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Maxwell Relation
file:Thermodynamic map.svg, 400px, Flow chart showing the paths between the Maxwell relations. P is pressure, T temperature, V volume, S entropy, \alpha coefficient of thermal expansion, \kappa compressibility, C_V heat capacity at constant volume, C_P heat capacity at constant pressure. Maxwell's relations are a set of equations in thermodynamics which are derivable from the symmetry of second derivatives and from the definitions of the thermodynamic potentials. These relations are named for the nineteenth-century physicist James Clerk Maxwell. Equations The structure of Maxwell relations is a statement of equality among the second derivatives for continuous functions. It follows directly from the fact that the order of differentiation of an analytic function of two variables is irrelevant ( Schwarz theorem). In the case of Maxwell relations the function considered is a thermodynamic potential and x_i and x_j are two different natural variables for that potential, we have ...
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Thermodynamic Equations
Thermodynamics is expressed by a mathematical framework of ''thermodynamic equations'' which relate various thermodynamic quantities and physical properties measured in a laboratory or production process. Thermodynamics is based on a fundamental set of postulates, that became the laws of thermodynamics. Introduction One of the fundamental thermodynamic equations is the description of thermodynamic work in analogy to mechanical work, or weight lifted through an elevation against gravity, as defined in 1824 by French physicist Sadi Carnot. Carnot used the phrase motive power for work. In the footnotes to his famous ''On the Motive Power of Fire'', he states: “We use here the expression ''motive power'' to express the useful effect that a motor is capable of producing. This effect can always be likened to the elevation of a weight to a certain height. It has, as we know, as a measure, the product of the weight multiplied by the height to which it is raised.” With the in ...
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Enthalpy
Enthalpy () is the sum of a thermodynamic system's internal energy and the product of its pressure and volume. It is a state function in thermodynamics used in many measurements in chemical, biological, and physical systems at a constant external pressure, which is conveniently provided by the large ambient atmosphere. The pressure–volume term expresses the work (physics), work W that was done against constant external pressure P_\text to establish the system's physical dimensions from V_\text=0 to some final volume V_\text (as W=P_\text\Delta V), i.e. to make room for it by displacing its surroundings. The pressure-volume term is very small for solids and liquids at common conditions, and fairly small for gases. Therefore, enthalpy is a stand-in for energy in chemical systems; Bond energy, bond, Lattice energy, lattice, solvation, and other chemical "energies" are actually enthalpy differences. As a state function, enthalpy depends only on the final configuration of internal e ...
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Grand Potential
The grand potential or Landau potential or Landau free energy is a quantity used in statistical mechanics, especially for irreversible processes in open systems. The grand potential is the characteristic state function for the grand canonical ensemble. Definition The grand potential is defined by \Phi_\text \stackrel U - T S - \mu N where ''U'' is the internal energy, ''T'' is the temperature of the system, ''S'' is the entropy, ''μ'' is the chemical potential, and ''N'' is the number of particles in the system. The change in the grand potential is given by \begin d\Phi_\text & = dU - T\,dS - S\,dT - \mu d\,N - N\,d\mu \\ & = - P\,dV - S\,dT - N\,d\mu \end where ''P'' is pressure and ''V'' is volume, using the fundamental thermodynamic relation (combined first and second thermodynamic laws); dU = T\,dS - P\,dV + \mu\,dN When the system is in thermodynamic equilibrium, ΦG is a minimum. This can be seen by considering that ''d''ΦG is zero if the volume is fixed and the ...
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Chemical Potential
In thermodynamics, the chemical potential of a Chemical specie, species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species in a mixture is defined as the rate of change of Thermodynamic free energy, free energy of a thermodynamic system with respect to the change in the number of atoms or molecules of the species that are added to the system. Thus, it is the partial derivative of the free energy with respect to the amount of the species, all other species' concentrations in the mixture remaining constant. When both temperature and pressure are held constant, and the number of particles is expressed in moles, the chemical potential is the partial Molar concentration, molar Gibbs free energy. At chemical equilibrium or in phase equilibrium, the total sum of the product of chemical potentials and stoichiometric coefficients is zero, as the free en ...
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Particle Number
In thermodynamics, the particle number (symbol ) of a thermodynamic system is the number of constituent particles in that system. The particle number is a fundamental thermodynamic property which is conjugate to the chemical potential. Unlike most physical quantities, the particle number is a dimensionless quantity, specifically a countable quantity. It is an extensive property, as it is directly proportional to the size of the system under consideration and thus meaningful only for closed systems. A constituent particle is one that cannot be broken into smaller pieces at the scale of energy involved in the process (where is the Boltzmann constant and is the temperature). For example, in a thermodynamic system consisting of a piston containing water vapour, the particle number is the number of water molecules in the system. The meaning of constituent particles, and thereby of particle numbers, is thus temperature-dependent. Determining the particle number The concept of pa ...
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Exterior Derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The resulting calculus, known as exterior calculus, allows for a natural, metric-independent generalization of Stokes' theorem, Gauss's theorem, and Green's theorem from vector calculus. If a differential -form is thought of as measuring the flux through an infinitesimal - parallelotope at each point of the manifold, then its exterior derivative can be thought of as measuring the net flux through the boundary of a -parallelotope at each point. Definition The exterior derivative of a differential form of degree (also differential -form, or just -form for brevity here) is a differential form of degree . If is a smooth function (a -form), then the exterior derivative of is the differential of . That is, is the unique -form such that ...
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Gibbs Equations
Gibbs or GIBBS is a surname and acronym. It may refer to: People * Gibbs (surname) Places * Gibbs (crater), on the Moon * Gibbs, Missouri, US * Gibbs, Tennessee, US * Gibbs Island (South Shetland Islands), Antarctica * 2937 Gibbs, an asteroid * Mount Gibbs Science Mathematics and statistics * Gibbs phenomenon * Gibbs' inequality * Gibbs sampling Physics * Gibbs phase rule * Gibbs free energy * Gibbs entropy * Gibbs paradox * Gibbs–Helmholtz equation * Gibbs algorithm * Gibbs state * Gibbs-Marangoni effect * Gibbs phenomenon, an MRI artifact Organisations * Gibbs & Cox naval architecture firm * Gothenburg International Bioscience Business School * Gibbs College, several US locations * Gibbs Technologies, developer and manufacturer of amphibious vehicles * Gibbs High School (other), several schools of this name exist * Antony Gibbs & Sons, British trading company, established in London in 1802 Other uses * Gibbs SR, former name of the toothpaste Mentaden ...
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Symmetry Of Second Derivatives
In mathematics, the symmetry of second derivatives (also called the equality of mixed partials) is the fact that exchanging the order of partial derivatives of a multivariate function :f\left(x_1,\, x_2,\, \ldots,\, x_n\right) does not change the result if some continuity conditions are satisfied (see below); that is, the second-order partial derivatives satisfy the identities :\frac \left( \frac \right) \ = \ \frac \left( \frac \right). In other words, the matrix of the second-order partial derivatives, known as the Hessian matrix, is a symmetric matrix. Sufficient conditions for the symmetry to hold are given by Schwarz's theorem, also called Clairaut's theorem or Young's theorem. In the context of partial differential equations, it is called the Schwarz integrability condition. Formal expressions of symmetry In symbols, the symmetry may be expressed as: :\frac \left( \frac \right) \ = \ \frac \left( \frac \right) \qquad\text\qquad \frac \ = ...
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Total Derivative
In mathematics, the total derivative of a function at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with respect to all of its arguments, not just a single one. In many situations, this is the same as considering all partial derivatives simultaneously. The term "total derivative" is primarily used when is a function of several variables, because when is a function of a single variable, the total derivative is the same as the ordinary derivative of the function. The total derivative as a linear map Let U \subseteq \R^n be an open subset. Then a function f:U \to \R^m is said to be (totally) differentiable at a point a\in U if there exists a linear transformation df_a:\R^n \to \R^m such that :\lim_ \frac=0. The linear map df_a is called the (total) derivative or (total) differential of f at a. Other notations for the total derivative inc ...
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Chain Rule
In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , then the chain rule is, in Lagrange's notation, h'(x) = f'(g(x)) g'(x). or, equivalently, h'=(f\circ g)'=(f'\circ g)\cdot g'. The chain rule may also be expressed in Leibniz's notation. If a variable depends on the variable , which itself depends on the variable (that is, and are dependent variables), then depends on as well, via the intermediate variable . In this case, the chain rule is expressed as \frac = \frac \cdot \frac, and \left.\frac\_ = \left.\frac\_ \cdot \left. \frac\_ , for indicating at which points the derivatives have to be evaluated. In integral, integration, the counterpart to the chain rule is the substitution rule. Intuitive explanation Intuitively, the chain rule states that knowing t ...
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Mnemonic
A mnemonic device ( ), memory trick or memory device is any learning technique that aids information retention or retrieval in the human memory, often by associating the information with something that is easier to remember. It makes use of elaborative encoding, retrieval cues and imagery as specific tools to encode information in a way that allows for efficient storage and retrieval. It aids original information in becoming associated with something more accessible or meaningful—which in turn provides better retention of the information. Commonly encountered mnemonics are often used for lists and in auditory system, auditory form such as Acrostic, short poems, acronyms, initialisms or memorable phrases. They can also be used for other types of information and in visual or kinesthetic forms. Their use is based on the observation that the human mind more easily remembers spatial, personal, surprising, physical, sexual, humorous and otherwise "relatable" information rather tha ...
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Thermodynamic Square
The thermodynamic square (also known as the thermodynamic wheel, Guggenheim scheme or Born square) is a mnemonic diagram attributed to Max Born and used to help determine thermodynamic relations. Born presented the thermodynamic square in a 1929 lecture. The symmetry of thermodynamics appears in a paper by F.O. Koenig. The corners represent common conjugate variables (thermodynamics), conjugate variables while the sides represent thermodynamic potentials. The placement and relation among the variables serves as a key to recall the relations they constitute. A mnemonic used by students to remember the Maxwell relations (in thermodynamics) is "Good Physicists Have Studied Under Very Fine Teachers", which helps them remember the order of the variables in the square, in clockwise direction. Another mnemonic used here is "Valid Facts and Theoretical Understanding Generate Solutions to Hard Problems", which gives the letter in the normal left-to-right writing direction. Both times A h ...
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