Closed system defined

U (or ) was previously teniied the internal energy (no longer used). For a closed" system (defined as one in which there is no exchange of mass with the sunoundings ) at rest, A(7=Q+ W if there is no other mechanism of exchange of energy. By convention,... 

In a closed system the rate of reaction is properly defined by a total time derivative of the concentration, if concentration is based on the closed total volume of the system or on a volume liquid of constant density. 

Adiabatic Reaction Temperature (T ). The concept of adiabatic or theoretical reaction temperature (T j) plays an important role in the design of chemical reactors, gas furnaces, and other process equipment to handle highly exothermic reactions such as combustion. T is defined as the final temperature attained by the reaction mixture at the completion of a chemical reaction carried out under adiabatic conditions in a closed system at constant pressure. Theoretically, this is the maximum temperature achieved by the products when stoichiometric quantities of reactants are completely converted into products in an adiabatic reactor. In general, T is a function of the initial temperature (T) of the reactants and their relative amounts as well as the presence of any nonreactive (inert) materials. T is also dependent on the extent of completion of the reaction. In actual experiments, it is very unlikely that the theoretical maximum values of T can be realized, but the calculated results do provide an idealized basis for comparison of the thermal effects resulting from exothermic reactions. Lower feed temperatures (T), presence of inerts and excess reactants, and incomplete conversion tend to reduce the value of T. The term theoretical or adiabatic flame temperature (T,, ) is preferred over T in dealing exclusively with the combustion of fuels. 

In actual experiments we do not usually observe directly the desorbed amount, but rather the derived read-out quantities, as is the time dependence of the pressure in most cases. In a closed system, this pressure is obviously a monotonously increasing function of time. In a flow or pumped system, the pressure-time dependence can exert a maximum, which is a function of the maximum desorption rate, but need not necessarily occur at the same time due to the effect of the pumping speed S. If there are particles on the surface which require different activation energies Ed for their desorption, several maxima (peaks) appear on the time curve of the recorded quantity reflecting the desorption process (total or partial pressure, weight loss). Thereby, the so-called desorption spectrum arises. It is naturally advantageous to evaluate the required kinetic parameters of the desorption processes from the primarily registered read-out curves, particularly from their maxima which are the best defined points. 

Figure 8. ° Pb7 Pb vs. Th/Us (derived using Eqn. 5 in the text) diagram for mid-ocean ridge and ocean island basalt based on a recent data set with mostly mass spectrometry measurements (Turner et al. 1997 Bourdon et al. 1996 Dosso et al. 1999 Claude-lvanaj et al. 1998, 2001 Sims et al. 2002). The data show a relatively well defined array that intersect a closed-system hne for the bulk Earth starting with an initial lead isotope composition equal to Canyon Diablo (T = 4.55 Ga). This intersect was used by Allegre et al. (1986) to define the Th/U ratio of the Earth.

Entropy can be described by considering a closed system undergoing a reversible process. The entropy change, dS, of the system is defined by the relationship... 

Consider a closed system (i.e., one in which there is no exchange of matter between the system and its surroundings) where a single chemical reaction may occur according to equation 1.1.3. Initially there are ni0 moles of constituent At present in the system. At some later time there are n moles of species At present. At this time the molar extent of reaction is defined as... 

Thus, given the weights and abscissas, the micromixing term for the moments is closed. Applying DQMOM, the micromixing source terms (which are added to the right-hand sides of Eqs. (133)—(135)) can be shown to obey for each n — 1,..., N the linear system defined by... 

The first and most critical step in developing a geochemical model is conceptualizing the system or process of interest in a useful manner. By system, we simply mean the portion of the universe that we decide is relevant. The composition of a closed system is fixed, but mass can enter and leave an open system. A system has an extent, which the modeler defines when he sets the amounts of fluid and mineral considered in the calculation. A system s extent might be a droplet of rainfall, the groundwater and sediments contained in a unit volume of an aquifer, or the world s oceans. 

The rate of feed vf (in mol 1 1 s ) of the reactant into the reaction medium is now the critical parameter to adjust for a favourable outcome of a cyclisa-tion experiment. A kinetic treatment of the open system under influxion incurs the same difficulties already discussed for the closed system in Section 2. However, when the higher-order polymerisation terms are relatively unimportant and the overall process is described to a useful approximation by (6), an exact mathematical solution is possible (Galli and Mandolini, 1975). After a relatively short initial time9, the concentration of M reaches a steady value [M]st given by (72), where (3 defined by (73) is a dimensionless parameter... 

In the Schrodinger picture operators in the case of a closed system do not depend explicitly on the time, but the state vector is time dependent. However, the expectation values are generally functions of the time. The commutator of the Hamiltonian operator H= —(h/2iri)(d/dt) and another operator A, is defined by... 

As for classical systems, measurement of the properties of macroscopic quantum systems is subject to experimental error that exceeds the quantum-mechanical uncertainty. For two measurable quantities F and G the inequality is defined as AFAG >> (5F6G.The state vector of a completely closed system described by a time-independent Hamiltonian H, with eigenvalues En and eigenfunctions is represented by... 

The microcanonical ensemble in quantum statistics describes a macroscopi-cally closed system in a state of thermodynamic equilibrium. It is assumed that the energy, number of particles and the extensive parameters are known. The Hamiltonian may be defined as... 

Supercritical fluids represent a different type of alternative solvent to the others discussed in this book since they are not in the liquid state. A SCF is defined as a substance above its critical temperature (Tc) and pressure (Pc)1, but below the pressure required for condensation to a solid, see Figure 6.1 [1], The last requirement is often omitted since the pressure needed for condensation to occur is usually unpractically high. The critical point represents the highest temperature and pressure at which the substance can exist as a vapour and liquid in equilibrium. Hence, in a closed system, as the boiling point curve is ascended, increasing both temperature and pressure, the liquid becomes less dense due to thermal expansion and the gas becomes denser as the pressure rises. The densities of both phases thus converge until they become identical at the critical point. At this point, the two phases become indistinguishable and a SCF is obtained. 

In a closed system, that is one in which there is no addition of V nor any removal of p , the reaction will come to a perfect balance the point of equilibrium . A common misunderstanding of the concept of this point of equilibrium is that it implies an equal concentration of r and p. This is not true. The point of equilibrium defines the relative concentrations of r and p when the rate of formation of p is exactly equal to the rate of formation of r. The point of equilibrium value for a chemical reaction can be determined experimentally. If the starting concentration of the reactant is known, then it follows that the relative concentrations of r and p when equilibrium has been reached must reflect the relative rates of the forward and reverse reactions. For a given reaction, under defined conditions, the point of equilibrium is a constant and given the symbol Keq. 

We briefly review processes in which isotopic fractionations may be recorded in isotopically distinct reservoirs that are preserved in nature. These concepts have been extensively covered in the H, C, O, and S isotope literature, and we illustrate several examples for the non-traditional stable isotope systems discussed in this volume. One of the simplest processes that produces isotopically distinct reservoirs would be slow reaction of substance A to B, where A and B remain open to complete isotopic exchange during the process. This is commonly referred to as closed system equilibrium, and the changes in isotopic compositions that occur may be defined by the exact relation ... 

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