- Clausius theorem
of Clausius s theorem url http theory.ph.man.ac.uk judith stat therm node30.html accessdate Octorber 4,2010 author Judith McGovern date 2004 03 17 cite web title The Clausius Inequality And The Mathematical ... October 5, 2010 DEFAULTSORT Clausius Theorem Category Thermodynamics Category Physics theorems eo Neegala o de Klausius it Teorema di Clausius pl Nier wno Clausiusa ru sv Clausius olikhet ...Lead rewrite date October 2010 Thermodynamics cTopic Thermodynamic equations Equations The Rudolf Clausius Clausius theorem 1855 states that in a cyclic process math oint frac delta Q T leq 0. math The equality holds in the reversible case ref http scienceworld.wolfram.com physics ClausiusTheorem.html Clausius theorem at Wolfram Research ref and the is in the irreversible case. The reversible case is used to introduce the state function entropy . This is because in cyclic process the variation of a state function is zero. Proof Image Proving Clausius Inequality.svg thumb 300px Proving Clausius Inequality Suppose a system absorbs heat math delta Q math at temperature math T math . Since the value of math frac delta Q T math does not depend on the details of how the heat is transferred, we can assume it is from a Carnot engine , which in turn absorbs heat math delta Q 0 math from a heat reservoir with constant temperature math T 0 math . According to the nature of Carnot cycle , math frac delta Q T frac delta Q 0 T 0 math math Rightarrow delta Q 0 T 0 frac delta Q T math Therefore in one cycle, the total heat absorbed from the reservoir is math Q 0 T 0 oint frac delta Q T math Since after a cycle, the system and the Carnot engine as a whole return to its initial status, the difference of the internal energy is zero. Thus according to First Law of Thermodynamics , math Q 0 Delta U ..., so math W total leq 0 math Combine all the above and we get Clausius inequality math oint frac delta ... more details
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- 29246 Clausius
Infobox planet minorplanet yes width 25em bgcolour FFFFC0 apsis name Clausius symbol image caption discovery yes discovery ref discoverer F. Borngen and L. D. Schmadel discovery site Tautenburg discovered September 2, 1992 designations yes mp name 29246 alt names 1992 RV named after Rudolf Clausius mp category orbit ref epoch May 14, 2008 aphelion 3.2046757 perihelion 2.2758841 semimajor eccentricity 0.1694702 period 1656.8791135 avg speed inclination 8.67760 asc node 15.99566 mean anomaly 148.43695 arg peri 1.69961 satellites physical characteristics yes dimensions mass density surface grav escape velocity sidereal day axial tilt pole ecliptic lat pole ecliptic lon albedo temperatures temp name1 mean temp 1 max temp 1 temp name2 max temp 2 spectral type abs magnitude 13.5 29246 Clausius 1992 RV is a Asteroid belt main belt asteroid discovered on September 2, 1992 by F. Borngen and L. D. Schmadel at Tautenburg . References Reflist External links http ssd.jpl.nasa.gov sbdb.cgi?sstr 29246 Clausius JPL Small Body Database Browser on 29246 Clausius MinorPlanets Navigator 29245 1992 PZ 29247 1992 RC4 MinorPlanets Footer DEFAULTSORT Clausius Category Main Belt asteroids Category Asteroids named for people Category Discoveries by Freimut B rngen Category Discoveries by Lutz D. Schmadel Category Astronomical objects discovered in 1992 beltasteroid stub fa it 29246 Clausius pl 29246 Clausius pt 29246 Clausius ... more details
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- Clausius (crater)
lunar crater data latitude 36.9 N or S S longitude 43.3 E or W W diameter 25 km depth 2.5 km colong 44 eponym Rudolf Clausius Rudolf J. E. Clausius Clausius is a moon lunar Impact crater crater that is located in the southwest part of the Moon , in the small lunar mare designated Lacus Excellentiae . It is completely enclosed by mare material, although the tiny satellite crater Clausius A lies just to the north. The rim of Clausius is low and sharp, with a slightly oval shape that is longer in the north south direction. The interior floor has been flooded by basalt ic lava , and appears level and featureless with a darker surface that matches the mare surface that surrounds the crater exterior. Satellite craters By convention these features are identified on lunar maps by placing the letter on the side of the crater mid point that is closest to Clausius. class wikitable width 25 style background eeeeee Clausius width 25 style background eeeeee Latitude width 25 style background eeeeee Longitude width 25 style background eeeeee Diameter align center A align center 36.3 S align center 43.9 W align center 7 km align center B align center 36.0 S align center 40.1 W align center 23 km align center BA align center 35.7 S align center 40.1 W align center 17 km align center C align center 35.4 S align center 38.9 W align center 15 km align center D align center 38.2 S align center 44.6 W align center 18 km align center E align center 36.4 S align center 45.5 W align center 6 km align center F align center 36.5 S align center 38.1 W align center 26 km align center G align center 37.1 S align center 41.0 W align center 6 km align center H align center 37.8 S align center 39.6 W align center 7 km align center J align center 37.2 S align center 42.7 W align center 4 km References Lunar crater references Category Impact craters on the Moon da Clausius m nekrater it Clausius cratere ... more details
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- Clausius?Duhem inequality
continuum mechanics cTopic Laws The Clausius Duhem inequality ref Citation last Truesdell first Clifford authorlink Clifford Truesdell year 1952 title The Mechanical foundations of elasticity and fluid dynamics journal Journal of Rational Mechanics and Analysis volume 1 issue pages 125 300 doi . ref ref Citation last Truesdell first Clifford lastauthoramp yes first2 Richard last2 Toupin year 1960 chapter The Classical Field Theories of Mechanics title Handbuch der Physik volume III location Berlin publisher Springer isbn . ref is a way of expressing the second law of thermodynamics that is used in continuum mechanics . This inequality is particularly useful in determining whether the constitutive relation of a material is thermodynamically allowable. ref Citation last Fr mond first M. year 2006 chapter The Clausius Duhem Inequality, an Interesting and Productive Inequality title Nonsmooth Mechanics and Analysis series Advances in mechanics and mathematics volume 12 pages 107 118 location New York publisher Springer doi 10.1007 0 387 29195 4 10 isbn 0387291962 . ref This inequality is a statement concerning the irreversibility of natural processes, especially when energy dissipation is involved. It was named after the German physicist Rudolf Clausius and French physicist Pierre Duhem . Clausius Duhem inequality in terms of the specific entropy The Clausius Duhem inequality can be expressed ... form the Clausius Duhem inequality can be written as math rho dot eta ge boldsymbol nabla ... q cdot mathbf n T text dA int Omega cfrac rho s T text dV . math Using the divergence theorem ... boldsymbol nabla cdot left cfrac mathbf q T right cfrac rho s T . math Clausius Duhem inequality in terms ... and angular momentum into the expression for the Clausius Duhem inequality. class toccolours ... in the Clausius Duhem inequality, we get math rho dot eta ge boldsymbol nabla cdot left cfrac mathbf ... . Hence the Clausius Duhem inequality is also called the dissipation inequality . In a real material ... more details
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- Clausius?Mossotti relation
Cleanup date July 2009 The Clausius Mossotti relation is named after the Italian physicist Ottaviano Fabrizio Mossotti , whose 1850 book analyzed the relationship between the dielectric constant s of two different media, and the German physicist Rudolf Clausius , who gave the formula explicitly in his 1879 book in the context not of dielectric constant s but of indices of refraction. The same formula also arises in the context of electrical conductivity conductivity , in which it is known as Maxwell s formula . It arises yet again in the context of refraction refractivity , in which it is known as the Lorentz Lorenz equation . Clausius Mossotti factor The Clausius Mossotti factor can be expressed in terms of complex permittivities math K omega frac epsilon p epsilon m epsilon p 2 epsilon m math math epsilon epsilon frac sigma i omega epsilon frac i sigma omega math where math epsilon math is the permittivity the subscript p refers to a lossless dielectric sphere suspended in a medium m ... of electrokinetic manipulation, the real part of the Clausius Mossotti factor is a determining ... of the particle to be manipulated and the electric field. Richard Feynman on the Clausius Mossotti ... the Clausius Mosotti Equation, in reference to the Index of Refraction for dense materials. He ... for a dense material, by taking into account the local fields of nearby atoms, you end up with the Clausius ... 0 201 50064 7 1989 commemorative hardcover three volume set ref Feynman states the Clausius Mosotti ... the dielectric constant to the polarizability, which is what the Clausius Mosotti equation or relation ... Clausius Mossotti Relation Category Electrodynamics Category Electromagnetism Category Equations Category Fundamental physics concepts ca Equaci de Clausius Mossoti de Clausius Mossotti Gleichung es Ecuaci n de Clausius Mossoti fa fr Formule de Clausius Mossotti it Equazione di Clausius Mossotti pt Equa o de Clausius Mossoti sl Clausius Mossottijeva zveza uk ... more details
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- Clausius?Clapeyron relation
The Clausius Clapeyron relation , named after Rudolf Clausius and Beno t Paul mile Clapeyron , who defined it sometime after 1834, is a way of characterizing a discontinuous phase transition between two phases of matter. On a pressure temperature P T diagram, the line separating the two phases is known as the coexistence curve. The Clausius Clapeyron relation gives the slope of this curve. Mathematically, math frac mathrm d P mathrm d T frac L T , Delta V math where math mathrm d P mathrm d T math is the slope of the coexistence curve, math L math is the latent heat , math T math is the temperature ... a less general form is sometimes called the Clausius Clapeyron equation. The less general form neglects ... green line gives the anomalous behavior of water. The Clausius Clapeyron relation can be used ... www.chem.arizona.edu salzmanr 480a 480ants clapeyro clapeyro.html title Clapeyron and Clausius Clapeyron ... to capital letters indicates a shift to extensive variable s. This last equation is called the Clausius ... to a version of the Clausius Clapeyron equation that is simpler to integrate ref name Wark1988 rp .... The Clausius Clapeyron equation is now obtained by inserting the molar volume of an ideal gas ... and chemical engineering The Clausius Clapeyron equation for the liquid vapor boundary may be used ... , math Delta H mathrm sub math Clausius Clapeyron equations is given for typical atmospheric conditions .... ISBN 0 471 86256 8 DEFAULTSORT Clausius Clapeyron Relation Category Thermodynamics Category Atmospheric thermodynamics Category Chemical engineering ca Equaci de Clausius Clapeyron cs Clausiusova Clapeyronova rovnice de Clausius Clapeyron Gleichung es Relaci n de Clausius Clapeyron eo Ekvacio de Clausius ... hu Clausius Clapeyron egyenlet nl Clausius Clapeyronvergelijking ja pl R wnanie Clausiusa Clapeyrona ru sl Clausius Clapeyronova ena ba sv Clausius Clapeyron ekvationen uk zh ... more details
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- Theorem
File Pythagorean Proof 3 .PNG thumb 200px right The Pythagorean theorem has at least 370 known proofs ... of Teachers of Mathematics. ref In mathematics , a theorem is a statement logic statement that has been ..., and previously accepted statements, such as axiom s. The derivation of a theorem is often interpreted ..., called the premise mathematics hypotheses and the conclusions. The proof of a mathematical theorem ... any further assumptions. The concept of a theorem is therefore fundamentally deductive , in contrast ... theorem from to investigate ref Although they can be written in a completely symbolic ... would express a preference for a proof that not only demonstrates the validity of a theorem ... to prove a theorem. Because theorems lie at the core of mathematics, they are also central to its ..., as a proof is simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a deep theorem may be simply stated, but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat s Last Theorem is a particularly well known example of such a theorem. Informal accounts of theorems Logically , many theorems are of the form of an indicative conditional if A, then B . Such a theorem does not state that B is always true, only that B must be true if A is true. In this case A is called the hypothesis of the theorem ... can also be denoted the antecedent and consequent . The theorem If n is an even natural number ... number and the conclusion is that n 2 is also a natural number . In order to be proven, a theorem ... with only four colors. The four color theorem states that such colorings are possible for any planar ... the statement of the theorem itself, or show surprising connections between disparate areas of mathematics. ref MathWorld title Deep Theorem urlname DeepTheorem ref A theorem might be simple to state and yet be deep. An excellent example is Fermat s Last Theorem, and there are many other examples of simple ... more details
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- Connectedness theorem
Deligne s connectedness theorem Fulton Hansen connectedness theorem Grothendieck s connectedness theorem Zariski s connectedness theorem Zariski s main theorem disambig ... more details
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- Dilation theorem
Dilation theorem may refer to Sz. Nagy s dilation theorem Stinespring factorization theorem Stinespring dilation theorem Naimark s dilation theorem disambig ... more details
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- Wiener's theorem
Wiener s theorem is any of several theorems named after Norbert Wiener Paley Wiener theorem Wiener s 1 f theorem Wiener s 1 &fnof theorem about functions with absolutely convergent Fourier series. Wiener Ikehara theorem Wiener Khinchin theorem Wiener s tauberian theorem Wiener Wintner theorem mathdab ... more details
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- Recursion theorem
Recursion theorem can refer to The Recursion recursion theorem in set theory Kleene s recursion theorem , also called the fixed point theorem, in computability theory disambig ... more details
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- Pappus's theorem
Pappus& 39 s theorem may refer to Pappus s centroid theorem Pappus s hexagon theorem mathdab ... more details
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- Remainder theorem
PAGENAME may refer to Polynomial remainder theorem Chinese remainder theorem B zout s theorem Disambig ... more details
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- Bombieri's theorem
Bombieri Vinogradov theorem Schneider Lang theorem for Bombieri s theorem on transcendental numbers mathdab ... more details
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- Bing's theorem
Bing s theorem may refer to Bing s recognition theorem Bing metrization theorem See also R. H. Bing mathdab ... more details
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- Morley's theorem
Morley s theorem may refer to Morley s trisector theorem , a theorem related to geometry, discovered by Frank Morley Morley s categoricity theorem , a theorem related to model theory, discovered by Michael D. Morley disambig ko ... more details
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- Density theorem
In mathemtics, density theorem may refer to Density conjecture for Kleinian groups Chebotarev s density theorem in algebraic number theory Jacobson density theorem in algebra Kaplansky density theorem in algebra Lebesgue s density theorem mathdab ... more details
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- Rado's theorem
In mathematics, Rado s theorem or Rad s theorem may refer to Tibor Rad s theorem harmonic functions Tibor Rad s theorem Riemann surfaces Richard Rado s theorem Ramsey theory mathdab fr Th or me de Rado ... more details
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- No-go theorem
Unreferenced stub auto yes date December 2009 In theoretical physics , a no go theorem is a theorem that states that a particular situation is not physically possible. Examples of no go theorems Bell s theorem Coleman Mandula theorem Haag Lopuszanski Sohnius theorem Earnshaw s theorem Uncertainty principle Heisenberg uncertainty principle Weinberg Witten theorem Antidynamo theorems e.g. Cowling s Theorem No communication theorem No teleportation theorem No cloning theorem No broadcast theorem DEFAULTSORT No Go Theorem Category Quantum field theory Category Supersymmetry Physics stub es Teorema de imposibilidad pt Teorema de impossibilidade sq Teorem pamund sie zh No go ... more details
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- Limit theorem
Limit theorem may refer to Central limit theorem , in probability theory Edgeworth s limit theorem , in economics disambig Category Mathematical disambiguation ... more details
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- Mason's theorem
disambig Mason s theorem may refer to either of the following The Mason Stothers theorem , a mathematics mathematical theorem about polynomial s Mason s rule ... more details
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- Skorokhod's theorem
Skorokhod s theorem may refer to Skorokhod s embedding theorem Skorokhod s representation theorem disambig Long comment to avoid being listed on short pages ... more details
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- Hasse's theorem
In mathematics , three different theorems of Helmut Hasse are sometimes called Hasse s theorem The Hasse norm theorem Hasse s theorem on elliptic curves The Hasse Minkowski theorem category Mathematical theorems disambig ... more details
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- Toy theorem
In mathematics , a toy theorem is a simplified version of a more general theorem . For instance, by introducing some simplifying assumptions in a theorem, one obtains a toy theorem. Usually, a toy theorem is used to illustrate the claim of a theorem. It can also be insightful to study proofs of a toy theorem derived from a non trivial theorem. Toy theorems can also have education value. After presenting a theorem with, say, a highly non trivial proof , one can sometimes give some assurance that the theorem really holds, by proving a toy version of the theorem. For instance, a toy theorem of the Brouwer fixed point theorem is obtained by restricting the dimension to one. In this case, the Brouwer fixed point theorem follows almost immediately from the intermediate value theorem . See also toy model lemma mathematics planetmath id 4684 title toy theorem Category Mathematical theorems Category Mathematical terminology eo Luda teoremo ... more details
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- Area theorem
For Hawking s area theorem, see Black hole thermodynamics The laws of black hole mechanics . For the area theorem in conformal mapping theory, see area theorem conformal mapping . Area theorem conformal mapping disambig ... more details
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