Case stationary conditions

Figure 6.6. Evolution of the calorimeter s response in case stationary conditions have been reached...

The idea is to construct a Lagrange function which has the same energy as the non-variational wave function, but which is variational in all parameters. Consider for example a CL wave function, which is variational in the state coefficients (a) but not in the MO coefficients (c) (note that we employ lower case c for the MO coefficients, but capital C to denote all wave function parameters, i.e. C contains both a and c), since they are determined by the stationary condition for the HF wave function. 

In this case, of course. A is not consistent with the stationary condition of Eqn. (1). Model (II) eliminates an apparent unlinked term, so is not completely size-extensive. On the other hand, the models (IT) and (III) do not contain any unlinked terms and are therefore exactly extensive. However, unlike EOM-CCSD itself, none of these models is exact for two-electron systems. 

Denote the forward and backward rate constants of this reaction by ka and kb- When the reaction proceeds under stationary conditions, the rates of the chemical and of the electron-transfer reaction are equal. Derive the current-potential relationship for this case. Assume that the concentrations of A and of the oxidized species are constant. 

Consider the reaction scheme of Eq. (11.1) and assume that the intermediate can diffuse away from the electrode surface. In the simplest case the current density of particles diffusing away is proportional to the concentration of the intermediate c-mt at the surface jditt = fccmt. Derive an expression for c nt under stationary conditions. 

The above expression contains the three rate constants involved in initiation, k (in C), kc and k, whose values are not known. (Recent attempts failed to determine the value of k due to experimental difficulties [10]). The value of [M] depends on both the relative and the absolute addition rates A and Aj, which means that at a given [M]/[I] in the feed, the [M] would increase with increasing addition rate. It is important therefore that the addition rate be sufficiently low so as to achieve stationary conditions rapidly, i.e., the time to reach stationary conditions should be negligible compared to the polymerization time. Unless this precaution is taken monomer/ inifer may accumulate in the system which may lead to short or in worse cases even to an absence of stationary periods which in turn... 

If we neglect migration, experiments can be performed under conditions of minimal convection, which are thus dominated by diffusion. Since S increases with time t in such a case, nonstationary conditions exist. On the other hand, if convection dominates in the electrolyte bulk, S 7 /( ), and we approach stationary conditions, as far as diffusion is concerned. 

The effort to carry out all these balances is high, but it significantly increases the reliability of the results, that should be based not only on single measurements (analyses). Usually, incorrect data are only detectable on the basis of at least two independent values or balances. If various balances are found, often an error can be identified as a false measurement or analysis mistake and not a real failure. As far as possible, several samples should be taken during each experiment for improved reliability. For continuous operation under stationary conditions, the average of some measurements and analyses will be used (any tendency in the individual values shows that the stationary state is not yet achieved). In case of batch operation a consistent change with time confirms reasonable results (here, in the mass balance the decrease of the cell liquid by the sampling has to be considered). 

The gas ballast pump has the function of pumping the fraction of air, which is often only a small part of the water-vapor mixture concerned, without simultaneously pumping much water vapor. It is, therefore, understandable that, within the combination of condenser and gas ballast pump in the stationary condition, the ratios of flow, which occur in the region of rough vacuum, are not easily assessed without further consideration. The simple application of the continuity equation is not adequate because one is no longer concerned with a source or sink-free field of flow (the condenser is, on the basis of condensation processes, a sink). This is emphasized especially at this point. In a practical case of non-functioning of the condenser - gas ballast pump combination, it might be unjustifiable to blame the condenser for the failure. 

The conditions under which the above stationary-state solution loses its stability can be determined following the recipe of 2.6. Again we find that instability may arise, and hence oscillatory behaviour is possible, in this reversible case. The condition for the onset of instability can be expressed in terms of the reactant concentration p < p p, where... 

The modeling of the electrochemical response corresponding to the application of a constant potential to an RDE is similar to that discussed in the case of a DME since in this electrode it is imperative to consider the convection caused by the rotation of the electrode. This problem was solved by Levich under stationary conditions [76]. To do this, the starting point is the diffusive-convective differential... 

When the characteristic dimension of the electrode fulfills that Q -C fnDt (with qG being equal to rd, rs, w, or rc for discs, spheres, bands, or cylinders, respectively), stationary or pseudo-stationary conditions are attained. Under these conditions, the expressions for the signal obtained in SWV as well as those corresponding to differential techniques DSCVC and DMPV previously discussed simplify greatly. In the case of discs or spheres, a true steady-state response can be obtained (see Eqs. (7.20) and Table 7.1) ... 

The excitation operator does not have to adhere to the unitary condition, as is the case for orthogonal orbitals. Each Brillouin matrix element (Eq. (4)) represents the stationary condition for the mixing of orbitals iffj and iffj according to Wi -> Vi + Vj- The wavefunction consisting of Vo and all singly excited states... 

A method is presented here which yields the polymer size distribution for arbitrary rates of radical arrival and termination. Furthermore, from this analysis one can see when each of the limiting cases is applicable. The computations are all carried out under stationary conditions with the rates of radical arrival, propagation, and termination constant. Under transient conditions the computations would be much more difficult. For the limiting cases, however, the moments of the polymer size distributions under transient conditions can be found (4). 

When the active centre concentrations change during propagation, the whole polymerization is non-stationary. Kinetically the process becomes more complicated and often even experimental control of the process becomes more difficult. On the other hand, a non-stationary condition can be utilized in studies of the elementary polymerization steps. To this end, the non-stationary phases of radical polymerizations are suitable, where outside these phases the process is essentially stationary [23-25]. Hayes and Pepper [26] called attention to the existence and solution of a simple non-stationary case caused by slower decay of rapidly generated cationic centres. In more complicated cases, exact analysis of the causes of a non-stationary condition is often beyond present possibilities. Information from the process kinetics is often not conclusive. It should be mentioned that, even when the condition d[Ac]/dt = 0 is strictly valid, polymerizations may be non-stationary, particularly in those cases when during propagation the more active form of the centres is slowly transformed to the less active form or vice versa. 

Smoluchowskl ) has given a simple derivation of this case. See fig. 4.3. As in the previous case, under stationary conditions the sum of the forces exerted on the particle and the solution around it must be zero. For an infinitesimal slice of thickness dx and area A at distance x from the surface (hatched in fig. 4.3) the force caused by momentum transport, Arj(doydx) - Arj dv /d =... 

It should be mentioned that the XCC formalism belongs to a broad category of alternative CC theories, which are obtained if we study the stationary conditions of the CC energy functional [74,75]. Other examples of alternative CC ansatze include the aforementioned unitary CC (UCC) approach [15,16,28, 70-73] (Refs. 15, 16, 28 discuss the MRCC variants of the UCC method), the symmetric expectation value CC (SXCC) method [70,74,75], the strongly-connected XCC (SC-XCC) theory and its UCC (SC-UCC) analog [75], and the extended CC (ECC) approach [76,77]. A possibility to extend the EOMCC theory to the ECC case via the new EOMECC formalism is considered in this study as well. 

The implicit assumption in the determinahon of Eqs. (5.42) and (5.45) is that the mass-and heat-transfer resistances within the droplet are null. This is a common assumphon that is based on the idea of quasi-stationary conditions within the droplet that often goes under the name of the rapid-mixing model. In the special case in which Bm is approximately constant, the right-hand side of Eq. (5.45) is constant, and thus the rate of change of the surface area is independent of the droplet diameter (i.e. the d evaporahon law). 

This system in many cases can be simplified further. For example, if we have a broad spectral line excitation with a not very intense laser radiation, we have a situation for an open transition when 7 Ti, H. In practical cases this condition is often fulfilled at excitation with cw lasers operating in a multimode regime. If the homogeneous width of spectral transition usually is in the range of 10 MHz, then the laser radiation spectral width broader than 100 MHz usually can be considered as a broad line excitation. In this case we can use a procedure known as adiabatic elimination. It means that we are assuming that optical coherence pi2 decays much faster than the populations of the levels puJ = 1,2. Then we can find stationary solution for off-diagonal elements for the density matrix and afterwards find a rate equations for populations in this limit. For the two level system we will have... 

Since the bulk of the adsorption is accomplished in the second phase under stationary conditions, the adsorbent was developed to obtain high sulfur dioxide-to-sulfuric acid conversion rates for a large portion of its inner surface. The relationship between pore structure and sulfur dioxide adsorption is shown in Figure 1. The ordinate is the time, in hours, after which 10% of the inlet sulfur dioxide will pass through the carbon without being adsorbed. The mean pore diameter of adsorption pores was selected for the abscissa as the parameter to characterize the adsorbent structure (3). Adsorbents produced from bituminous coal with and without catalyst impregnation were tested. In both cases, the sulfur... 

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