limits the events in the future that we will be able to influence. The presence of either type of horizon depends on the details of the Friedmann–Lemaître–Robertson–Walker (FLRW) metric that describes the expansion of the universe. Our understanding of the universe back to very early times suggests that there is a past horizon, though in practice our view is also limited by the opacity of the universe at early times. So our view cannot extend further backward in time, though the horizon recedes in space. If the expansion of the universe continues to accelerate, there is a future horizon as well.
Thermalization. Some processes in the early universe occurred too slowly, compared to the expansion rate of the universe, to reach approximate thermodynamic equilibrium. Others were fast enough to reach thermalization. The parameter usually used to find out whether a process in the very early universe has reached thermal equilibrium is the ratio between the rate of the process (usually rate of collisions between particles) and the Hubble parameter. The larger the ratio, the more time particles had to thermalize before they were too far away from each other. Timeline. According to the Big Bang models, the universe at the beginning was very hot and very compact, and since then it has been expanding and cooling. Singularity. Existing theories of physics cannot tell us about the moment of the Big Bang. Extrapolation of the expansion of the universe backwards in time using only general relativity yields a gravitational singularity with infinite density and temperature at a finite time in the past, but the meaning of this extrapolation in the context of the Big Bang is unclear. Moreover, classical gravitational theories are expected to be inadequate to describe physics under these conditions. Quantum gravity effects are expected to be dominant during the Planck epoch, when the temperature of the universe was close to the Planck scale (around 1032 K or 1028 eV).
Even below the Planck scale, undiscovered physics could greatly influence the expansion history of the universe. The Standard Model of particle physics is only tested up to temperatures of order 1017K (10 TeV) in particle colliders, such as the Large Hadron Collider. Moreover, new physical phenomena decoupled from the Standard Model could have been important before the time of neutrino decoupling, when the temperature of the universe was only about 1010K (1 MeV). Inflation and baryogenesis. The earliest phases of the Big Bang are subject to much speculation, given the lack of available data. In the most common models the universe was filled homogeneously and isotropically with a very high energy density and huge temperatures and pressures, and was very rapidly expanding and cooling. The period up to 10−43 seconds into the expansion, the Planck epoch, was a phase in which the four fundamental forces—the electromagnetic force, the strong nuclear force, the weak nuclear force, and the gravitational force, were unified as one. In this stage, the characteristic scale length of the universe was the Planck length, , and consequently had a temperature of approximately 1032 degrees Celsius. Even the very concept of a particle breaks down in these conditions. A proper understanding of this period awaits the development of a theory of quantum gravity. The Planck epoch was succeeded by the grand unification epoch beginning at 10−43 seconds, where gravitation separated from the other forces as the universe's temperature fell.
At approximately 10−37 seconds into the expansion, a phase transition caused a cosmic inflation, during which the universe grew exponentially, unconstrained by the light speed invariance, and temperatures dropped by a factor of 100,000. This concept is motivated by the flatness problem, where the density of matter and energy is very close to the critical density needed to produce a flat universe. That is, the shape of the universe has no overall geometric curvature due to gravitational influence. Microscopic quantum fluctuations that occurred because of Heisenberg's uncertainty principle were "frozen in" by inflation, becoming amplified into the seeds that would later form the large-scale structure of the universe. At a time around 10−36 seconds, the electroweak epoch begins when the strong nuclear force separates from the other forces, with only the electromagnetic force and weak nuclear force remaining unified. Inflation stopped locally at around 10−33 to 10−32 seconds, with the observable universe's volume having increased by a factor of at least 1078. Reheating followed as the inflaton field decayed, until the universe obtained the temperatures required for the production of a quark–gluon plasma as well as all other elementary particles. Temperatures were so high that the random motions of particles were at relativistic speeds, and particle–antiparticle pairs of all kinds were being continuously created and destroyed in collisions. At some point, an unknown reaction called baryogenesis violated the conservation of baryon number, leading to a very small excess of quarks and leptons over antiquarks and antileptons—of the order of one part in 30 million. This resulted in the predominance of matter over antimatter in the present universe.
Cooling. The universe continued to decrease in density and fall in temperature, hence the typical energy of each particle was decreasing. Symmetry-breaking phase transitions put the fundamental forces of physics and the parameters of elementary particles into their present form, with the electromagnetic force and weak nuclear force separating at about 10−12 seconds. After about 10−11 seconds, the picture becomes less speculative, since particle energies drop to values that can be attained in particle accelerators. At about 10−6 seconds, quarks and gluons combined to form baryons such as protons and neutrons. The small excess of quarks over antiquarks led to a small excess of baryons over antibaryons. The temperature was no longer high enough to create either new proton–antiproton or neutron–antineutron pairs. A mass annihilation immediately followed, leaving just one in 108 of the original matter particles and none of their antiparticles. A similar process happened at about 1 second for electrons and positrons. After these annihilations, the remaining protons, neutrons and electrons were no longer moving relativistically and the energy density of the universe was dominated by photons (with a minor contribution from neutrinos).
A few minutes into the expansion, when the temperature was about a billion kelvin and the density of matter in the universe was comparable to the current density of Earth's atmosphere, neutrons combined with protons to form the universe's deuterium and helium nuclei in a process called Big Bang nucleosynthesis (BBN). Most protons remained uncombined as hydrogen nuclei. As the universe cooled, the rest energy density of matter came to gravitationally dominate over that of the photon and neutrino radiation at a time of about 50,000 years. At a time of about 380,000 years, the universe cooled enough that electrons and nuclei combined into neutral atoms (mostly hydrogen) in an event called recombination. This process made the previously opaque universe transparent, and the photons that last scattered during this epoch comprise the cosmic microwave background. Structure formation. After the recombination epoch, the slightly denser regions of the uniformly distributed matter gravitationally attracted nearby matter and thus grew even denser, forming gas clouds, stars, galaxies, and the other astronomical structures observable today. The details of this process depend on the amount and type of matter in the universe. The four possible types of matter are known as cold dark matter (CDM), warm dark matter, hot dark matter, and baryonic matter. The best measurements available, from the Wilkinson Microwave Anisotropy Probe (WMAP), show that the data is well-fit by a Lambda-CDM model in which dark matter is assumed to be cold. This CDM is estimated to make up about 23% of the matter/energy of the universe, while baryonic matter makes up about 4.6%.
Cosmic acceleration. Independent lines of evidence from Type Ia supernovae and the CMB imply that the universe today is dominated by a mysterious form of energy known as dark energy, which appears to homogeneously permeate all of space. Observations suggest that 73% of the total energy density of the present day universe is in this form. When the universe was very young it was likely infused with dark energy, but with everything closer together, gravity predominated, braking the expansion. Eventually, after billions of years of expansion, the declining density of matter relative to the density of dark energy allowed the expansion of the universe to begin to accelerate. Dark energy in its simplest formulation is modeled by a cosmological constant term in Einstein field equations of general relativity, but its composition and mechanism are unknown. More generally, the details of its equation of state and relationship with the Standard Model of particle physics continue to be investigated both through observation and theory.
All of this cosmic evolution after the inflationary epoch can be rigorously described and modeled by the lambda-CDM model of cosmology, which uses the independent frameworks of quantum mechanics and general relativity. There are no easily testable models that would describe the situation prior to approximately 10−15 seconds. Understanding this earliest of eras in the history of the universe is one of the greatest unsolved problems in physics. Concept history. Etymology. English astronomer Fred Hoyle is credited with coining the term "Big Bang" during a talk for a March 1949 BBC Radio broadcast, saying: "These theories were based on the hypothesis that all the matter in the universe was created in one big bang at a particular time in the remote past." However, it did not catch on until the 1970s. It is popularly reported that Hoyle, who favored an alternative "steady-state" cosmological model, intended this to be pejorative, but Hoyle explicitly denied this and said it was just a striking image meant to highlight the difference between the two models. Helge Kragh writes that the evidence for the claim that it was meant as a pejorative is "unconvincing", and mentions a number of indications that it was not a pejorative.
A primordial singularity is sometimes called "the Big Bang", but the term can also refer to a more generic early hot, dense phase. The term itself has been argued to be a misnomer because it evokes an explosion. The argument is that whereas an explosion suggests expansion into a surrounding space, the Big Bang only describes the intrinsic expansion of the contents of the universe. Another issue pointed out by Santhosh Mathew is that bang implies sound, which is not an important feature of the model. However, an attempt to find a more suitable alternative was not successful. According to Timothy Ferris: The term 'big bang' was coined with derisive intent by Fred Hoyle, and its endurance testifies to Sir Fred's creativity and wit. Indeed, the term survived an international competition in which three judges — the television science reporter Hugh Downs, the astronomer Carl Sagan, and myself — sifted through 13,099 entries from 41 countries and concluded that none was apt enough to replace it. No winner was declared, and like it or not, we are stuck with 'big bang'.
Before the name. Early cosmological models developed from observations of the structure of the universe and from theoretical considerations. In 1912, Vesto Slipher measured the first Doppler shift of a "spiral nebula" (spiral nebula is the obsolete term for spiral galaxies), and soon discovered that almost all such nebulae were receding from Earth. He did not grasp the cosmological implications of this fact, and indeed at the time it was highly controversial whether or not these nebulae were "island universes" outside our Milky Way. Ten years later, Alexander Friedmann, a Russian cosmologist and mathematician, derived the Friedmann equations from the Einstein field equations, showing that the universe might be expanding in contrast to the static universe model advocated by Albert Einstein at that time. In 1924, American astronomer Edwin Hubble's measurement of the great distance to the nearest spiral nebulae showed that these systems were indeed other galaxies. Starting that same year, Hubble painstakingly developed a series of distance indicators, the forerunner of the cosmic distance ladder, using the Hooker telescope at Mount Wilson Observatory. This allowed him to estimate distances to galaxies whose redshifts had already been measured, mostly by Slipher. In 1929, Hubble discovered a correlation between distance and recessional velocity—now known as Hubble's law.
Independently deriving Friedmann's equations in 1927, Georges Lemaître, a Belgian physicist and Roman Catholic priest, proposed that the recession of the nebulae was due to the expansion of the universe. He inferred the relation that Hubble would later observe, given the cosmological principle. In 1931, Lemaître went further and suggested that the evident expansion of the universe, if projected back in time, meant that the further in the past the smaller the universe was, until at some finite time in the past all the mass of the universe was concentrated into a single point, a "primeval atom" where and when the fabric of time and space came into existence. In the 1920s and 1930s, almost every major cosmologist preferred an eternal steady-state universe, and several complained that the beginning of time implied by an expanding universe imported religious concepts into physics; this objection was later repeated by supporters of the steady-state theory. This perception was enhanced by the fact that the originator of the expanding universe concept, Lemaître, was a Roman Catholic priest. Arthur Eddington agreed with Aristotle that the universe did not have a beginning in time, "viz"., that matter is eternal. A beginning in time was "repugnant" to him. Lemaître, however, disagreed:
During the 1930s, other ideas were proposed as non-standard cosmologies to explain Hubble's observations, including the Milne model, the oscillatory universe (originally suggested by Friedmann, but advocated by Albert Einstein and Richard C. Tolman) and Fritz Zwicky's tired light hypothesis. After World War II, two distinct possibilities emerged. One was Fred Hoyle's steady-state model, whereby new matter would be created as the universe seemed to expand. In this model the universe is roughly the same at any point in time. The other was Lemaître's expanding universe theory, advocated and developed by George Gamow, who used it to develop a theory for the abundance of chemical elements in the universe. and whose associates, Ralph Alpher and Robert Herman, predicted the cosmic background radiation. As a named model. Ironically, it was Hoyle who coined the phrase that came to be applied to Lemaître's theory, referring to it as "this "big bang" idea" during a BBC Radio broadcast in March 1949. For a while, support was split between these two theories. Eventually, the observational evidence, most notably from radio source counts, began to favor Big Bang over steady state. The discovery and confirmation of the CMB in 1964 secured the Big Bang as the best theory of the origin and evolution of the universe.
In 1968 and 1970, Roger Penrose, Stephen Hawking, and George F. R. Ellis published papers where they showed that mathematical singularities were an inevitable initial condition of relativistic models of the Big Bang. Then, from the 1970s to the 1990s, cosmologists worked on characterizing the features of the Big Bang universe and resolving outstanding problems. In 1981, Alan Guth made a breakthrough in theoretical work on resolving certain outstanding theoretical problems in the Big Bang models with the introduction of an epoch of rapid expansion in the early universe he called "inflation". Meanwhile, during these decades, two questions in observational cosmology that generated much discussion and disagreement were over the precise values of the Hubble Constant and the matter-density of the universe (before the discovery of dark energy, thought to be the key predictor for the eventual fate of the universe). Significant progress in Big Bang cosmology has been made since the late 1990s as a result of advances in telescope technology as well as the analysis of data from satellites such as the Cosmic Background Explorer (COBE), the Hubble Space Telescope and WMAP. Cosmologists now have fairly precise and accurate measurements of many of the parameters of the Big Bang model, and have made the unexpected discovery that the expansion of the universe appears to be accelerating.
Observational evidence. The Big Bang models offer a comprehensive explanation for a broad range of observed phenomena, including the abundances of the light elements, the cosmic microwave background, large-scale structure, and Hubble's law. The earliest and most direct observational evidence of the validity of the theory are the expansion of the universe according to Hubble's law (as indicated by the redshifts of galaxies), discovery and measurement of the cosmic microwave background and the relative abundances of light elements produced by Big Bang nucleosynthesis (BBN). More recent evidence includes observations of galaxy formation and evolution, and the distribution of large-scale cosmic structures. These are sometimes called the "four pillars" of the Big Bang models. Precise modern models of the Big Bang appeal to various exotic physical phenomena that have not been observed in terrestrial laboratory experiments or incorporated into the Standard Model of particle physics. Of these features, dark matter is currently the subject of most active laboratory investigations. Remaining issues include the cuspy halo problem and the dwarf galaxy problem of cold dark matter. Dark energy is also an area of intense interest for scientists, but it is not clear whether direct detection of dark energy will be possible. Inflation and baryogenesis remain more speculative features of current Big Bang models. Viable, quantitative explanations for such phenomena are still being sought. These are unsolved problems in physics.
Hubble's law and the expansion of the universe. Observations of distant galaxies and quasars show that these objects are redshifted: the light emitted from them has been shifted to longer wavelengths. This can be seen by taking a frequency spectrum of an object and matching the spectroscopic pattern of emission or absorption lines corresponding to atoms of the chemical elements interacting with the light. These redshifts are uniformly isotropic, distributed evenly among the observed objects in all directions. If the redshift is interpreted as a Doppler shift, the recessional velocity of the object can be calculated. For some galaxies, it is possible to estimate distances via the cosmic distance ladder. When the recessional velocities are plotted against these distances, a linear relationship known as Hubble's law is observed: formula_1 where Hubble's law implies that the universe is uniformly expanding everywhere. This cosmic expansion was predicted from general relativity by Friedmann in 1922 and Lemaître in 1927, well before Hubble made his 1929 analysis and observations, and it remains the cornerstone of the Big Bang model as developed by Friedmann, Lemaître, Robertson, and Walker.
The theory requires the relation formula_5 to hold at all times, where formula_3 is the proper distance, formula_2 is the recessional velocity, and formula_2, formula_9, and formula_3 vary as the universe expands (hence we write formula_4 to denote the present-day Hubble "constant"). For distances much smaller than the size of the observable universe, the Hubble redshift can be thought of as the Doppler shift corresponding to the recession velocity formula_2. For distances comparable to the size of the observable universe, the attribution of the cosmological redshift becomes more ambiguous, although its interpretation as a kinematic Doppler shift remains the most natural one. An unexplained discrepancy with the determination of the Hubble constant is known as Hubble tension. Techniques based on observation of the CMB suggest a lower value of this constant compared to the quantity derived from measurements based on the cosmic distance ladder. Cosmic microwave background radiation. In 1964, Arno Penzias and Robert Wilson serendipitously discovered the cosmic background radiation, an omnidirectional signal in the microwave band. Their discovery provided substantial confirmation of the big-bang predictions by Alpher, Herman and Gamow around 1950. Through the 1970s, the radiation was found to be approximately consistent with a blackbody spectrum in all directions; this spectrum has been redshifted by the expansion of the universe, and today corresponds to approximately 2.725 K. This tipped the balance of evidence in favor of the Big Bang model, and Penzias and Wilson were awarded the 1978 Nobel Prize in Physics.
The "surface of last scattering" corresponding to emission of the CMB occurs shortly after "recombination", the epoch when neutral hydrogen becomes stable. Prior to this, the universe comprised a hot dense photon-baryon plasma sea where photons were quickly scattered from free charged particles. Peaking at around , the mean free path for a photon becomes long enough to reach the present day and the universe becomes transparent. In 1989, NASA launched COBE, which made two major advances: in 1990, high-precision spectrum measurements showed that the CMB frequency spectrum is an almost perfect blackbody with no deviations at a level of 1 part in 104, and measured a residual temperature of 2.726 K (more recent measurements have revised this figure down slightly to 2.7255 K); then in 1992, further COBE measurements discovered tiny fluctuations (anisotropies) in the CMB temperature across the sky, at a level of about one part in 105. John C. Mather and George Smoot were awarded the 2006 Nobel Prize in Physics for their leadership in these results.
During the following decade, CMB anisotropies were further investigated by a large number of ground-based and balloon experiments. In 2000–2001, several experiments, most notably BOOMERanG, found the shape of the universe to be spatially almost flat by measuring the typical angular size (the size on the sky) of the anisotropies. In early 2003, the first results of the Wilkinson Microwave Anisotropy Probe were released, yielding what were at the time the most accurate values for some of the cosmological parameters. The results disproved several specific cosmic inflation models, but are consistent with the inflation theory in general. The "Planck" space probe was launched in May 2009. Other ground and balloon-based cosmic microwave background experiments are ongoing. Abundance of primordial elements. Using Big Bang models, it is possible to calculate the expected concentration of the isotopes helium-4 (4He), helium-3 (3He), deuterium (2H), and lithium-7 (7Li) in the universe as ratios to the amount of ordinary hydrogen. The relative abundances depend on a single parameter, the ratio of photons to baryons. This value can be calculated independently from the detailed structure of CMB fluctuations. The ratios predicted (by mass, not by abundance) are about 0.25 for 4He:H, about 10−3 for 2H:H, about 10−4 for 3He:H, and about 10−9 for 7Li:H.
The measured abundances all agree at least roughly with those predicted from a single value of the baryon-to-photon ratio. The agreement is excellent for deuterium, close but formally discrepant for 4He, and off by a factor of two for 7Li (this anomaly is known as the cosmological lithium problem); in the latter two cases, there are substantial systematic uncertainties. Nonetheless, the general consistency with abundances predicted by BBN is strong evidence for the Big Bang, as the theory is the only known explanation for the relative abundances of light elements, and it is virtually impossible to "tune" the Big Bang to produce much more or less than 20–30% helium. Indeed, there is no obvious reason outside of the Big Bang that, for example, the young universe before star formation, as determined by studying matter supposedly free of stellar nucleosynthesis products, should have more helium than deuterium or more deuterium than 3He, and in constant ratios, too. Galactic evolution and distribution. Detailed observations of the morphology and distribution of galaxies and quasars are in agreement with the current Big Bang models. A combination of observations and theory suggest that the first quasars and galaxies formed within a billion years after the Big Bang, and since then, larger structures have been forming, such as galaxy clusters and superclusters.
Populations of stars have been aging and evolving, so that distant galaxies (which are observed as they were in the early universe) appear very different from nearby galaxies (observed in a more recent state). Moreover, galaxies that formed relatively recently appear markedly different from galaxies formed at similar distances but shortly after the Big Bang. These observations are strong arguments against the steady-state model. Observations of star formation, galaxy and quasar distributions and larger structures, agree well with Big Bang simulations of the formation of structure in the universe, and are helping to complete details of the theory. Primordial gas clouds. In 2011, astronomers found what they believe to be pristine clouds of primordial gas by analyzing absorption lines in the spectra of distant quasars. Before this discovery, all other astronomical objects have been observed to contain heavy elements that are formed in stars. Despite being sensitive to carbon, oxygen, and silicon, these three elements were not detected in these two clouds. Since the clouds of gas have no detectable levels of heavy elements, they likely formed in the first few minutes after the Big Bang, during BBN.
Other lines of evidence. The age of the universe as estimated from the Hubble expansion and the CMB is now in agreement with other estimates using the ages of the oldest stars, both as measured by applying the theory of stellar evolution to globular clusters and through radiometric dating of individual Population II stars. It is also in agreement with age estimates based on measurements of the expansion using Type Ia supernovae and measurements of temperature fluctuations in the cosmic microwave background. The agreement of independent measurements of this age supports the Lambda-CDM (ΛCDM) model, since the model is used to relate some of the measurements to an age estimate, and all estimates turn agree. Still, some observations of objects from the relatively early universe (in particular quasar APM 08279+5255) raise concern as to whether these objects had enough time to form so early in the ΛCDM model. The prediction that the CMB temperature was higher in the past has been experimentally supported by observations of very low temperature absorption lines in gas clouds at high redshift. This prediction also implies that the amplitude of the Sunyaev–Zel'dovich effect in clusters of galaxies does not depend directly on redshift. Observations have found this to be roughly true, but this effect depends on cluster properties that do change with cosmic time, making precise measurements difficult.
Future observations. Future gravitational-wave observatories might be able to detect primordial gravitational waves, relics of the early universe, up to less than a second after the Big Bang. Problems and related issues in physics. As with any theory, a number of mysteries and problems have arisen as a result of the development of the Big Bang models. Some of these mysteries and problems have been resolved while others are still outstanding. Proposed solutions to some of the problems in the Big Bang model have revealed new mysteries of their own. For example, the horizon problem, the magnetic monopole problem, and the flatness problem are most commonly resolved with inflation theory, but the details of the inflationary universe are still left unresolved and many, including some founders of the theory, say it has been disproven. What follows are a list of the mysterious aspects of the Big Bang concept still under intense investigation by cosmologists and astrophysicists. Baryon asymmetry. It is not yet understood why the universe has more matter than antimatter. It is generally assumed that when the universe was young and very hot it was in statistical equilibrium and contained equal numbers of baryons and antibaryons. However, observations suggest that the universe, including its most distant parts, is made almost entirely of normal matter, rather than antimatter. A process called baryogenesis was hypothesized to account for the asymmetry. For baryogenesis to occur, the Sakharov conditions must be satisfied. These require that baryon number is not conserved, that C-symmetry and CP-symmetry are violated and that the universe depart from thermodynamic equilibrium. All these conditions occur in the Standard Model, but the effects are not strong enough to explain the present baryon asymmetry.
Dark energy. Measurements of the redshift–magnitude relation for type Ia supernovae indicate that the expansion of the universe has been accelerating since the universe was about half its present age. To explain this acceleration, cosmological models require that much of the energy in the universe consists of a component with large negative pressure, dubbed "dark energy". Dark energy, though speculative, solves numerous problems. Measurements of the cosmic microwave background indicate that the universe is very nearly spatially flat, and therefore according to general relativity the universe must have almost exactly the critical density of mass/energy. But the mass density of the universe can be measured from its gravitational clustering, and is found to have only about 30% of the critical density. Since theory suggests that dark energy does not cluster in the usual way it is the best explanation for the "missing" energy density. Dark energy also helps to explain two geometrical measures of the overall curvature of the universe, one using the frequency of gravitational lenses, and the other using the characteristic pattern of the large-scale structure--baryon acoustic oscillations--as a cosmic ruler.
Negative pressure is believed to be a property of vacuum energy, but the exact nature and existence of dark energy remains one of the great mysteries of the Big Bang. Results from the WMAP team in 2008 are in accordance with a universe that consists of 73% dark energy, 23% dark matter, 4.6% regular matter and less than 1% neutrinos. According to theory, the energy density in matter decreases with the expansion of the universe, but the dark energy density remains constant (or nearly so) as the universe expands. Therefore, matter made up a larger fraction of the total energy of the universe in the past than it does today, but its fractional contribution will fall in the far future as dark energy becomes even more dominant. The dark energy component of the universe has been explained by theorists using a variety of competing theories including Einstein's cosmological constant but also extending to more exotic forms of quintessence or other modified gravity schemes. A cosmological constant problem, sometimes called the "most embarrassing problem in physics", results from the apparent discrepancy between the measured energy density of dark energy, and the one naively predicted from Planck units.
Dark matter. During the 1970s and the 1980s, various observations showed that there is not sufficient visible matter in the universe to account for the apparent strength of gravitational forces within and between galaxies. This led to the idea that up to 90% of the matter in the universe is dark matter that does not emit light or interact with normal baryonic matter. In addition, the assumption that the universe is mostly normal matter led to predictions that were strongly inconsistent with observations. In particular, the universe today is far more lumpy and contains far less deuterium than can be accounted for without dark matter. While dark matter has always been controversial, it is inferred by various observations: the anisotropies in the CMB, the galaxy rotation problem, galaxy cluster velocity dispersions, large-scale structure distributions, gravitational lensing studies, and X-ray measurements of galaxy clusters. Indirect evidence for dark matter comes from its gravitational influence on other matter, as no dark matter particles have been observed in laboratories. Many particle physics candidates for dark matter have been proposed, and several projects to detect them directly are underway.
Additionally, there are outstanding problems associated with the currently favored cold dark matter model which include the dwarf galaxy problem and the cuspy halo problem. Alternative theories have been proposed that do not require a large amount of undetected matter, but instead modify the laws of gravity established by Newton and Einstein; yet no alternative theory has been as successful as the cold dark matter proposal in explaining all extant observations. Horizon problem. The horizon problem results from the premise that information cannot travel faster than light. In a universe of finite age this sets a limit—the particle horizon—on the separation of any two regions of space that are in causal contact. The observed isotropy of the CMB is problematic in this regard: if the universe had been dominated by radiation or matter at all times up to the epoch of last scattering, the particle horizon at that time would correspond to about 2 degrees on the sky. There would then be no mechanism to cause wider regions to have the same temperature.
A resolution to this apparent inconsistency is offered by inflation theory in which a homogeneous and isotropic scalar energy field dominates the universe at some very early period (before baryogenesis). During inflation, the universe undergoes exponential expansion, and the particle horizon expands much more rapidly than previously assumed, so that regions presently on opposite sides of the observable universe are well inside each other's particle horizon. The observed isotropy of the CMB then follows from the fact that this larger region was in causal contact before the beginning of inflation. Heisenberg's uncertainty principle predicts that during the inflationary phase there would be quantum thermal fluctuations, which would be magnified to a cosmic scale. These fluctuations served as the seeds for all the current structures in the universe. Inflation predicts that the primordial fluctuations are nearly scale invariant and Gaussian, which has been confirmed by measurements of the CMB. A related issue to the classic horizon problem arises because in most standard cosmological inflation models, inflation ceases well before electroweak symmetry breaking occurs, so inflation should not be able to prevent large-scale discontinuities in the electroweak vacuum since distant parts of the observable universe were causally separate when the electroweak epoch ended.
Magnetic monopoles. The magnetic monopole objection was raised in the late 1970s. Grand unified theories (GUTs) predicted topological defects in space that would manifest as magnetic monopoles. These objects would be produced efficiently in the hot early universe, resulting in a density much higher than is consistent with observations, given that no monopoles have been found. This problem is resolved by cosmic inflation, which removes all point defects from the observable universe, in the same way that it drives the geometry to flatness. Flatness problem. The flatness problem (also known as the oldness problem) is an observational problem associated with a FLRW. The universe may have positive, negative, or zero spatial curvature depending on its total energy density. Curvature is negative if its density is less than the critical density; positive if greater; and zero at the critical density, in which case space is said to be "flat". Observations indicate the universe is consistent with being flat. The problem is that any small departure from the critical density grows with time, and yet the universe today remains very close to flat. Given that a natural timescale for departure from flatness might be the Planck time, 10−43 seconds, the fact that the universe has reached neither a heat death nor a Big Crunch after billions of years requires an explanation. For instance, even at the relatively late age of a few minutes (the time of nucleosynthesis), the density of the universe must have been within one part in 1014 of its critical value, or it would not exist as it does today.
Misconceptions. One of the common misconceptions about the Big Bang model is that it fully explains the origin of the universe. However, the Big Bang model does not describe how energy, time, and space were caused, but rather it describes the emergence of the present universe from an ultra-dense and high-temperature initial state. It is misleading to visualize the Big Bang by comparing its size to everyday objects. When the size of the universe at Big Bang is described, it refers to the size of the observable universe, and not the entire universe. Another common misconception is that the Big Bang must be understood as the expansion of space and not in terms of the contents of space exploding apart. In fact, either description can be accurate. The expansion of space (implied by the FLRW metric) is only a mathematical convention, corresponding to a choice of coordinates on spacetime. There is no generally covariant sense in which space expands. The recession speeds associated with Hubble's law are not velocities in a relativistic sense (for example, they are not related to the spatial components of 4-velocities). Therefore, it is not remarkable that according to Hubble's law, galaxies farther than the Hubble distance recede faster than the speed of light. Such recession speeds do not correspond to faster-than-light travel.
Many popular accounts attribute the cosmological redshift to the expansion of space. This can be misleading because the expansion of space is only a coordinate choice. The most natural interpretation of the cosmological redshift is that it is a Doppler shift. Implications. Given current understanding, scientific extrapolations about the future of the universe are only possible for finite durations, albeit for much longer periods than the current age of the universe. Anything beyond that becomes increasingly speculative. Likewise, at present, a proper understanding of the origin of the universe can only be subject to conjecture. Pre–Big Bang cosmology. The Big Bang explains the evolution of the universe from a starting density and temperature that is well beyond humanity's capability to replicate, so extrapolations to the most extreme conditions and earliest times are necessarily more speculative. Lemaître called this initial state the ""primeval atom" while Gamow called the material "ylem"". How the initial state of the universe originated is still an open question, but the Big Bang model does constrain some of its characteristics. For example, if specific laws of nature were to come to existence in a random way, inflation models show, some combinations of these are far more probable, partly explaining why our Universe is rather stable. Another possible explanation for the stability of the Universe could be a hypothetical multiverse, which assumes every possible universe to exist, and thinking species could only emerge in those stable enough. A flat universe implies a balance between gravitational potential energy and other energy forms, requiring no additional energy to be created.
The Big Bang theory is built upon the equations of classical general relativity, which are not expected to be valid at the origin of cosmic time, as the temperature of the universe approaches the Planck scale. Correcting this will require the development of a correct treatment of quantum gravity. Certain quantum gravity treatments, such as the Wheeler–DeWitt equation, imply that time itself could be an emergent property. As such, physics may conclude that time did not exist before the Big Bang. While it is not known what could have preceded the hot dense state of the early universe or how and why it originated, or even whether such questions are sensible, speculation abounds on the subject of "cosmogony". Some speculative proposals in this regard, each of which entails untested hypotheses, are: Proposals in the last two categories see the Big Bang as an event in either a much larger and older universe or in a multiverse. Ultimate fate of the universe. Before observations of dark energy, cosmologists considered two scenarios for the future of the universe. If the mass density of the universe were greater than the critical density, then the universe would reach a maximum size and then begin to collapse. It would become denser and hotter again, ending with a state similar to that in which it started—a Big Crunch.
Alternatively, if the density in the universe were equal to or below the critical density, the expansion would slow down but never stop. Star formation would cease with the consumption of interstellar gas in each galaxy; stars would burn out, leaving white dwarfs, neutron stars, and black holes. Collisions between these would result in mass accumulating into larger and larger black holes. The average temperature of the universe would very gradually asymptotically approach absolute zero—a Big Freeze. Moreover, if protons are unstable, then baryonic matter would disappear, leaving only radiation and black holes. Eventually, black holes would evaporate by emitting Hawking radiation. The entropy of the universe would increase to the point where no organized form of energy could be extracted from it, a scenario known as heat death. Modern observations of accelerating expansion imply that more and more of the currently visible universe will pass beyond our event horizon and out of contact with us. The eventual result is not known. The ΛCDM model of the universe contains dark energy in the form of a cosmological constant. This theory suggests that only gravitationally bound systems, such as galaxies, will remain together, and they too will be subject to heat death as the universe expands and cools. Other explanations of dark energy, called phantom energy theories, suggest that ultimately galaxy clusters, stars, planets, atoms, nuclei, and matter itself will be torn apart by the ever-increasing expansion in a so-called Big Rip. Religious and philosophical interpretations. As a description of the origin of the universe, the Big Bang has significant bearing on religion and philosophy. As a result, it has become one of the liveliest areas in the discourse between science and religion. Some believe the Big Bang implies a creator, while others argue that Big Bang cosmology makes the notion of a creator superfluous.
Bock Bock () is a strong German beer, usually a dark lager. History. The style now known as "Bock" was first brewed in the 14th century in the Hanseatic town of Einbeck in Lower Saxony. The style was later adopted in Bavaria by Munich brewers in the 17th century. Due to their Bavarian accent, citizens of Munich pronounced "Einbeck" as "ein Bock" ("a billy goat"), and thus the beer became known as "Bock". A goat often appears on bottle labels. Bock is historically associated with special occasions, often religious festivals such as Christmas, Easter, or Lent (""). Bock has a long history of being brewed and consumed by Bavarian monks as a source of nutrition during times of fasting. Styles. Substyles of Bock include: Traditionally Bock is a sweet, relatively strong (6.3–7.6% by volume), lightly hopped lager registering between 20 and 30 International Bitterness Units (IBUs). The beer should be clear, with colour ranging from light copper to brown, and a bountiful, persistent off-white head. The aroma should be malty and toasty, possibly with hints of alcohol, but no detectable hops or fruitiness. The mouthfeel is smooth, with low to moderate carbonation and no astringency. The taste is rich and toasty, sometimes with a bit of caramel. The low-to-undetectable presence of hops provides just enough bitterness so that the sweetness is not cloying and the aftertaste is muted.
Maibock. The Maibock style – also known as Heller Bock or Lente Bock in the Netherlandsis a strong pale lager, lighter in colour and with more hop presence. Colour can range from deep gold to light amber with a large, creamy, persistent white head, and moderate to moderately high carbonation, while alcohol content ranges from 6.3% to 8.1% by volume. The flavour is typically less malty than a traditional Bock, and may be drier, hoppier, and more bitter, but still with a relatively low hop flavour, with a mild spicy or peppery quality from the hops, increased carbonation and alcohol content. Doppelbock. "Doppelbock" or "Double Bock" is a stronger version of traditional Bock that was first brewed in Munich by the Paulaner Friars, a Franciscan order founded by St. Francis of Paula. Historically, Doppelbock was high in alcohol and sweetness. The story is told that it served as "liquid bread" for the Friars during times of fasting when solid food was not permitted. However, historian Mark Dredge, in his book "A Brief History of Lager", says that this story is myth and that the monks produced Doppelbock to supplement their order's vegetarian diet all year.
Today, Doppelbock is still strongranging from 7% to 12% or more by volume. It is clear, with colour ranging from dark gold, for the paler version, to dark brown with ruby highlights for a darker version. It has a large, creamy, persistent head (although head retention may be impaired by alcohol in the stronger versions). The aroma is intensely malty, with some toasty notes, and possibly some alcohol presence as well; darker versions may have a chocolate-like or fruity aroma. The flavour is very rich and malty, with noticeable alcoholic strength, and little or no detectable hops (16–26 IBUs). Paler versions may have a drier finish. The monks who originally brewed Doppelbock named their beer "Salvator" (literally "Savior", but actually a malapropism for "Sankt Vater", "St. Father", originally brewed for the feast of St. Francis of Paola on 2 April which often falls in Lent), which today is trademarked by Paulaner. Brewers of modern Doppelbock often add "-ator" to their beer's name as a signpost of the style; there are 200 "-ator" Doppelbock names registered with the German patent office.
The following are representative examples of the style: Paulaner Salvator, Ayinger Celebrator, Weihenstephaner Korbinian, Andechser Doppelbock Dunkel, Spaten Optimator, Augustiner Brau Maximator, Tucher Bajuvator, Weltenburger Kloster Asam-Bock, Capital Autumnal Fire, EKU 28, Eggenberg Urbock 23º, Bell's Consecrator, Moretti La Rossa, Samuel Adams Double Bock, Tröegs Tröegenator Double Bock, Wasatch Brewery Devastator, Great Lakes Doppelrock, Abita Andygator, Wolverine State Brewing Company Predator, Burly Brewing's Burlynator, Monteith's Doppel Bock, and Christian Moerlein Emancipator Doppelbock. Eisbock. Eisbock is a traditional specialty beer of the Kulmbach district of Bavaria, made by partially freezing a Doppelbock and removing the water ice to concentrate the flavour and alcohol content, which ranges from 8.6% to 14.3% by volume. It is clear, with a colour ranging from deep copper to dark brown in colour, often with ruby highlights. Although it can pour with a thin off-white head, head retention is frequently impaired by the higher alcohol content. The aroma is intense, with no hop presence, but frequently can contain fruity notes, especially of prunes, raisins, and plums. Mouthfeel is full and smooth, with significant alcohol, although this should not be hot or sharp. The flavour is rich and sweet, often with toasty notes, and sometimes hints of chocolate, always balanced by a significant alcohol presence.
The following are representative examples of the style: Colorado Team Brew "Warning Sign", Kulmbacher Reichelbräu Eisbock, Eggenberg, Schneider Aventinus Eisbock, Urbock Dunkel Eisbock, Franconia Brewing Company Ice Bock 17%. The strongest ice beer, Strength in Numbers, was a one-time collaboration in 2020 between Schorschbrau of Germany and BrewDog of Scotland, who had competed with each other in the early years of the 21st century to produce the world's strongest beer. "Strength in Numbers" was created using traditional ice distillation, reaching a final strength of 57.8% ABV. Weizenbock. Weizenbock is a style that replaces some of the barley in the grain bill with 40–60% wheat. It was first produced in Bavaria in 1907 by G. Schneider & Sohn and was named "Aventinus" after 16th-century Bavarian historian Johannes Aventinus. The style combines darker Munich malts and top-fermenting wheat beer yeast, brewed at the strength of a Doppelbock.
Bantu languages The Bantu languages (English: , Proto-Bantu: *bantʊ̀) are a language family of about 600 languages that are spoken by the Bantu peoples of Central, Southern, Eastern and Southeast Africa. They form the largest branch of the Southern Bantoid languages. The total number of Bantu languages is estimated at between 440 and 680 distinct languages, depending on the definition of "language" versus "dialect". Many Bantu languages borrow words from each other, and some are mutually intelligible. Some of the languages are spoken by a very small number of people, for example the Kabwa language was estimated in 2007 to be spoken by only 8500 people but was assessed to be a distinct language. The total number of Bantu speakers is estimated to be around 350 million in 2015 (roughly 30% of the population of Africa or 5% of the world population). Bantu languages are largely spoken southeast of Cameroon, and throughout Central, Southern, Eastern, and Southeast Africa. About one-sixth of Bantu speakers, and one-third of Bantu languages, are found in the Democratic Republic of the Congo.
The most widely spoken Bantu language by number of speakers is Swahili, with 16 million native speakers and 80 million L2 speakers (2015). Most native speakers of Swahili live in Tanzania, where it is a national language, while as a second language, it is taught as a mandatory subject in many schools in East Africa, and is a lingua franca of the East African Community. Other major Bantu languages include Lingala with more than 20 million speakers (Congo, DRC), followed by Zulu with 13.56 million speakers (South Africa), Xhosa at a distant third place with 8.2 million speakers (South Africa and Zimbabwe), and Shona with less than 10 million speakers (if Manyika and Ndau are included), while Sotho-Tswana languages (Sotho, Tswana and Pedi) have more than 15 million speakers (across Botswana, Lesotho, South Africa, and Zambia). Zimbabwe has Kalanga, Matebele, Nambiya, and Xhosa speakers. "Ethnologue" separates the largely mutually intelligible Kinyarwanda and Kirundi, which together have 20 million speakers. Name.
The similarity among dispersed Bantu languages had been observed as early as the 17th century. The term "Bantu" as a name for the group was not coined but "noticed" or "identified" (as "Bâ-ntu") by Wilhelm Bleek as the first European in 1857 or 1858, and popularized in his "Comparative Grammar" of 1862. He noticed the term to represent the word for "people" in loosely reconstructed Proto-Bantu, from the plural noun class prefix "ba-" categorizing "people", and the root "ntʊ̀-" "some (entity), any" (e.g. Xhosa "umntu" "person", "abantu" "people"; Zulu "umuntu" "person", "abantu" "people"). There is no native term for the people who speak Bantu languages because they are not an ethnic group. People speaking Bantu languages refer to their languages by ethnic endonyms, which did not have an indigenous concept prior to European contact for the larger ethnolinguistic phylum named by 19th-century European linguists. Bleek's identification was inspired by the anthropological observation of groups frequently self-identifying as "people" or "the true people" (as is the case, for example, with the term "Khoikhoi", but this is a "kare" "praise address" and not an ethnic name).
The term "narrow Bantu", excluding those languages classified as Bantoid by Malcolm Guthrie (1948), was introduced in the 1960s. The prefix "ba-" specifically refers to people. Endonymically, the term for cultural objects, including language, is formed with the "ki-" noun class (Nguni "ísi-"), as in KiSwahili (Swahili language and culture), IsiZulu (Zulu language and culture) and KiGanda (Ganda religion and culture). In the 1980s, South African linguists suggested referring to these languages as "KiNtu." The word "kintu" exists in some places, but it means "thing", with no relation to the concept of "language". In addition, delegates at the African Languages Association of Southern Africa conference in 1984 reported that, in some places, the term "Kintu" has a derogatory significance. This is because "kintu" refers to "things" and is used as a dehumanizing term for people who have lost their dignity. In addition, "Kintu" is a figure in some mythologies. In the 1990s, the term "Kintu" was still occasionally used by South African linguists. But in contemporary decolonial South African linguistics, the term "Ntu languages" is used.

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