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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_B is Boltzmann's constant, T is the absolute temperature, \lambda =\exp mu/(k_BT) 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^ Here H(1,2,\ld ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Virial Expansion
The classical virial expansion expresses the pressure P of a many-particle system in equilibrium as a power series in the density: Z \equiv \frac = A + B\rho + C\rho^2 + \cdots where Z is called the compressibility factor. This is the virial equation of state, the most general function relating pressure, , density, , and temperature, , of fluids. It was first proposed by Kamerlingh Onnes.Kamerlingh Onnes H., Expression of state of gases and liquids by means of series, KNAW Proceedings, 4, 1901-1902, Amsterdam, 125-147 (1902). The compressibility factor is a dimensionless quantity, indicating how much a real fluid deviates from an ideal gas. ''A'' is the first virial coefficient, which has a constant value of 1. It makes the statement that at low density, all fluids behave like ideal gases. The virial coefficients , , , etc., are temperature-dependent, and are generally presented as Taylor series in terms of . Second and third virial coefficients The second, , and third, , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maria Goeppert-Mayer
Maria Goeppert Mayer (; June 28, 1906 – February 20, 1972) was a German-born American theoretical physicist, and Nobel laureate in Physics for proposing the nuclear shell model of the atomic nucleus The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford based on the 1909 Geiger–Marsden gold foil experiment. After the discovery of the neutron i .... She was the second woman to win a Nobel Prize in physics, the first being Marie Curie. 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 sect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Excess Molar Quantity
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 the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boyle Temperature
The Boyle temperature 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. Expanding the van der Waals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leiden University
Leiden University (abbreviated as ''LEI''; nl, Universiteit Leiden) is a Public university, public research university in Leiden, Netherlands. The university was founded as a Protestant university in 1575 by William the Silent, William, Prince of Orange, as a reward to the city of Leiden for its Siege of Leiden, defence against Spanish attacks during the Eighty Years' War. As the oldest institution of higher education in the Netherlands, it enjoys a reputation across Europe and the world. Known for its historic foundations and emphasis on the social sciences, the university came into particular prominence during the Dutch Golden Age, when scholars from around Europe were attracted to the Dutch Republic due to its climate of intellectual tolerance and Leiden's international reputation. During this time, Leiden became the home to individuals such as René Descartes, Rembrandt, Christiaan Huygens, Hugo Grotius, Baruch Spinoza and Baron d'Holbach. The university has seven academic f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leonard Ornstein
Leonard Salomon Ornstein (November 12, 1880 in Nijmegen, the Netherlands – May 20, 1941 in Utrecht, the Netherlands) was a Dutch physicist. Biography Ornstein studied theoretical physics with Hendrik Antoon Lorentz at University of Leiden. He subsequently carried out Ph.D. research under the supervision of Lorentz, concerning an application of the statistical mechanics of Gibbs to molecular problems. In 1914, Ornstein was appointed professor of physics, as successor of Peter Debye, at University of Utrecht. Among his doctoral students was Jan Frederik Schouten. In 1922, he became director of Physical Laboratory (''Fysisch Laboratorium'') and extended his research interests to experimental subjects. His precision measurements concerning intensities of spectral lines brought Physical Laboratory in the international limelight. Ornstein is also remembered for the Ornstein-Zernike theory (named after himself and Frederik Zernike) concerning correlation functions, and the ... [...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 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 *Statistical graphics Statistical graphics, also known as statistical graphical techniques, are graphic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Cluster Integral 1
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 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 *Statistical graphics Statistical graphics, also known as statistical graphical techniques, are graphic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set. Permutations differ from combinations, which are selections of some members of a set regardless of order. For example, written as tuples, there are six permutations of the set , namely (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). These are all the possible orderings of this three-element set. Anagrams of words whose letters are different are also permutations: the letters are already ordered in the original word, and the anagram is a reordering of the letters. The study of permutations of finite sets is an important topic in the fields of combinatorics and group theory. Permutations are used in almost every branch of mathematics, and in many other fields of scie ... [...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 (Temperature times the Boltzmann constant k_)−1 . See also *Virial coefficient *Cluster expansion *Excluded vol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex Labeling
In the mathematical discipline of graph theory, a graph labelling is the assignment of labels, traditionally represented by integers, to edges and/or vertices of a graph. Formally, given a graph , a vertex labelling 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 labelling 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] |
