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Louvre Pyramid
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The Louvre Pyramid
The Louvre Pyramid (Pyramide du Louvre) is a large glass and metal pyramid designed by Chinese-American architect I.M. Pei, surrounded by three smaller pyramids, in the main courtyard (Cour Napoléon) of the Louvre Palace (Palais du Louvre) in Paris. The large pyramid serves as the main entrance to the Louvre Museum. Completed in 1989,[1] it has become a landmark of the city of Paris.
Contents
[hide]
1 Design and construction
2 History
3 Controversy
4 Urban legend of 666 panes
5 La Pyramide Inversée
6 Renovation
7 See also
8 References
9 External links
Design and construction[edit]
Inside the Pyramid: the view of the Louvre Museum in Paris from the underground lobby of the pyramid.
Commissioned by the President of France, François Mitterrand, in 1984, it was designed by the architect I. M. Pei. The structure, which was constructed entirely with glass segments and metal poles, reaches a height of 21.6 metres (71 ft).[2] Its square base has sides of 34 metres (112 ft) and a base surface area of 1,000 square metres (11,000 sq ft).[3] It consists of 603 rhombus-shaped and 70 triangular glass segments.[2] The pyramid structure was engineered by Nicolet Chartrand Knoll Ltd. of Montreal (Pyramid Structure / Design Consultant) and Rice Francis Ritchie of Paris (Pyramid Structure / Construction Phase).[4]
The pyramid and the underground lobby beneath it were created because of a series of problems with the Louvre's original main entrance, which could no longer handle the enormous number of visitors on an everyday basis.[citation needed] Visitors entering through the pyramid descend into the spacious lobby then ascend into the main Louvre buildings.[citation needed]
For design historian Mark Pimlott, "I.M. Pei’s plan distributes people effectively from the central concourse to myriad destinations within its vast subterranean network... the architectonic framework evokes, at gigantic scale, an ancient atrium of a Pompeiian villa; the treatment of the opening above, with its tracery of engineered castings and cables, evokes the atria of corporate office buildings; the busy movement of people from all directions suggests the concourses of rail termini or international airports."[5]
Several other museums have duplicated this concept, most notably the Museum of Science and Industry in Chicago. The Dolphin Centre, featuring a similar pyramid, was opened in April 1982, by Prince Richard, Duke of Gloucester.[6] The construction work on the pyramid base and underground lobby was carried out by the Vinci construction company.[7]
The large glass pyramid seen at night
The large glass pyramid seen by day
History[edit]
In 1839, according to one newspaper account, in ceremonies commemorating the "glorious revolution" of 1830, "The tombs of the Louvre were covered with black hangings and adorned with tricolored flags. In front and in the middle was erected an expiatory monument of a pyramidical shape, and surmounted by a funeral vase."[8]
Controversy[edit]
The construction of the pyramid triggered many years of strong and lively aesthetic and political debate.[9] Criticisms tended to fall into four areas: (1) the modernist style of the edifice being inconsistent with the classic French Renaissance style and history of the Louvre; (2) the pyramid being an unsuitable symbol of death from ancient Egypt; (3) the project being an immodest, pretentious, megalomaniacal folly imposed by then-President François Mitterrand; and (4) Chinese-American architect I.M. Pei being insufficiently French to be entrusted with the task of updating the treasured Parisian landmark.[10]
Those criticizing the aesthetics said it was "sacrilegious" to tamper with the Louvre's majestic old French Renaissance architecture, and called the pyramid an anachronistic intrusion of an Egyptian death symbol in the middle of Paris.[11] Meanwhile, political critics referred to the structure as Pharaoh Francois' Pyramid.[10] Many still continue to feel the harsh modernism of the edifice is out of place.[12][13][14]
During the design phase, there was a proposal[by whom?] that the design include a spire on the pyramid to simplify window washing. Pei objected, however, and this proposal was eliminated.
Urban legend of 666 panes[edit]
It has been claimed by some that the glass panes in the Louvre Pyramid number exactly 666, "the number of the beast", often associated with Satan. Dominique Stezepfandt's book François Mitterrand, Grand Architecte de l'Univers declares that "the pyramid is dedicated to a power described as the Beast in the Book of Revelation (...) The entire structure is based on the number 6."
The story of the 666 panes originated in the 1980s, when the official brochure published during construction did indeed cite this number (even twice, though a few pages earlier the total number of panes was given as 672 instead). The number 666 was also mentioned in various newspapers. The Louvre museum, however, states that the finished pyramid contains 673 glass panes (603 rhombi and 70 triangles).[2] A higher figure was obtained by David A. Shugarts, who reports that the pyramid contains 689 pieces of glass.[15] Shugarts obtained the figure from the Pei's offices.
Elementary arithmetic allows for easy counting of the panes: each of the three sides of the pyramid without an entrance has 18 triangular panes and 17 rows of rhombic ones arranged in a triangle, thus giving 17 ⋅ ( 17 + 1 ) 2 = 153 {\displaystyle \textstyle {\frac {17\cdot (17+1)}{2}}=153} rhombic panes (171 panes total). The side with the entrance has 11 panes fewer (9 rhombic, 2 triangular), so the whole pyramid consists of 4 ⋅ 153 − 9 = 603 {\displaystyle 4\cdot 153-9=603} rhombi and 4 ⋅ 18 − 2 = 70 {\displaystyle 4\cdot 18-2=70} triangles, 673 panes total.
The myth resurfaced in 2003, when Dan Brown incorporated it in his best-selling novel The Da Vinci Code, in which the protagonist reflects that "this pyramid, at President Mitterrand's explicit demand, had been constructed of exactly 666 panes of glass — a bizarre request that had always been a hot topic among conspiracy buffs who claimed 666 was the number of Satan."[16] However, David A. Shugarts reports that according to a spokeswoman of the offices of Pei, the French President never specified the number of panes to be used in the pyramid. Noting how the 666 rumor circulated in some French newspapers in the mid-1980s, she commented: "If you only found those old articles and didn't do any deeper fact checking, and were extremely credulous, you might believe the 666 story."[15]
Comparison of approximate profiles of the Louvre Pyramid with other notable pyramidal or near-pyramidal buildings. Dotted lines indicate original heights, where data are available. In its SVG file, hover over a pyramid to highlight and click for its article.
La Pyramide Inversée[edit]
Main article: La Pyramide Inversée
La Pyramide Inversée (The Inverted Pyramid) is a skylight in the Carrousel du Louvre shopping mall in front of the Louvre Museum. It looks like an upside-down and smaller version of the Louvre Pyramid.
Renovation[edit]
Designed for a museum that attracted 4.5 million visitors a year, the pyramid proved inadequate by the time the Louvre's attendance had doubled in 2014. Between 2014 and 2017, the layout of the foyer area in the Cour Napoleon beneath the glass pyramid is undergoing a thorough redesign, including better access to the pyramid and the Passage Richelieu.[17]
See also[edit]
Paris portal
References[edit]
Jump up ^ Simons, Marlise (28 March 1993). "5 Pieces of Europe's Past Return to Life: France; A vast new exhibition space as the Louvre renovates". The New York Times. Retrieved 7 October 2008.
^ Jump up to: a b c "Architecture: Louvre Pyramid". Glass on the Web. June 2005. Archived from the original on 12 January 2002. Retrieved 16 January 2011.
Jump up ^ Official Press Release, Louvre. ""Pyramid" Project Launch: The Musée du Louvre is improving visitor reception (2014-2016)" (PDF). Louvre. p. 10. Retrieved 16 May 2016.
Jump up ^ "Grand Louvre: Phase I". Pei Cobb Freed & Partners. Retrieved 16 January 2011.
Jump up ^ Pimlott, Mark (2007). "The Grand Louvre & I.M. Pei". Without and Within: Essays on Territory and the Interior (Excerpt). Rotterdam: Episode Publishers. Retrieved 13 August 2012 – via artdesigncafe.
Jump up ^ Steer, Phil. "Dolphin Centre: Brief History". Romford Now & Then. Archived from the original on 23 November 2011. [self-published source]
Jump up ^ "History". Vinci. Archived from the original on 1 October 2011. Retrieved 16 January 2011.
Jump up ^ "The Paris Sketch Book Of Mr. M. A. Titmarsh". Project Gutenberg.
Jump up ^ Tempest, Rone. "Controversial New Pyramid Entrance to the Louvre Opens in Paris". Los Angeles Times. Retrieved 16 May 2016.
^ Jump up to: a b Bernstein, Richard. "I.M. Pei's Pyramid: A Provocative Plan for The Louvre". The New York Times. The New York Times. Retrieved 16 May 2016.
Jump up ^ Goldberger, Paul. "Pei Pyramid and New Louvre Open Today". The New York Times. The New York Times. Retrieved 16 May 2016.
Jump up ^ Stamberg, Susan. "Landmark At The Louvre: The Pyramid Turns 20". NPR. National Public Radio. Retrieved 16 May 2016. ,
Jump up ^ Carbone, Ken. "Viva Le Louvre! At 20, I.M. Pei's Controversial Pyramid Defies Critics". Fast Company. Mansueto Ventures, LLC. Retrieved 16 May 2016.
Jump up ^ Souza, Eduardo. "AD Classics: Le Grande Louvre / I.M. Pei". Arch Daily. Plataforma Networks Broadcasting Architecture Worldwide. Retrieved 16 May 2016.
^ Jump up to: a b Secrets of the Code, edited by Dan Burstein, p. 259.[full citation needed]
Jump up ^ Dan Brown. The Da Vinci Code, p. 21.[full citation needed]
Jump up ^ Pes, Javier (28 April 2014). "Louvre's Director Makes Unblocking Pyramid Bottleneck a Priority". The Art Newspaper. Archived from the original on 29 April 2014. CS1 maint: Unfit url (link)
External links[edit]
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Education in Poland
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This article is about education in modern Poland. For other periods in Polish history, see Education in Poland (disambiguation).
Education in Poland
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The Collegium Novum of the Jagiellonian University in Kraków
Compulsory education in Poland starts at the age of six from the mandatory "0" reception class (Polish zerówka or klasa 0, literally Year 0). At the age of seven kids start the 1st grade of primary school (Polish szkoła podstawowa) lasting for 8 (6 until 2017) years and finished with an exam. Afterwards, until 2017, pupils have joined the mandatory junior high school for three years (lower secondary education) and at the end, take another compulsory exam.[1]
Polish Ministry of Education established by King Stanisław August Poniatowski in 1773 was the first ministry of education in the world,[2][3] and the traditions continue. The international PISA 2012 praised the progresses made by Polish education in Mathematics, Science and Literacy; the number of top-performers having increased since 2003 while the number of low-performers decreased again.[4] In 2014, the Pearson/Economist Intelligence Unit rated Polish education as 5th best in Europe and 10th best in the world.[5]
There are several alternatives for the upper secondary education later on, the most common being the four (three until 2017) years of a liceum or five (four until 2017) years in a technikum. Both end with a maturity exam (matura, similar to French baccalauréat), and may be followed by several forms of upper education, leading to Bachelor: licencjat or inżynier (the Polish Bologna Process first cycle qualification), Master: magister (the Polish Bologna Process second cycle qualification) and eventually PhD: doktor (the Polish Bologna Process third cycle qualification). The system of education in Poland allows for 22 years of continuous, uninterrupted schooling.[6]
Contents
1 Compulsory education
1.1 Primary school
1.2 Junior high school
2 Upper secondary education
3 Tertiary education
3.1 Grading system at University level
4 Foreign languages
5 History
5.1 PiS reforms of 2016
6 Violence
7 See also
8 References
9 External links
Compulsory education[edit]
Primary school[edit]
Primary school usually starts at seven years old.[7] Following changes introduced by 2016 act "Law on School Education" primary school has been extended to 8 years.[8] The first three year cycle is "integrated", with one teacher handling alone all or most of the subjects, while the following five years offer a subject-based teaching. At the end of primary school, pupils write a compulsory national competence test. If completed, the examination grants a primary-school leaving certificate. The Polish Education system for primary school aged children is centred around a pedogogical style that focuses on the learning of facts and does not involve the concept of exploration or investigation, individuality and opinions are inconsequential.
Junior high school[edit]
The junior high school (gimnazjum) covers lower secondary education and ends general basic education and lasts three years. The subjects taught are: Polish language, history, civic education, two foreign languages, mathematics, physics and astronomy, chemistry, biology, geography, fine arts/music, technology, information technology, physical education and religion or ethics.[7] At the end of the curriculum, pupils are evaluated based on their continuing results and on an examination in humanities, science and foreign languages.[7]
Following 2016 reform by PiS ruling party changes to Polish education system were gradually introduced. Starting with the school year of 2017/18, middle schools are scheduled to be disbanded, primary schools to be extended to eight years and upper secondary schools to be given one year more, as it was before 1999.[9] Institutions are either set to close, be changed into primary schools or high schools by 1st September 2019.
Upper secondary education[edit]
The upper secondary education begins at the end of full-time compulsory education, preparing students for entry directly into the labour-market and/or tertiary (i.e. higher) education. Upper secondary education takes many forms.
General education can be pursued in general secondary schools (liceum): after three years, students can pass the "Matura", which grants access to higher education.[7] Vocational and technical education is mainly provided by Technical schools (technikum) and/or basic vocational schools (zasadnicza szkoła zawodowa). Technical schools last 4 years and lead to the Matura. Their primary goal is to teach occupations and trades, the most popular being: accountant, mechanic, electronics specialist, and salesperson.[10] Basic vocational schools also provide a vocational education lasting 2 years and grant a certificate of competence in various fields, the most popular being: shop-assistant, cook, gardener, automobile mechanic, hairdresser and baker.[10] Graduates from basic vocational schools can pass the Matura after an extra-curriculum of two years in a general secondary school, or, since 2004, of three years in a Technical school.[7] Profiled general secondary schools (liceum profilowane) provide a vocational education in three years, but only in fields described by the Polish Classification of Activities (PKD).[10] In addition, mentally and/or physically handicapped students can join special schools (szkoła specjalna) which prepare to the Matura in three years.[10]
Tertiary education[edit]
University of Łódź, Faculty of Management
Poland follows the Bologna scheme and most of its tertiary level programmes are made of two cycles: a three-year bachelor's degree followed by a two-year master's degree.[7] Some master's degrees are however granted after a unique long-cycle programme, lasting between four and six years (Ex: five years for pharmacy, six year for medicine).[7] Doctoral programmes are achieved in about three years. The diploma of primary school teachers requires three years of study within a teacher training college.[7] Vocational education is handled by post-secondary school(szkola policealna) with programmes lasting two and a half years.[10]
Grading system at University level[edit]
The university-level education uses a numeric system of grades from two to five, with most grades including 0.5 point increments: 2.0 is the failing grade, 3.0 is the lowest passing grade, followed by 3.5, 4.0 and 4.5, with 5.0 being the highest grade. There is no grade 2.5. Also, 5.5 or 6.0 is sometimes given as an "exceeds expectations" grade, but this differs among various universities and may be equivalent to 5.0 for some purposes. Similarly "3-" is occasionally (but very rarely) given as a "barely passing" grade, but for all official purposes it is equivalent to 3.0.
The grading is done every semester (twice a year), not just once in a school year. Depending on the subject, the final grade may be based on the result of a single exam, or on the student's performance during the whole semester. In the latter case, usually a point system, not the 2–5 scale is used. The points accumulated during the semester are added and converted to a final grade according to some scale.
As a failing grade means merely having to repeat the failed subject, and can usually be corrected on a retake exam (and in some cases also on a special "committee exam"), it is used much more liberally, and it is quite common for a significant number of students to fail a class on the first attempt.
Foreign languages[edit]
Students in Polish schools typically learn one or two foreign languages. In 2005/06, the fractions of students studying foreign languages in Polish schools included: English – 67.9%, German – 33.3%, French – 13.3%, Spanish – 10.2%, Russian – 6.1%, Italian – 4.3%, Latin – 0.6%.
In 2005/06, there were 49,200 students in schools for national minorities, most of them in German, Kashubian, Ukrainian and Belarusian language schools[11].
Under the education reform introduced by Polish education minister Katarzyna Hall, students of Polish lower secondary schools had to learn two different foreign languages. The first foreign language (usually English) was taught three times a week. The second foreign language was taught twice a week. The reform introduced two different levels of the exam – a higher level (if a student learnt the same language in primary school) and a standard level (if a student learnt the first language in lower secondary school). The results of lower secondary school language examinations contributed to the criteria of applying to enter upper secondary level school.
History[edit]
The education of Polish society was a goal of rulers as early as the 12th century, and Poland soon became one of the most educated countries in Europe. The library catalog of the Cathedral Chapter of Kraków dating back to 1110 shows that in the early 12th-century Polish intellectuals had access to European literature. The Jagiellonian University, founded in 1364 by King Casimir III in Kraków, is one of Europe's oldest universities. In 1773 King Stanisław August Poniatowski established the Commission of National Education (Komisja Edukacji Narodowej), the world's first state ministry of education.
The first university in Poland, Kraków's Jagiellonian University, was established in 1364 by Casimir III the Great in Kraków. It is the oldest university in Poland. It is the second oldest university in Central Europe (after Prague University) and one of the oldest universities in the world. Casimir III realized that the nation needed a class of educated people, especially lawyers, who could codify the country's laws and administer the courts and offices. His efforts to found an institution of higher learning in Poland were finally rewarded when Pope Urban V granted him permission to open the University of Kraków.
In the 16th century, 90% of all parishes in Lesser Poland and Greater Poland had schools where grammar and Latin were taught at the basic level, while secondary education was available in cities and larger towns. At the Jagiellonian University, 65% of students came from urban backgrounds, 25% from the nobility and 10% from peasant families.[12]
The idea of compulsory education was put forward by Andrzej Frycz Modrzewski in 1555. After the partitions of Poland, compulsory education was introduced by Prussian authorities in Polish provinces which belonged to Prussia (1825), and Austrian authorities in Galicia (1873). In the Russian Empire compulsory education did not exist. As a result, in 1921, after Poland regained independence, one-third of the population of the Second Polish Republic was illiterate. Illiteracy was very high in the east, but almost non-existent in western provinces. Compulsory education in Poland was introduced by a decree in February 1919. This covered all children aged 7 to 14. At the beginning, however, the newly created Polish state faced several problems of implementation - a lack of qualified teachers, buildings and funds. After World War Two, compulsory education remained as one of the priorities of the state. By 1978, only 1.2 percent of the Polish population was illiterate. In Poland, compulsory education ends at the age of 18. It usually starts when children are 6 years old and ends after 12 years of learning (usually in a high school). Contemporary Polish law distinguishes between compulsory school (obowiązek szkolny) and compulsory education (obowiązek nauki).
A major reform of the education system took place in 1999, changing the overall organisational structure from 8 (primary school) + 4 (high school/lyceum) years of education to 6 (primary) + 3 (junior high school, gymnasium) + 3 (senior high school, lyceum) years.[6] The reform increased time spent on core subjects and delayed vocational study (lyceum) by one year. Poland's OECD educational rankings for reading and science shifted from being below average to being in the top 10, and to the top 15 for mathematics.[9]
PiS reforms of 2016[edit]
The PiS government of Poland introduced a major educational reform in 2016, for successive implementation over the three school years starting with 2017/2018. The reform reverts the overall organisational structure from 6 (primary) + 3 (junior high school) + 3 (high school) years of education to 8 (primary school) + 4 (high school) years.[citation needed] According to history teacher Anna Dzierzgowska, the reform inherits the Polish-centred and Eurocentric focuses of the previous history syllabus, removes the Non-Aligned Movement from the syllabus, and focuses on political and military leaders and the nobility, neglecting the historical role of lower social classes.[13] The term communism was removed from teaching about the nineteenth century, during which it is called socialism, and only appears later in association with the Polish People's Republic. Dzierzgowska argues that the notion of nationalism historically dates only back to the nineteenth century, but is used much too frequently in the new curriculum, giving pupils insufficient geographical context.[13]
Violence[edit]
In 2006, in response to the suicide of a girl after she was sexually molested in school, the Polish Minister of Education, Roman Giertych, launched a "zero tolerance" school reform.[14] Under this plan, teachers would have the legal status of civil servants, making violent crimes against them punishable by higher penalties. Head teachers (equivalent to principals in the US) will be, in theory, able to send aggressive pupils to perform community service and these students' parents may also be fined. Teachers who fail to report violent acts in school could face a prison sentence.[15]
See also[edit]
Grade (education)
General education liceum
Technikum
List of universities in Poland
Polish science and technology
Underground Education in Poland During World War II
Education in the People's Republic of Poland
Open access in Poland
References[edit]
Jump up ^ "Polish System of Education". Bureau for Academic Recognition and International Exchange. April 2005. Archived from the original (Internet Archive) on January 29, 2009. Retrieved December 2, 2012.
Jump up ^ Jan IJ. van der Meer (2002). Literary Activities and Attitudes in the Stanislavian Age in Poland (1764–1795): A Social System?. Rodopi. p. 233. ISBN 978-90-420-0933-2. Retrieved 26 April 2012.
Jump up ^ Norman Davies, God's Playground: A History of Poland, Columbia University Press, 2005, ISBN 0-231-12819-3, Google Print, p.167
Jump up ^ PISA 2012 Results in Focus (PDF), OECD, 3 December 2013, retrieved 15 January 2014
Jump up ^ Top 20 Education Systems BBC. Source: Pearson/Economist Intelligence Unit.
^ Jump up to: a b "The education system in Poland before and after the reform of 1999". Internet Archive. Bureau for Academic Recognition and International Exchange. June 2002. Archived from the original (Graphs) on March 27, 2009. Retrieved August 13, 2012.
^ Jump up to: a b c d e f g h "World Data on education: Poland" (PDF). UNESCO-IBE. 2012. Retrieved 5 August 2014.
Jump up ^ "THE POLISH EDUCATION SYSTEM IN BRIEF" (PDF). Eurydice.org.pl. 2018. Retrieved 2 April 2018.
^ Jump up to: a b Swinford, Steven (2015-01-19). "Poland is leading the way for England's schools, Education Secretary says". The Telegraph. Archived from the original on 2017-09-01. Retrieved 2017-09-01.
^ Jump up to: a b c d e "TVET in Poland". UNESCO-UNEVOC. 2013. Retrieved 5 August 2014.
Jump up ^ [1]
Jump up ^ Friedrich, Karin; Pendzich, Barbara M., eds. (2009). Citizenship and Identity in a Multinational Commonwealth, Poland-Lithuania in Context, 1550-1772. Brill. p. 4.
^ Jump up to: a b Gostkiewicz, Michał; Dzierzgowska, Anna (2017-09-01). "Czego dzieci nauczą się na historii? Nauczycielka nie ma złudzeń. 'Od mamuta do Bieruta'" [What do children learn from history? This teacher has no illusions. "From the mammoth to Bierut"] (in Polish). Gazeta Wyborcza. Archived from the original on 2017-09-01. Retrieved 2017-09-01.
Jump up ^ Easton, A. (2006, November 3). Polish drug use and suicide sparks school plan. BBC News Online, London.
Jump up ^ The shadow in our schools (2006, December 20). Warsaw Voice.
External links[edit]
Study in Poland
Education in Poland on Poland.pl
Recognition of Foreign Diplomas in Poland Source: Ordinance of the Minister of National Education on the Rules and Procedure. Retrieved December 2, 2012.
Ministry of Education (Poland)
Structure of education in Poland
Official register of educational institutions by type (Wykaz szkół i placówek oświatowych według typów). Centrum Informatyczne Edukacji, 03.01.2014. (in Polish)
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text:
Fugacity
From Wikipedia, the free encyclopedia
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In chemical thermodynamics, the fugacity of a real gas is an effective partial pressure which replaces the mechanical partial pressure in an accurate computation of the chemical equilibrium constant. It is equal to the pressure of an ideal gas which has the same chemical potential as the real gas. For example, nitrogen gas (N2) at 0 °C and a pressure of P = 100 atm has a fugacity of f = 97.03 atm.[1] This means that the chemical potential of real nitrogen at a pressure of 100 atm is less than if nitrogen were an ideal gas at 100 atm. Ideal-gas nitrogen at 97.03 atm would have the same chemical potential as real nitrogen at 100 atm.
Fugacities are determined experimentally or estimated from various models such as a Van der Waals gas that are closer to reality than an ideal gas. The ideal gas pressure and fugacity are related through the dimensionless fugacity coefficient φ.[2]
φ = f P {\displaystyle \varphi ={\frac {f}{P}}\,}
For nitrogen at 100 atm, the fugacity coefficient is 97.03 atm / 100 atm = 0.9703. For an ideal gas, fugacity and pressure are equal so φ is 1.
The contribution of nonideality to the chemical potential of a real gas is equal to RT ln φ. Again for nitrogen at 100 atm, the chemical potential is μ = μid + RT ln 0.9703, which is less than the ideal value μid because of intermolecular attractive forces.
The fugacity is closely related to the thermodynamic activity. For a gas, the activity is simply the fugacity divided by a reference pressure to give a dimensionless quantity. This reference pressure is called the standard state and normally chosen as 1 atmosphere or 1 bar, Again using nitrogen at 100 atm as an example, since the fugacity is 97.03 atm, the activity is just 97.03 without units.
Accurate calculations of chemical equilibrium for real gases should use the fugacity rather than the pressure. The thermodynamic condition for chemical equilibrium is that the total chemical potential of reactants is equal to that of products. If the chemical potential of each gas is expressed as a function of fugacity, the equilibrium condition may be transformed into the familiar reaction quotient form (or law of mass action) except that the pressures are replaced by fugacities.
For a condensed phase (liquid or solid) in equilibrium with its vapor phase, the chemical potential is equal to that of the vapor, and therefore the fugacity is equal to the fugacity of the vapor. This fugacity is approximately equal to the vapor pressure when the vapor pressure is not too high.
The word fugacity is derived from the Latin for "fleetness", which is often interpreted as "the tendency to flee or escape". The concept of fugacity was introduced by American chemist Gilbert N. Lewis in 1901.[3]
Contents
[hide]
1 Definition in terms of chemical potential
2 Evaluation for a real gas
3 Of a pure liquid
4 See also
5 References
6 External links
6.1 Video lectures
Definition in terms of chemical potential[edit]
The fugacity of a real gas is formally defined by an equation analogous to the relation between the chemical potential and the pressure of an ideal gas.
In general, the chemical potential μ is defined as the partial molar Gibbs energy. However, for any pure substance it is equal to the molar Gibbs energy, and its variation with temperature (T) and pressure (P) is given by
d μ = d G = − S d T + V d P {\displaystyle {d\mu }=dG=-SdT+VdP} .
At constant temperature, this expression can be integrated as a function of P. We must also set a reference state. For an ideal gas the reference state depends only on pressure, and we set P = 1 bar so that
∫ μ ∘ μ d μ = ∫ P ∘ P V ¯ d P {\displaystyle \int _{\mu ^{\circ }}^{\mu }{d\mu }=\int _{P^{\circ }}^{P}{{\bar {V}}dP}}
Now, for an ideal gas:
V ¯ = R T P {\displaystyle {\bar {V}}={\frac {RT}{P}}}
therefore
μ − μ ∘ = ∫ P ∘ P R T P d P = R T ln  P P ∘ {\displaystyle \mu -\mu ^{\circ }=\int _{P^{\circ }}^{P}{{\frac {RT}{P}}dP}=RT\ln {\frac {P}{P^{\circ }}}}
Reordering, we get
μ = μ ∘ + R T ln  P P ∘ {\displaystyle \mu =\mu ^{\circ }+RT\ln {\frac {P}{P^{\circ }}}}
This gives the chemical potential for an ideal gas in an isothermal process, with a reference state is P = 1 bar.
For a real gas, the integral
∫ P ∘ P V ¯ d P {\displaystyle \int _{P^{\circ }}^{P}{{\bar {V}}dP}}
cannot be calculated because there is no simple expression for a real gas's molar volume. Even if using an approximate expression such as the van der Waals equation, the Redlich–Kwong or any other equation of state, it would depend on the substance being studied and would be therefore of very limited utility.
Additionally, chemical potential is not mathematically well behaved. It approaches negative infinity as pressure approaches zero and this creates problems in doing real calculations.
It is desirable that the expression for a real gas's chemical potential to be similar to the one for an ideal gas. Therefore, one can define a quantity, called fugacity, so that the chemical potential for a real gas becomes
μ = μ ∘ + R T ln  f f ∘ {\displaystyle \mu =\mu ^{\circ }+RT\ln {\frac {f}{f^{\circ }}}}
with a given reference state to be discussed later. This is the usual form of the definition, but it may be solved for f to obtain the equivalent explicit form
f = f ∘ exp  ( μ − μ ∘ R T ) {\displaystyle f={f^{\circ }}\exp \left({\frac {\mu -\mu ^{\circ }}{RT}}\right)}
Evaluation for a real gas[edit]
Fugacity is used to better approximate the chemical potential of real gases than estimations made using the ideal gas law. Yet fugacity allows the use of many of the relationships developed for an idealized system.
In the real world, gases approach ideal gas behavior at low pressures and high temperatures; under such conditions the value of fugacity approaches the value of pressure. Yet no substance is truly ideal. At moderate pressures real gases have attractive interactions and at high pressures intermolecular repulsions become important. Both interactions result in a deviation from "ideal" behavior for which interactions between gas atoms or molecules are ignored.
For a given temperature T, the fugacity f satisfies the following differential relation:
d ln  f f 0 = d G R T = V ¯ d P R T {\displaystyle d\ln {f \over f_{0}}={dG \over RT}={{{\bar {V}}dP} \over RT}\,}
where G is the Gibbs energy, R is the gas constant, V is the fluid's molar volume, and f0 is a reference fugacity which is generally taken as 1 bar. For an ideal gas, when f = P, this equation reduces to the ideal gas law.
Thus, for any two physical states at the same temperature, represented by subscripts 1 and 2, the ratio of the two fugacities is as follows:
f 2 f 1 = exp  ( 1 R T ∫ G 1 G 2 d G ) = exp  ( 1 R T ∫ P 1 P 2 V ¯ d P ) {\displaystyle {f_{2} \over f_{1}}=\exp \left({1 \over RT}\int _{G_{1}}^{G_{2}}dG\right)=\exp \left({1 \over RT}\int _{P_{1}}^{P_{2}}{\bar {V}}\,dP\right)\,}
For an ideal gas, this becomes simply
f 2 f 1 = p 2 p 1 {\displaystyle {f_{2} \over f_{1}}={p_{2} \over p_{1}}} or f = P {\displaystyle f=P}
But for P → 0, every gas is an ideal gas. Therefore, fugacity must obey the limit equation
lim P → 0 f P = 1 {\displaystyle \lim _{P\to 0}{\frac {f}{P}}=1}
We determine f by defining a new function Φ, which is the difference between the actual molar volume and the ideal-gas molar volume:
Φ = P V ¯ − R T P {\displaystyle \Phi ={\frac {P{\bar {V}}-RT}{P}}}
We can obtain values for Φ experimentally easily by measuring V, T and P.
From the expression above we have
V ¯ = R T P + Φ {\displaystyle {\bar {V}}={\frac {RT}{P}}+\Phi }
(It is more obvious here that Φ is the "extra" molar volume of a non-ideal gas.) We can then write
∫ μ ∘ μ d μ = ∫ P ∘ P V ¯ d P = ∫ P ∘ P R T P d P + ∫ P ∘ P Φ d P {\displaystyle \int _{\mu ^{\circ }}^{\mu }{d\mu }=\int _{P^{\circ }}^{P}{{\bar {V}}dP}=\int _{P^{\circ }}^{P}{{\frac {RT}{P}}dP}+\int _{P^{\circ }}^{P}{\Phi dP}}
Where
μ = μ ∘ + R T ln  P P ∘ + ∫ P ∘ P Φ d P {\displaystyle \mu =\mu ^{\circ }+RT\ln {\frac {P}{P^{\circ }}}+\int _{P^{\circ }}^{P}{\Phi dP}}
Since the expression for an ideal gas was chosen to be
μ = μ ∘ + R T ln  f f ∘ {\displaystyle \mu =\mu ^{\circ }+RT\ln {\frac {f}{f^{\circ }}}} ,
we must have
μ ∘ + R T ln  f f ∘ = μ ∘ + R T ln  P P ∘ + ∫ P ∘ P Φ d P {\displaystyle \mu ^{\circ }+RT\ln {\frac {f}{f^{\circ }}}=\mu ^{\circ }+RT\ln {\frac {P}{P^{\circ }}}+\int _{P^{\circ }}^{P}{\Phi dP}}
⇒ R T ln  f f ∘ − R T ln  P P ∘ = ∫ P ∘ P Φ d P {\displaystyle \Rightarrow RT\ln {\frac {f}{f^{\circ }}}-RT\ln {\frac {P}{P^{\circ }}}=\int _{P^{\circ }}^{P}{\Phi dP}}
R T ln  f P ∘ P f ∘ = ∫ P ∘ P Φ d P {\displaystyle RT\ln {\frac {fP^{\circ }}{Pf^{\circ }}}=\int _{P^{\circ }}^{P}{\Phi dP}}
Suppose we choose P° → 0. Since
lim P ∘ → 0 f ∘ = P ∘ {\displaystyle {\mathop {\lim }}_{P^{\circ }\to 0}f^{\circ }=P^{\circ }} ,
we obtain
R T ln  f P = ∫ 0 P Φ d P {\displaystyle RT\ln {\frac {f}{P}}=\int _{0}^{P}{\Phi dP}}
The fugacity coefficient is defined as φ = f/P (note that for an ideal gas, φ = 1), and it will then verify
ln  φ = 1 R T ∫ 0 P Φ d P {\displaystyle \ln \varphi ={\frac {1}{RT}}\int _{0}^{P}{\Phi dP}}
The integral can be evaluated via graphical integration if we experimentally measure values for Φ while varying P.
We can then find the fugacity coefficient of a gas at a given pressure P and calculate
f = φ P {\displaystyle f=\varphi P\,} .
The reference state for the expression of a real gas's chemical potential is taken to be "ideal gas, at P = 1 bar and temperature T". Since in the reference state the gas is considered to be ideal (it is an hypothetical reference state), we can write that for the real gas
μ = μ ∘ + R T ln  f P ∘ {\displaystyle \mu =\mu ^{\circ }+RT\ln {\frac {f}{P^{\circ }}}}
For a gas obeying the van der Waals equation, the explicit formula [4] for the fugacity is
R T ln  f p = R T b V ¯ [ p ] − b − 2 a V ¯ [ p ] − R T ln  ( 1 − a ( V ¯ [ p ] − b ) R T V ¯ [ p ] 2 ) {\displaystyle RT\ln {\frac {f}{p}}={\frac {RTb}{{\bar {V}}[p]-b}}-{\frac {2a}{{\bar {V}}[p]}}-RT\ln \left(1-{\frac {a({\bar {V}}[p]-b)}{RT{\bar {V}}[p]^{2}}}\right)}
This formula is difficult to use, since the pressure depends on the molar volume V[p]; so one must choose a volume, calculate the pressure, and then use these two values on the right-hand side of the equation.
Of a pure liquid[edit]
Fugacity vs. pressure of ethane at −60 °F (−51 °C). Note the "breakpoint" at the saturation pressure and the curvature at higher pressures as the Poynting correction factor becomes important.
For a pure fluid in vapor–liquid equilibrium, the vapor phase fugacity is equal to the liquid phase fugacity. At pressures above the saturation pressure, the liquid phase fugacity is:[5]
f l i q ( T , P ) = φ s a t P s a t e x p [ ∫ P s a t P v l i q R T d P ] {\displaystyle f_{\rm {liq}}(T,P)=\varphi _{\rm {sat}}P_{\rm {sat}}{\rm {exp}}\left[\int _{P_{\rm {sat}}}^{P}{v_{\rm {liq}} \over RT}dP\right]}
where vliq is the molar volume of the liquid.
The fugacity correction factor for the vapor, φsat, should be evaluated at the saturation pressure and is unity when Psat is low. The exponential term represents the Poynting correction factor and is usually near 1 unless pressures are very high. Frequently, the fugacity of the pure liquid is used as a reference state when defining and using mixture activity coefficients.
See also[edit]
Activity (chemistry), the measure of the "effective concentration" of a species in a mixture
Chemical equilibrium
Electrochemical potential
Excess chemical potential
Partial molar property
Thermodynamic equilibrium
Fugacity capacity
Multimedia fugacity model
References[edit]
Jump up ^ Atkins, Peter; De Paula, Julio (2006). Atkins' Physical Chemistry (8th ed.). W. H. Freeman. p. 112.
Jump up ^ Atkins, Peter; Locke, John W. (January 2002). Atkins' Physical Chemistry (7th ed.). Oxford University Press. ISBN 0-19-879285-9.
Jump up ^ Lewis, Gilbert Newton (June 1901). "The Law of Physico-Chemical Change". Proceedings of the American Academy of Arts and Sciences. 37 (4): 49–69. doi:10.2307/20021635.
Jump up ^ David, Carl W. (2015). "Fugacity Examples 2: The fugacity of a "hard-sphere" semi-ideal gas and the van der Waals gas". Chemistry Education Materials. 91.
Jump up ^ Prausnitz; et al. (1999). Molecular Thermodynamics of Fluid Phase Equilibria (3rd ed.). Prentice-Hall. pp. 40–43.
External links[edit]
Video lectures[edit]
Thermodynamics, University of Colorado-Boulder, 2011
Introduction to fugacity: Where did it come from?
What is fugacity?
What is fugacity in mixtures?
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