text stringlengths 2 4.67k | source dict |
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\{(\mathbf {a} _{i},\mathbf {b} _{i}+\langle \mathbf {a} _{i},\mathbf {t} \rangle )/q\}} and feed these samples to A {\displaystyle {\mathcal {A}}} . Since S {\displaystyle {\mathcal {S}}} comprises a large fraction of Z q n {\displaystyle \mathbb {Z} _{q}^{n}} , with high probability, if we choose a polynomial number ... | {
"page_id": 23864530,
"title": "Learning with errors"
} |
ϕ {\displaystyle DGS_{\phi }} is given by an n {\displaystyle n} -dimensional lattice L {\displaystyle L} and a number r ≥ ϕ ( L ) {\displaystyle r\geq \phi (L)} . The goal is to output a sample from D L , r {\displaystyle D_{L,r}} . Regev shows that there is a reduction from GapSVP 100 n γ ( n ) {\displaystyle \operat... | {
"page_id": 23864530,
"title": "Learning with errors"
} |
(n)\geq n/(\alpha {\sqrt {\log n}})} , ζ ( n ) ≥ γ ( n ) {\displaystyle \zeta (n)\geq \gamma (n)} and q ≥ ( ζ / n ) ω log n ) {\displaystyle q\geq (\zeta /{\sqrt {n}})\omega {\sqrt {\log n}})} . == Use in cryptography == The LWE problem serves as a versatile problem used in construction of several cryptosystems. In 2... | {
"page_id": 23864530,
"title": "Learning with errors"
} |
( n ) {\displaystyle \chi =\Psi _{\alpha (n)}} for α ( n ) ∈ o ( 1 / n log n ) {\displaystyle \alpha (n)\in o(1/{\sqrt {n}}\log n)} , where Ψ β {\displaystyle \Psi _{\beta }} is a probability distribution obtained by sampling a normal variable with mean 0 {\displaystyle 0} and standard variation β 2 π {\displaystyle ... | {
"page_id": 23864530,
"title": "Learning with errors"
} |
{s} \rangle /q} is closer to 0 {\displaystyle 0} than to 1 2 {\displaystyle {\frac {1}{2}}} , and 1 {\displaystyle 1} otherwise. The proof of correctness follows from choice of parameters and some probability analysis. The proof of security is by reduction to the decision version of LWE: an algorithm for distinguishing... | {
"page_id": 23864530,
"title": "Learning with errors"
} |
papers laid the groundwork for a variety of recent signature algorithms some based directly on the ring learning with errors problem and some which are not tied to the same hard RLWE problems. == See also == Post-quantum cryptography Ring learning with errors Lattice-based cryptography Ring learning with errors key exc... | {
"page_id": 23864530,
"title": "Learning with errors"
} |
In genetic engineering, floxing refers to the insertion of a DNA sequence (which is then said to be floxed) between two LoxP sequences, creating an artificial gene cassette which can then be conditionally deleted (knocked out), translocated, or inverted in a process called Cre-Lox recombination. Recombination between L... | {
"page_id": 30483666,
"title": "Floxing"
} |
of the segment of interest with loxP sites which face the same direction. The Cre recombinase will detect the unidirectional loxP sites and excise the floxed segment of DNA. The successfully edited clones can be selected using a selection marker which can be removed using the same Cre-LoxP system. The same mechanism ca... | {
"page_id": 30483666,
"title": "Floxing"
} |
These recombinations are capable of disrupting genes in a manner that is specific to heart tissue in vivo and allows for the creation of conditional knockouts of the heart, mostly for use as controls. For example, using the Cre recombinase with the α {\displaystyle \alpha } -MyHC promoter causes the floxed gene to be i... | {
"page_id": 30483666,
"title": "Floxing"
} |
ACS Energy Letters is a monthly peer-reviewed scientific journal published by the American Chemical Society. It was established in 2016 and the editor-in-chief is Prashant V. Kamat (University of Notre Dame). It covers research on all aspects of energy and aims for rapid publication. == Abstracting and indexing == The ... | {
"page_id": 70919377,
"title": "ACS Energy Letters"
} |
SIMION is an ion optics simulation program that calculates electric fields for electrodes of defined voltages and ion trajectories in those fields. == Program development == The program was developed in the late 1970s by Don C. McGilvery at La Trobe University, Melbourne, Australia as part of his Ph.D. research working... | {
"page_id": 20718802,
"title": "SIMION"
} |
the trajectories of charged particles in these fields can be calculated. Particle trajectory calculations are a result of three interdependent computations. First, electrostatic forces must be calculated at the current position of the ion. These forces are then used to compute the current ion acceleration and then by n... | {
"page_id": 20718802,
"title": "SIMION"
} |
In chemistry, the capped trigonal prismatic molecular geometry describes the shape of compounds where seven atoms or groups of atoms or ligands are arranged around a central atom defining the vertices of an augmented triangular prism. This shape has C2v symmetry and is one of the three common shapes for heptacoordinate... | {
"page_id": 58467542,
"title": "Capped trigonal prismatic molecular geometry"
} |
In electromagnetism), Gauss's law, also known as Gauss's flux theorem or sometimes Gauss's theorem, is one of Maxwell's equations. It is an application of the divergence theorem, and it relates the distribution of electric charge to the resulting electric field. == Definition == In its integral form, it states that the... | {
"page_id": 74964,
"title": "Gauss's law"
} |
of gravity, both of which are inverse-square laws. The law can be expressed mathematically using vector calculus in integral form and differential form; both are equivalent since they are related by the divergence theorem, also called Gauss's theorem. Each of these forms in turn can also be expressed two ways: In terms... | {
"page_id": 74964,
"title": "Gauss's law"
} |
components of the electromagnetic tensor; g {\displaystyle g} is the determinant of metric tensor; d S κ = d S i j = d x i d x j {\displaystyle \mathrm {d} S_{\kappa }=\mathrm {d} S^{ij}=\mathrm {d} x^{i}\mathrm {d} x^{j}} is an orthonormal element of the two-dimensional surface surrounding the charge Q {\displaystyle ... | {
"page_id": 74964,
"title": "Gauss's law"
} |
symmetry. See the article Gaussian surface for examples where these symmetries are exploited to compute electric fields. === Differential form === By the divergence theorem, Gauss's law can alternatively be written in the differential form: ∇ ⋅ E = ρ ε 0 {\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsil... | {
"page_id": 74964,
"title": "Gauss's law"
} |
form: Φ D = Q f r e e {\displaystyle \Phi _{D}=Q_{\mathrm {free} }} where ΦD is the D-field flux through a surface S which encloses a volume V, and Qfree is the free charge contained in V. The flux ΦD is defined analogously to the flux ΦE of the electric field E through S: Φ D = {\displaystyle \Phi _{D}=} S {\displayst... | {
"page_id": 74964,
"title": "Gauss's law"
} |
the electric field obeys the superposition principle. The superposition principle states that the resulting field is the vector sum of fields generated by each particle (or the integral, if the charges are distributed smoothly in space). Since Coulomb's law only applies to stationary charges, there is no reason to expe... | {
"page_id": 74964,
"title": "Gauss's law"
} |
Hydroids are a life stage for most animals of the class Hydrozoa, small predators related to jellyfish. Some hydroids such as the freshwater Hydra are solitary, with the polyp attached directly to the substrate. When these produce buds, they become detached and grow on as new individuals. The majority of hydroids are c... | {
"page_id": 37954776,
"title": "Hydroid (zoology)"
} |
The molecular formula C13H11NO3 (molar mass: 229.23 g/mol, exact mass: 229.073894 u) may refer to: γ-Fagarine Phenopicolinic acid | {
"page_id": 34874584,
"title": "C13H11NO3"
} |
The tables below provides information on the variation of solubility of different substances (mostly inorganic compounds) in water with temperature, at one atmosphere pressure. Units of solubility are given in grams of substance per 100 millilitres of water (g/100 ml), unless shown otherwise. The substances are listed ... | {
"page_id": 5973210,
"title": "Solubility table"
} |
RECODE is a database of "programmed" frameshifts, bypassing and codon redefinition used for gene expression. == See also == Translational frameshift == References == == External links == http://recode.ucc.ie/ | {
"page_id": 30483675,
"title": "Recode (database)"
} |
A cholecystocyte is an epithelial cell found in the gallbladder tightly joined by junctional complexes. == See also == List of human cell types derived from the germ layers == References == | {
"page_id": 31859934,
"title": "Cholecystocyte"
} |
The molecular formula C17H17N3O2 (molar mass: 295.336 g/mol, exact mass: 295.1321 u) may refer to: Divaplon (RU-32698) GYKI-52895 | {
"page_id": 61482208,
"title": "C17H17N3O2"
} |
A constrained conditional model (CCM) is a machine learning and inference framework that augments the learning of conditional (probabilistic or discriminative) models with declarative constraints. The constraint can be used as a way to incorporate expressive prior knowledge into the model and bias the assignments made ... | {
"page_id": 28255458,
"title": "Constrained conditional model"
} |
making use of multiple pre-learned components, such as summarization, textual entailment and question answering. In all these cases, it is natural to formulate the decision problem as a constrained optimization problem, with an objective function that is composed of learned models, subject to domain- or problem-specifi... | {
"page_id": 28255458,
"title": "Constrained conditional model"
} |
the penalty incurred for violating the constraints. Constraints assigned an infinite penalty are known as hard constraints, and represent unfeasible assignments to the optimization problem. == Training paradigms == === Learning local vs. global models === The objective function used by CCMs can be decomposed and learne... | {
"page_id": 28255458,
"title": "Constrained conditional model"
} |
representation decision is available to the learner. Identifying the correct (or optimal) learning representation is viewed as a structured prediction process and therefore modeled as a CCM. This problem was covered in several papers, in both supervised and unsupervised settings. In all cases research showed that expli... | {
"page_id": 28255458,
"title": "Constrained conditional model"
} |
The Cuvier–Geoffroy debate of 1830 was a scientific debate between the two French naturalists Georges Cuvier and Étienne Geoffroy Saint-Hilaire. For around two months the debate occurred in front of the French Academy of Sciences. The debate centered primarily on animal structure; Cuvier asserted that animal structure ... | {
"page_id": 38872295,
"title": "Cuvier–Geoffroy debate"
} |
to accompany Napoleon on an expedition to Egypt in 1798 where he studied mummies and made hypotheses on change over time in humans and other organisms. Geoffroy agreed with Buffon that any classification system was arbitrary and thus somewhat empty, but he nonetheless attempted to find the general laws that applied to ... | {
"page_id": 38872295,
"title": "Cuvier–Geoffroy debate"
} |
original members of anatomy and zoology. Cuvier was only twenty-six years old (three years Geoffroy's junior) and the youngest member at the time, while Geoffroy was not given admittance to the Academy for another twelve years. == Differing viewpoints == Prior to the debate, biological scientists were generally split i... | {
"page_id": 38872295,
"title": "Cuvier–Geoffroy debate"
} |
animals be referred to as one uniform type?" Cuvier on the other hand was known as Lamarck's greatest critic, as Cuvier felt Lamarck was too speculative without enough facts to back up his theories. Cuvier grew up reading the works of Linnaeus and was mostly self-taught through dissection and measurements of organisms.... | {
"page_id": 38872295,
"title": "Cuvier–Geoffroy debate"
} |
1817, "Memoir on the Cephalopods and on Their Anatomy", though the author and actual manuscript name were not mentioned. Geoffroy ridiculed the old-fashioned way of describing nature by focusing on differences instead of similarities, presenting his unity of composition theory as a better alternative. Cuvier disagreed ... | {
"page_id": 38872295,
"title": "Cuvier–Geoffroy debate"
} |
Practical Utility as an Instrument". In response to Cuvier's comments about unity of composition being poorly defined, Geoffroy claimed that he was seeking more "philosophical resemblances" other than actual, observable similarities between animals. This vague explanation of unity of composition and the claim that othe... | {
"page_id": 38872295,
"title": "Cuvier–Geoffroy debate"
} |
continuous opportunity to bash each other publicly and academically. Geoffroy finally declined to provide further comment on April 5, 1830. His reasons for ending the debate are not entirely known, but it's generally believed Cuvier won the debate with his commanding presence and overwhelming amounts of evidence. == Re... | {
"page_id": 38872295,
"title": "Cuvier–Geoffroy debate"
} |
for laying the foundation with his genius and massive knowledge of the natural world. While neither Geoffroy or Cuvier's ideas were fully adopted in future science theory, both unity of composition and functional morphology can be seen as influences to further works on the natural world. Geoffroy's ideas of unity of co... | {
"page_id": 38872295,
"title": "Cuvier–Geoffroy debate"
} |
Human evolutionary developmental biology or informally human evo-devo is the human-specific subset of evolutionary developmental biology. Evolutionary developmental biology is the study of the evolution of developmental processes across different organisms. It is utilized within multiple disciplines, primarily evolutio... | {
"page_id": 52962536,
"title": "Human evolutionary developmental biology"
} |
H. erectus either matching or exceeding H. erectus. He makes the case that this finding had wide application and relevance to the overall study of human evolution. It is pertinent specifically to the connections between energy expenditure and brain development. These finding are of specific utility in studies on matern... | {
"page_id": 52962536,
"title": "Human evolutionary developmental biology"
} |
emotional and communicative nature of humans. He credits this factor as an integral facet of human evolution. However, there have also been cautions against the application of this aspect to group ranking during it inappropriate as a measure of evolutionary achievement. === Fossil record === Early comparative and human... | {
"page_id": 52962536,
"title": "Human evolutionary developmental biology"
} |
paths and presentations suggest significant environmental influence. They focus on plasticity between stages of development and the factors that shape it. Rate of maturation, fecundity, and fertility were all impacted by environmental circumstances. They argue that early maturation can be positive, reflecting opportuni... | {
"page_id": 52962536,
"title": "Human evolutionary developmental biology"
} |
Chomsky, who argued against language as a human specific adaptation. Adaptation and adaptive theory has been argued even separate from its utility in the study of language. Gould and Lewontin engage with what they saw as flaws in adaptive theory using the analogy of the spandrels of San Marco. Among the issues identifi... | {
"page_id": 52962536,
"title": "Human evolutionary developmental biology"
} |
Plasma cells, also called plasma B cells or effector B cells, are white blood cells that originate in the lymphoid organs as B cells and secrete large quantities of proteins called antibodies in response to being presented specific substances called antigens. These antibodies are transported from the plasma cells by th... | {
"page_id": 926954,
"title": "Plasma cell"
} |
considerably more stable. == Development == After leaving the bone marrow, the B cell acts as an antigen-presenting cell (APC) and internalizes offending antigens, which are taken up by the B cell through receptor-mediated endocytosis and processed. Pieces of the antigen (which are now known as antigenic peptides) are ... | {
"page_id": 926954,
"title": "Plasma cell"
} |
die or irrevocably differentiate into a mature, fully differentiated plasma cell. Differentiation of mature B cells into plasma cells is dependent upon the transcription factors Blimp-1/PRDM1, BCL6, and IRF4. == Function == Unlike their precursors, plasma cells cannot switch antibody classes, cannot act as antigen-pres... | {
"page_id": 926954,
"title": "Plasma cell"
} |
produce several thousand matching antibodies per second. This prolific production of antibodies is an integral part of the humoral immune response. === Long-lived plasma cells === The current findings suggest that after the process of affinity maturation in germinal centers, plasma cells develop into one of two types o... | {
"page_id": 926954,
"title": "Plasma cell"
} |
in nasal-associated lymphoid tissues (NALT), human tonsillar lymphoid tissues and human mucosa or mucosa-associated lymphoid tissues (MALT). Originally it was thought that the continuous production of antibodies is a result of constant replenishment of short-lived plasma cells by memory B cell re-stimulation. Recent fi... | {
"page_id": 926954,
"title": "Plasma cell"
} |
– Histology Learning System at Boston University Histology at wadsworth.org | {
"page_id": 926954,
"title": "Plasma cell"
} |
The Five Freedoms outline five aspects of animal welfare. They were developed in response to a 1965 UK Government report on livestock husbandry, and were formalised in 1979 press statement by the UK Farm Animal Welfare Council. They have been adopted internationally and endorsed by professional groups including veterin... | {
"page_id": 33629418,
"title": "Five freedoms"
} |
such as New Zealand, have proposed going further than the five freedoms to "five domains" to include a focus on the mental health of the animal following New Zealand becoming the fifth country to legally recognize animal sentience in 2015. As of 2021, animal sentience was also recognized in Chile, Colombia, Peru, Tanza... | {
"page_id": 33629418,
"title": "Five freedoms"
} |
This is a list of prices of chemical elements. Listed here are mainly average market prices for bulk trade of commodities. Data on elements' abundance in Earth's crust is added for comparison. As of 2020, the most expensive non-synthetic element by both mass and volume is rhodium. It is followed by caesium, iridium and... | {
"page_id": 13772012,
"title": "Prices of chemical elements"
} |
Resorcylic acid lactones are a group of estrogenic compounds. They are lactones of resorcylic acid. Examples include the mycoestrogens (and synthetic analogues) zearalenone, zearalanone, zeranol (α-zearalanol), taleranol (β-zearalanol), α-zearalenol, and β-zearalenol. == References == | {
"page_id": 50734317,
"title": "Resorcylic acid lactone"
} |
Egon Orowan FRS (Hungarian: Orován Egon) (2 August 1902 – 3 August 1989) was a Hungarian-British physicist and metallurgist. He was key in introducing crystal dislocation into physics and understanding of how materials plastically deform under stress. According to György Marx, he was one of The Martians, a group of Jew... | {
"page_id": 1385708,
"title": "Egon Orowan"
} |
discovery was neglected until after World War II, it was critical in developing the modern science of solid mechanics. In Hungary, he seemed to have experienced some difficulty in finding immediate employment and spent the next few years living with his mother and ruminating on his doctoral research. From 1936 to 1939,... | {
"page_id": 1385708,
"title": "Egon Orowan"
} |
in Western society, coming up with the term "socionomy". He also studied the Arab historian Ibn-Khaldun. == Honors == Fellow of the Royal Society (1947) Member of the American Academy of Arts and Sciences (1951) Bingham Medal of the Society of Rheology (1959) Honorary doctor of engineering degree from Technische Univer... | {
"page_id": 1385708,
"title": "Egon Orowan"
} |
Congress of Applied Mechanics. New York: ASME, 1953. pp. 453– 472. "Condition of High Velocity Ductile Fracture". Journal of Applied Physics, vol. 26 (1955): 900–902. "Continental Drift and the Origin of Mountains". Science, vol. 146 (1964): 1003–1010. "Dilatancy and the Seismic Focal Mechanism". Reviews of Geophysics,... | {
"page_id": 1385708,
"title": "Egon Orowan"
} |
The van der Waals equation is a mathematical formula that describes the behavior of real gases. It is an equation of state that relates the pressure, volume, number of molecules, and temperature in a fluid. The equation modifies the ideal gas law in two ways: first, it considers particles to have a finite diameter (whe... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
der Waals wrote it in an equivalent, explicit in temperature, form in his Thesis (although he could not denote absolute temperature by its modern form in 1873) This form is useful for plotting isobars (constant pressure curves). Writing v = V / n {\displaystyle v=V/n} , and multiplying both sides by n {\displaystyle n}... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
characterized by a single length, σ {\displaystyle \sigma } , and a minimum energy, − ε {\displaystyle -\varepsilon } (with ε ≥ 0 {\displaystyle \varepsilon \geq 0} ). Two of the many such functions that have been suggested are shown in Fig. B. A modern theory based on statistical mechanics produces the same resu... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
, or p v = R T {\displaystyle pv=RT} . With R = N A k {\displaystyle R=N_{\text{A}}k} , where k {\displaystyle k} is the Boltzmann constant, and using the definition v = V / n {\displaystyle v=V/n} given after Eq (1a), this becomes p V = N k T {\displaystyle pV=NkT} ; either of these forms expresses the ideal gas l... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
composed of particles in motion, with number density N / V {\displaystyle N/V} , mass m {\displaystyle m} , and mean square speed c 2 ¯ {\displaystyle {\overline {c^{2}}}} . He then noted that using the classical laws of Boyle and Charles, one could write m c 2 ¯ / 3 = k T {\displaystyle m{\overline {c^{2}}}/3... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
at low temperatures a jump in density at a certain pressure, while at higher temperatures there was no abrupt change (the figure can be seen here). Andrews called the isotherm at which the jump disappears the critical point. Given the similarity of the titles of this paper and van der Waals' subsequent thesis, one migh... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
seen the liquid phase. This faith arose out of the repeated success of the van der Waals theory, which is essentially a universal equation of state, independent of the details of any particular substance once it has been properly scaled. [...] As a result, not only was it possible to believe that hydrogen could be liqu... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
corresponding states == Figure 1 shows four isotherms of the van der Waals equation (abbreviated as vdW) on a p , v {\displaystyle p,v} (pressure, molar volume) plane. The essential character of these curves is that they come in three forms: At some critical temperature T c {\displaystyle T_{\text{c}}} (orange isotherm... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
/ p c ) v 2 + ( a / p c ) v − a b / p c = 0 {\displaystyle v^{3}-(b+RT_{\text{c}}/p_{\text{c}})v^{2}+(a/p_{\text{c}})v-ab/p_{\text{c}}=0} , or ( v − v c ) 3 = 0 {\displaystyle (v-v_{\text{c}})^{3}=0} ; two forms with the same coefficients. === Course of the isotherms === Above the critical temperature T c {\displaystyl... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
also a spinodal point. === Principle of corresponding states === Using the critical values to define reduced (dimensionless) variables p r = p / p c {\displaystyle p_{r}=p/p_{\text{c}}} , T r = T / T c {\displaystyle T_{r}=T/T_{\text{c}}} , and v r = v / v c {\displaystyle v_{r}=v/v_{\text{c}}} renders the vdW eq... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
3 = 3 dimensionless groups. Here v ∗ = b {\displaystyle v^{*}=b} is a characteristic molar volume, p ∗ = a / b 2 {\displaystyle p^{*}=a/b^{2}} a characteristic pressure, and T ∗ = a / ( R b ) {\displaystyle T^{*}=a/(Rb)} a characteristic temperature, and the 3 dimensionless groups are p / p ∗ , v / v ∗ , T / T ∗ {\di... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
on the Pitzer (acentric) factor, ω {\displaystyle \omega } , and the liquid metals (mercury and cesium) are well approximated by it. == Thermodynamic properties == The properties molar internal energy, u {\displaystyle u} , and entropy, s {\displaystyle s} , are defined by the first and second laws of thermodynamics... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
=== Internal energy, and entropy === The energetic equation of state gives the internal energy, and the entropic equation of state gives the entropy as u − C u = ∫ c v ( v , T ) d T + ∫ T 2 ∂ ( p / T ) ∂ T d v s − C s = ∫ c v ( T ) d T T + ∫ ∂ p ∂ T d v {\displaystyle {\begin{aligned}u-C_{u}&=\int c_{v}(v,T)\,dT+\int T... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
thermodynamic property can be expressed in terms of any three of them. A standard set is composed of α , κ T , c v {\displaystyle \alpha ,\kappa _{T},c_{v}} . For a vdW fluid c v ( T ) {\displaystyle c_{v}(T)} is a known function, and the other two are obtained from the first partial derivatives of the vdW equation as,... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
T {\displaystyle c_{p}-c_{v}=-T(\partial _{T}p)^{2}/\partial _{v}p=Tv\alpha ^{2}/\kappa _{T}} . Then the two partials of the vdW equation can be used to express c p {\displaystyle c_{p}} as, Here in the limit v → ∞ {\displaystyle v\to \infty } , c p − c v = R {\displaystyle c_{p}-c_{v}=R} , which is also the ideal gas ... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
v f {\displaystyle v_{\text{f}}} and vapor v g {\displaystyle v_{\text{g}}} , which lie outside the range of v {\displaystyle v} set by the spinodal points ( v f < v min {\displaystyle v_{\text{f}}<v_{\text{min}}} and v g > v max {\displaystyle v_{\text{g}}>v_{\text{max}}} ). Applying Eq (1a) to the saturated liquid an... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
= y cosh y − sinh y sinh y cosh y − y , g ( y ) = 1 + 2 f ( y ) cosh y + f ( y ) 2 {\displaystyle {\begin{aligned}f(y)&={\frac {y\cosh y-\sinh y}{\sinh y\cosh y-y}},&g(y)&=1+2f(y)\cosh y+f(y)^{2}\end{aligned}}} The parameter 0 ≤ y < ∞ {\displaystyle 0\leq y<\infty } is given physically by y = ( s g − s f ) / ... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
Waals equation in great detail, Boltzmann discussed these metastable states in a section titled "Undercooling, Delayed evaporation". (Today, these states are now denoted "subcooled vapor" and "superheated liquid".) Moreover, it has now become clear that these metastable states occur regularly in the phase transition pr... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
horizontal line bridging the discontinuous region of an isotherm (not shown in Fig. 1 because it is a different equation from the vdW equation). === Extended corresponding states === The idea of corresponding states originated when van der Waals cast his equation in the dimensionless form, p r = p ( v r , T r ) {\displ... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
r 3 + 3.68769 T r 6 + 17.92998 ln T r ) {\displaystyle {\begin{aligned}\ln p_{\text{rs}}=5.37270(1-1/T_{\text{r}})+\omega (&7.49408-11.181777\ {T_{\text{r}}}^{3}+\\&3.68769\ {T_{\text{r}}}^{6}+17.92998\,\ln T_{\text{r}})\end{aligned}}} which has an RMS error of ± 0.42 {\displaystyle \pm 0.42} over the range 1 ≤ T r ≤... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
ideal gases, for which h = h ( T ) {\displaystyle h=h(T)} , do not change temperature in such a process, real gases do, and it is important in applications to know whether they heat up or cool down. This coefficient can be found in terms of the previously derived α {\displaystyle \alpha } and c p {\displaystyle c_{p}} ... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
p p ∗ = − 1 + 4 ( T 2 T ∗ ) 1 / 2 − 3 ( T 2 T ∗ ) {\displaystyle {\frac {p}{p^{*}}}=-1+4\left({\frac {T}{2T^{*}}}\right)^{1/2}-3\left({\frac {T}{2T^{*}}}\right)} where, for simplicity, a , b , R {\displaystyle a,b,R} have been replaced by p ∗ , T ∗ {\displaystyle p^{*},T^{*}} . A plot of the curve, in reduced variables... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
{9\rho _{r}}{8T_{r}}}} where 0 ≤ ρ r = 1 / v r ≤ 3 {\displaystyle 0\leq \rho _{r}=1/v_{r}\leq 3} . At the critical point, T r = ρ r = 1 {\displaystyle T_{r}=\rho _{r}=1} and Z = Z c = 3 / 2 − 9 / 8 = 3 / 8 {\displaystyle Z=Z_{\text{c}}=3/2-9/8=3/8} . In the limit ρ → 0 {\displaystyle \rho \rightarrow 0} , Z = 1 {\displ... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
where the equation of state reduces to the ideal gas law. However, the fluid does not behave like an ideal gas there, because neither its derivatives ( α , κ T ) {\displaystyle (\alpha ,\kappa _{T})} nor c p {\displaystyle c_{p}} reduce to their ideal gas values, other than where b ρ B ≪ 1 , T B ∼ 27 T c / 8 {\displays... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
Z {\displaystyle Z} , Eq (9), into an infinite series, absolutely convergent for b ρ < 1 {\displaystyle b\rho <1} , produces Z ( ρ , T ) = 1 + ( 1 − a b R T ) b ρ + ∑ k = 3 ∞ ( b ρ ) k − 1 . {\displaystyle Z(\rho ,T)=1+\left(1-{a \over bRT}\right)b\rho +\sum _{k=3}^{\infty }(b\rho )^{k-1}.} The second virial coefficien... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
an enhanced version of the vdW equation: ( p + a v 2 ) ( v − b 3 ) = R T ( 1 + 2 b 3 v + 7 b 2 24 v 2 ) . {\displaystyle \left(p+{a \over v^{2}}\right)\left(v-{b \over 3}\right)=RT\left(1+{2b \over 3v}+{{7b^{2}} \over {24v^{2}}}\right).} On expanding ( v − b / 3 ) − 1 {\displaystyle (v-b/3)^{-1}} , this produced the co... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
/ N {\displaystyle x_{1}=N_{1}/N} and x 2 = N 2 / N {\displaystyle x_{2}=N_{2}/N} , with N = N 1 + N 2 {\displaystyle N=N_{1}+N_{2}} (so that x 1 + x 2 = 1 {\displaystyle x_{1}+x_{2}=1} ), are the mole fractions of the two fluid substances. Adding the equations for the two fluids shows that p ≠ p 1 + p 2 {\displaystyle... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
agreement with experimental results at high pressure. === Common tangent construction === In this article, van der Waals used the Helmholtz potential minimum principle to establish stability conditions. This principle states that in a system in diathermal contact with a heat reservoir T = T R {\displaystyle T=T_{R}} , ... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
curvature everywhere when T R ≥ T c {\displaystyle T_{R}\geq T_{\text{c}}} , the curve and the straight line will have a single tangent. However, for a subcritical T R , f ( T R , v ) {\displaystyle T_{R},\,f(T_{R},v)} is not everywhere convex. With p = p s ( T R ) {\displaystyle p=p_{\text{s}}(T_{R})} and a suitable v... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
in which the fluid cannot exist in a homogeneous equilibrium state (represented by the dotted grey curve). With a further increase in v {\displaystyle v} the curvature increases to a maximum at v g {\displaystyle v_{g}} , where the slope is p s {\displaystyle p_{\text{s}}} ; the region between this point and the second... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
the vdW equation, for the subcritical isotherm T r = 7 / 8 {\displaystyle T_{\text{r}}=7/8} is shown in Figure 8 along with the line tangent to it at its two coexisting saturation points. The data illustrated in Figure 8 is the same as that shown in Figure 1 for this isotherm. This double tangent construction thus prov... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
2 − ∂ 2 f ∂ x ∂ v = 0 {\displaystyle {\frac {\partial ^{2}f}{\partial v^{2}}}{\frac {\partial ^{2}f}{\partial x^{2}}}-{\frac {\partial ^{2}f}{\partial x\partial v}}=0} specifies the spinodal curves on the surface. For a binary mixture, the Euler equation can be written in the form f = − p v + μ 1 x 1 + μ 2 x 2 = − p v ... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
complex. The surface can develop a wave (called a plait or fold) in the x {\displaystyle x} direction as well as the one in the v {\displaystyle v} direction. Therefore, there can be two liquid phases that can be either miscible, or wholly or partially immiscible, as well as a vapor phase. Despite a great deal of both ... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
1 / 3 ) / 2. {\displaystyle a_{12}=(a_{11}a_{22})^{1/2}\qquad {\text{and}}\qquad b_{12}^{1/3}=(b_{11}^{1/3}+b_{22}^{1/3})/2.} These relations correspond to the empirical combining rules for the intermolecular force constants, ϵ 12 = ( ϵ 11 ϵ 22 ) 1 / 2 and σ 12 = ( σ 11 + σ 22 ) / 2 , {\displaystyle \epsilon _{12}=(\ep... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
1 n p c i x i . {\displaystyle T_{{\text{c}}x}=\sum _{i=1}^{n}T_{{\text{c}}i}x_{i}\qquad {\text{and}}\qquad p_{{\text{c}}x}=\sum _{i=1}^{n}\,p_{{\text{c}}i}x_{i}.} In terms of these quantities, the vdW mixture constants are a x = ( 3 4 ) 3 ( R T c x ) 2 p c x , b x = ( 1 2 ) 3 R T c x p c x {\displaystyle a_{x}=\left({... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
of this approximation, "It was, and indeed still is, hard to improve on the original van der Waals recipe when expressed in [this] form". == Validity of the equation == Since van der Waals presented his thesis, "[m]any derivations, pseudo-derivations, and plausibility arguments have been given" for it. However, no math... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
success. In fact, its success has been so great that not a single other model of the many proposed since, has equalled its quantitative predictions, let alone its simplicity." Engineers have made extensive use of this empirical validity, modifying the equation in numerous ways (by one account there have been some 400 c... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
States Correlation of Saturated and Metastable Properties". Can J Chem Eng. 64: 158–161. doi:10.1002/cjce.5450640123. Epstein, P.S. (1937). Textbook of Thermodynamics. NY: John Wiley and Sons. Gibbs, J.W. (1948) [1901]. The Collected Works of J. Willard Gibbs Volume II Part One Elementary Principles in Statistical Mech... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
the Van der Waals–Maxwell Theory of the Liquid–Vapor Transition". Jour Math Phys. 7 (1): 98–113. Bibcode:1966JMP.....7...98L. doi:10.1063/1.1704821. Lekner, J. (1982). "Parametric solution of the van der Waals liquid–vapor coexistence curve". Am. J. Phys. 50 (2): 161–163. Bibcode:1982AmJPh..50..161L. doi:10.1119/1.1287... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
incompatibility (help) Strutt, J.W., 3rd Baron Rayleigh (1891). "On the Virial of a System of Hard Colliding Bodies". Nature. 45 (1152): 80–82. Bibcode:1891Natur..45...80R. doi:10.1038/045080a0.{{cite journal}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link) Su, G.J. (1946)... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
== Chandler, David (1987). Introduction to Modern Statistical Mechanics. Oxford: Oxford University Press. pp. 287–295. ISBN 0195042778. Cross, Michael (2004), "Lecture 3: First Order Phase Transitions" (PDF), Physics 127: Statistical Physics, Second Term, Pasadena, California: Division of Physics, Mathematics, and Astr... | {
"page_id": 206064,
"title": "Van der Waals equation"
} |
Brian G. Wowk is a Canadian medical physicist and cryobiologist known for the discovery and development of synthetic molecules that mimic the activity of natural antifreeze proteins in cryopreservation applications, sometimes called "ice blockers". As a senior scientist at 21st Century Medicine, Inc., he was a co-devel... | {
"page_id": 4138226,
"title": "Brian Wowk"
} |
In the magnetosphere, the plasma sheet is a sheet-like region of denser (0.3-0.5 ions/cm3 versus 0.01-0.02 in the lobes) hot plasma and lower magnetic field located on the magnetotail and near the equatorial plane, between the magnetosphere's north and south lobes. The origin of the plasma sheet is still a subject of d... | {
"page_id": 36447476,
"title": "Plasma sheet"
} |
Surface runoff (also known as overland flow or terrestrial runoff) is the unconfined flow of water over the ground surface, in contrast to channel runoff (or stream flow). It occurs when excess rainwater, stormwater, meltwater, or other sources, can no longer sufficiently rapidly infiltrate in the soil. This can occur ... | {
"page_id": 4400374,
"title": "Surface runoff"
} |
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