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Virial Coefficient
Virial coefficients B_i appear as coefficients in the virial expansion of the pressure of a many-particle system in powers of the density, providing systematic corrections to the ideal gas law. They are characteristic of the interaction potential between the particles and in general depend on the temperature. The second virial coefficient B_2 depends only on the pair interaction between the particles, the third (B_3) depends on 2- and non-additive 3-body interactions, and so on. Derivation The first step in obtaining a closed expression for virial coefficients is a cluster expansion of the grand canonical partition function : \Xi = \sum_ = e^ Here p is the pressure, V is the volume of the vessel containing the particles, k_\text is the Boltzmann constant, T is the absolute temperature, \lambda =\exp mu/(k_\textT) is the fugacity, with \mu the chemical potential. The quantity Q_n is the canonical partition function of a subsystem of n particles: : Q_n = \operatorname e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Virial Expansion
The virial expansion is a model of thermodynamic equations of state. It expresses the pressure of a gas in local Thermodynamic equilibrium, equilibrium as a power series of the density. This equation may be represented in terms of the compressibility factor, , as Z \equiv \frac = A + B\rho + C\rho^2 + \cdots This equation was first proposed by Heike Kamerlingh Onnes, Kamerlingh Onnes.Kamerlingh Onnes, H."Expression of the equation of state of gases and liquids by means of series" ''KNAW, Proceedings'', 4, 1901-1902, Amsterdam, 125-147 (1902). The terms , , and represent the virial coefficients. The leading coefficient is defined as the constant value of 1, which ensures that the equation reduces to the ideal gas expression as the gas density approaches zero. Second and third virial coefficients The second, , and third, , virial coefficients have been studied extensively and tabulated for many fluids for more than a century. Two of the most extensive compilations are in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maria Goeppert-Mayer
Maria Goeppert Mayer (; ; June 28, 1906 – February 20, 1972) was a German-American theoretical physicist who shared the 1963 Nobel Prize in Physics with J. Hans D. Jensen and Eugene Wigner. One half of the prize was awarded jointly to Goeppert Mayer and Jensen for their model of the atomic nucleus. She was the second woman to win a Nobel Prize in Physics, the first being Marie Curie in 1903. In 1986, the Maria Goeppert-Mayer Award for early-career women physicists was established in her honor. A graduate of the University of Göttingen, Goeppert Mayer wrote her doctoral thesis on the theory of possible two-photon absorption by atoms. At the time, the chances of experimentally verifying her thesis seemed remote, but the development of the laser in the 1960s later permitted this. Today, the unit for the two-photon absorption cross section is named the Goeppert Mayer (GM) unit. Maria Goeppert married chemist Joseph Edward Mayer and moved to the United States, where he was an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Excess Property
In chemical thermodynamics, excess properties are properties of mixtures which quantify the non- ideal behavior of real mixtures. They are defined as the difference between the value of the property in a real mixture and the value that would exist in an ideal solution under the same conditions. The most frequently used excess properties are the excess volume, excess enthalpy, and excess chemical potential. The excess volume (), internal energy (), and enthalpy () are identical to the corresponding mixing properties; that is, :\begin V^E &= \Delta V_\text \\ H^E &= \Delta H_\text \\ U^E &= \Delta U_\text \end These relationships hold because the volume, internal energy, and enthalpy changes of mixing are zero for an ideal solution. Definition By definition, excess properties are related to those of the ideal solution by: :z^E = z - z^\text Here, the superscript IS denotes the value in the ideal solution, a superscript E denotes the excess molar property, and z denotes th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boyle Temperature
The Boyle temperature, named after Robert Boyle, is formally defined as the temperature for which the second virial coefficient, B_(T), becomes zero. It is at this temperature that the attractive forces and the repulsive forces acting on the gas particles balance out P = RT \left(\frac + \frac + \cdots \right) This is the virial equation of state and describes a real gas. Since higher order virial coefficients are generally much smaller than the second coefficient, the gas tends to behave as an ideal gas over a wider range of pressures when the temperature reaches the Boyle temperature (or when c = \frac or P are minimized). In any case, when the pressures are low, the second virial coefficient will be the only relevant one because the remaining concern terms of higher order on the pressure. Also at Boyle temperature the dip in a PV diagram tends to a straight line over a period of pressure. We then have :\frac = 0 \qquad\mbox~P \to 0 where Z is the compressibility factor. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leiden University
Leiden University (abbreviated as ''LEI''; ) is a Public university, public research university in Leiden, Netherlands. Established in 1575 by William the Silent, William, Prince of Orange as a Protestantism, Protestant institution, it holds the distinction of being the oldest university in the Netherlands of today. During the Dutch Golden Age scholars from around Europe were attracted to the Dutch Republic for its climate of intellectual tolerance. Individuals such as René Descartes, Rembrandt, Christiaan Huygens, Hugo Grotius, Benedictus Spinoza, and later Baron d'Holbach were active in Leiden and environs. The university has seven academic faculties and over fifty subject departments, housing more than forty national and international research institutes. Its historical primary campus consists of several buildings spread over Leiden, while a second campus located in The Hague houses a liberal arts college (Leiden University College The Hague) and several of its faculties. It i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leonard Ornstein
Leonard Salomon Ornstein (12 November 1880 in Nijmegen, the Netherlands – 20 May 1941 in Utrecht (city), Utrecht, the Netherlands) was a Dutch physicist. Biography Ornstein studied theoretical physics with Hendrik Lorentz, Hendrik Antoon Lorentz at the University of Leiden. He subsequently carried out Ph.D. research under the supervision of Hendrik Lorentz, Lorentz, concerning an application of the statistical mechanics of Josiah Willard Gibbs, Gibbs to molecular problems. In 1914, Ornstein was appointed professor of physics, as successor of Peter Debye, at the University of Utrecht. Among his doctoral students was Jan Frederik Schouten. In 1922, Ornstein became director of the Physical Laboratory (''Fysisch Laboratorium'') and extended his research interests to experimental subjects. His precision measurements concerning intensities of spectral lines brought the Physical Laboratory in the international limelight. Ornstein is also remembered for the Ornstein-Zernike eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Cluster Integral 2
Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties * Graph (topology), a topological space resembling a graph in the sense of discrete mathematics *Graph of a function * Graph of a relation *Graph paper *Chart, a means of representing data (also called a graph) Computing *Graph (abstract data type), an abstract data type representing relations or connections *graph (Unix), Unix command-line utility *Conceptual graph, a model for knowledge representation and reasoning *Microsoft Graph, a Microsoft API developer platform that connects multiple services and devices Other uses * HMS ''Graph'', a submarine of the UK Royal Navy See also * Complex network *Graf *Graff (other) *Graph database *Grapheme, in linguistics *Graphemics *Graphic (other) *-graphy (suffix from the Greek for "describe," "write" or "draw") *List of information graphics software *Stati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation
In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first meaning is the six permutations (orderings) of the set : written as tuples, they are (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). Anagrams of a word whose letters are all different are also permutations: the letters are already ordered in the original word, and the anagram reorders them. The study of permutations of finite sets is an important topic in combinatorics and group theory. Permutations are used in almost every branch of mathematics and in many other fields of science. In computer science, they are used for analyzing sorting algorithms; in quantum physics, for describing states of particles; and in biology, for describing RNA sequences. The number of permutations of distinct objects is factorial, us ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetry Number
The symmetry number or symmetry order of an object is the number of different but indistinguishable (or equivalent) arrangements (or views) of the object, that is, it is the order of its symmetry group. The object can be a molecule, crystal lattice, lattice, tiling, or in general any kind of mathematical object that admits symmetries. In statistical thermodynamics, the symmetry number corrects for any overcounting of equivalent molecular conformations in the partition function. In this sense, the symmetry number depends upon how the partition function is formulated. For example, if one writes the partition function of ethane so that the integral includes full rotation of a methyl, then the 3-fold rotational symmetry of the methyl group contributes a factor of 3 to the symmetry number; but if one writes the partition function so that the integral includes only one rotational energy well of the methyl, then the methyl rotation does not contribute to the symmetry number. Symmetry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mayer F-function
The Mayer f-function is an auxiliary function that often appears in the series expansion of thermodynamic quantities related to classical many-particle systems.Donald Allan McQuarrie, ''Statistical Mechanics'' (HarperCollins, 1976), page 228 It is named after chemist and physicist Joseph Edward Mayer. Definition Consider a system of classical particles interacting through a pair-wise potential :V(\mathbf,\mathbf) where the bold labels \mathbf and \mathbf denote the continuous degrees of freedom associated with the particles, e.g., :\mathbf=\mathbf_i for spherically symmetric particles and :\mathbf=(\mathbf_i,\Omega_i) for rigid non-spherical particles where \mathbf denotes position and \Omega the orientation parametrized e.g. by Euler angles. The Mayer f-function is then defined as :f(\mathbf,\mathbf)=e^-1 where \beta=(k_T)^ the inverse absolute temperature in units of energy−1 . See also *Virial coefficient Virial coefficients B_i appear as coefficients in the virial expan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex Labeling
In the mathematical discipline of graph theory, a graph labeling is the assignment of labels, traditionally represented by integers, to edges and/or vertices of a graph. Formally, given a graph , a vertex labeling is a function of to a set of labels; a graph with such a function defined is called a vertex-labeled graph. Likewise, an edge labeling is a function of to a set of labels. In this case, the graph is called an edge-labeled graph. When the edge labels are members of an ordered set (e.g., the real numbers), it may be called a weighted graph. When used without qualification, the term labeled graph generally refers to a vertex-labeled graph with all labels distinct. Such a graph may equivalently be labeled by the consecutive integers , where is the number of vertices in the graph. For many applications, the edges or vertices are given labels that are meaningful in the associated domain. For example, the edges may be assigned weights representing the "cost" of traver ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
