Conservative system, definition

Problem Solving Methods Most, if not aU, problems or applications that involve mass transfer can be approached by a systematic-course of action. In the simplest cases, the unknown quantities are obvious. In more complex (e.g., iTmlticomponent, multiphase, multidimensional, nonisothermal, and/or transient) systems, it is more subtle to resolve the known and unknown quantities. For example, in multicomponent systems, one must know the fluxes of the components before predicting their effective diffusivities and vice versa. More will be said about that dilemma later. Once the known and unknown quantities are resolved, however, a combination of conservation equations, definitions, empirical relations, and properties are apphed to arrive at an answer. Figure 5-24 is a flowchart that illustrates the primary types of information and their relationships, and it apphes to many mass-transfer problems. 

For conservative systems with time-independent Hamiltonian the density operator may be defined as a function of one or more quantum-mechanical operators A, i.e. g= tp( A). This definition implies that for statistical equilibrium of an ensemble of conservative systems, the density operator depends only on constants of the motion. The most important case is g= [Pg.463]

Consider the system before energy conservation. By definition ... 

There are a number of traps involved in even this straightforward definition. The use of the reference volume as that of the reaction mixture is necessary to account for the fact that in some cases the total volume will change in proportion to the molal balance between reactants and products. If the reaction were A —> 2B and involved ideal gases with no change in temperature or total pressure, the volume of product at the completion of the reaction would be twice that of initial reactant. To conserve our definition of reaction rate independent of the size of the system, the volume change must be accounted for. This is not difficult to do, as will be shown a little later. The problem is that one often sees reaction rates written in the following form ... 

This is another viewpoint on the wave equation that is generally presented as an ordinary second-order differential equation with respect to two variables, time and space vector. The eigenvalue is a function of the wave velocity u which represents the propagation speed of the wave and which is a concept that differs from the oscillator velocity Uy introduced in definition F5.8. However, although concepts are different, the modulus of the wave velocity becomes equal to the oscillator velocity in a conservative system (without dissipation) ... 

Note that the Liouville equation, formally, is identical with the first conservation equation, the so-called continuity equation of hydrodynamics, equation (la). The change of the mass density and the change of the phase-space-distribution can be derived based on the conservation of the total mass and the total number of systems, respectively.) The last step of equation (7) is a definition of the term A(/ ) called the phase-space compression factor. In the case of conservative systems (the most common example of which is Hamilton s equations), the Liouville equation describes an incompressible flow and the right-hand side of equation (7) is zero. (In many statistical mechanical texts, only this incompressible form is referred to as the Liouville equation.)... 

Standard for gauging flammability. The most common systems for designating flammability are the Department of Transportation (DOT) definitions, the National Fire Protection Association s (NFPA) system, and the Environmental Protection Agency s (EPA) Resource Conservation and Recovery Act s (RCRA) definition of ignitable wastes, all of which use flashpoint in their schemes. The NFPA diamond, which comprises the backbone of the NFPA Hazard Signal System, uses a four-quadrant diamond to display the... 

One molecule (or mole) of propane reacts with five molecules (or moles) of oxygen to produce three molecules (or moles) or carbon dioxide and four molecules (or moles) of water. These numbers are called stoichiometric coefficients (v.) of the reaction and are shown below each reactant and product in the equation. In a stoichiometrically balanced equation, the total number of atoms of each constituent element in the reactants must be the same as that in the products. Thus, there are three atoms of C, eight atoms of H, and ten atoms of O on either side of the equation. This indicates that the compositions expressed in gram-atoms of elements remain unaltered during a chemical reaction. This is a consequence of the principle of conservation of mass applied to an isolated reactive system. It is also true that the combined mass of reactants is always equal to the combined mass of products in a chemical reaction, but the same is not generally valid for the total number of moles. To achieve equality on a molar basis, the sum of the stoichiometric coefficients for the reactants must equal the sum of v. for the products. Definitions of certain terms bearing relevance to reactive systems will follow next. 

Based on the law of conservation of mass, mass is neither created nor destroyed, except in nuclear reactions. Therefore, in a definite system, mass in is equal to mass out. 

The operational interpretation of rA, as opposed to this verbal definition, does depend on the circumstances of the reaction.1 This is considered further in Chapter 2 as a consequence of the application of the conservation of mass to particular situations. Furthermore, rA depends on several parameters, and these are considered in Section 1.4.2. The rate with respect to any other species involved in the reacting system may be related to rA directly through reaction stoichiometry for a simple, single-phase system, or it may require additional kinetics information for a complex system. This aspect is considered in Section 1.4.4, following a preliminary discussion of the measurement of rate of reaction in Section 1.4.3. 

The number of equations and unknowns must balance. Thus, one can calculate the appropriate number of needed relationships from the degrees of freedom of a system, as shown for various systems by Newman. In terms of the relations, the equations can be broken down into five main types. The first are the conservation equations, the second are the transport relations, the third are the reactions, the fourth are equilibrium relationships, and the fifth are the auxiliary or supporting relations, which include variable definitions and such relations as Faraday s law. 

Although we know that the total force field / is conservative, we lack such knowledge concerning the separate pieces fx or A- Therefore some specific path of integration must be chosen to complete the definition (210). Seemingly the most natural path is along the radius on which the point ro lies. As the center of the coordinate system a position of the nucleus is chosen in the case of single-ion system, the center of symmetry for a symmetrical molecule, and some inner point for molecule or cluster with lower symmetry. Then... 

The first law of thermodynamics, which can be stated in various ways, enuciates the principle of the conservation of energy. In the present context, its most important application is in the calculation of the heat evolved or absorbed when a given chemical reaction takes place. Certain thermodynamic properties known as state functions are used to define equilibrium states and these properties depend only on the present state of the system and not on its history, that is the route by which it reached that state. The definition of a sufficient number of thermodynamic state functions serves to fix the state of a system for example, the state of a given mass of a pure gas is defined if the pressure and temperature are fixed. When a system undergoes some change from state 1 to state 2 in which a quantity of heat, Q, is absorbed and an amount of work, W, is done on the system, the first law may be written... 

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