Stationary condition

In this approach [51], the expectation value ( T // T ) / ( T ) is treated variationally and made stationary with respect to variations in the C and. coeflScients. The energy fiinctional is a quadratic function of the Cj coefficients, and so one can express the stationary conditions for these variables in the secular fonu... 

However, E is a quartic function of the Cy,i coefficients because each matrix element <

involves one- and two-electron integrals over the mos ( )i, and the two-electron integrals depend quartically on the Cyj coefficients. The stationary conditions with respect to these Cy i parameters must be solved iteratively because of this quartic dependence. 

The field phasor is a continuously rotating phasor in the space, whose angular position keeps changing with the position of the rotor with respect to the stationary stator. Let the rotor field displacement under the stationary condition with respect to the stator be denoted by angle /3as shown in Figure 6.11, This displacement will continue to change and will rotate the rotor (field frame). All the phasor quantities of the stator arc now expressed in terms ol the field frame. Figure 6.11 shows these two equivalent stator side phasors transformed to the rotor frame. 

The intrinsic drawback of LIBS is a short duration (less than a few hundreds microseconds) and strongly non-stationary conditions of a laser plume. Much higher sensitivity has been realized by transport of the ablated material into secondary atomic reservoirs such as a microwave-induced plasma (MIP) or an inductively coupled plasma (ICP). Owing to the much longer residence time of ablated atoms and ions in a stationary MIP (typically several ms compared with at most a hundred microseconds in a laser plume) and because of additional excitation of the radiating upper levels in the low pressure plasma, the line intensities of atoms and ions are greatly enhanced. Because of these factors the DLs of LA-MIP have been improved by one to two orders of magnitude compared with LIBS. 

For a variationally optimized wave function, the first term is again zero (eq. (10.24)). Furthermore, the second term, which involves calculation of the second derivative of the wave function with respect to the parameters, can be avoided. This can be seen by differentiating the stationary condition (10.24) with respect to the perturbation. 

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. 

When a perturbation is introduced, the stationary condition means that the orbitals must change, which may be described as a mixing of the unperturbed MOs. In other words. 

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. 

Assuming that the catalytic reaction takes place in a flow reactor under stationary conditions, we may use the steady state approximation to eliminate the fraction of adsorbed intermediate from the rate expressions to yield ... 

Considering only termination by reaction (72) and assuming that stationary conditions are attained, the rate of reduction of peroxydisulphate in the chain process is given by... 

It should be noted that the integral equations [2] determining the elements Kp B. aE derived as an energy-variational problem, correspond also to the stationary condition of the variational functional proposed by Newton (9). Thus the K-matrix elements obeying equation [2] guarantee a stationary value for the K-matrix on the energy shell. 

Each radical transfers an electron to the colloidal particles. One colloidal particle can accept and store a large number of electrons until the aqueous solvent is reduced. Under stationary conditions, a certain number x of electrons reside on a particle, thus producing a negative potential sufficiently high for Hj evolution (n = agglomeration number) ... 

Equations 6,71, 6.75 and 6.76 can be solved simultaneously to yield g(t) and G(t) when the initial state vector x0 and the parameter estimate vector k(J) are given. In order to determine ktl+l) the output vector (given by Equation 6.2) is inserted into the objective function (Equation 6.4) and the stationary condition yields,... 

When the output vector is nonlinearly related to the state vector (Equation 6.3) then substitution of x<,+l> from Equation 6.74 into the Equation 6.3 followed by substitution of the resulting equation into the objective function (Equation 6.4) yields the following equation after application of the stationary condition (Equation 6.78)... 

Substitution of the Equations 9.4 and 9.5 into the Lagrangian given by Equation 9.3 and use of the stationary conditions... 

Subsequent use of the stationary condition (3Sp/3k<-, 1))=0, yields the normal equations... 

Most results of in situ IR studies on Pt in acidic methanol solutions so far have been obtained using a relatively fast (8.5-13.6) Hz) modulation of electrode potential. As already pointed out by Bockris [27], collection of spectral data alternatively at two potentials is not appropriate for processes which are not reversible to follow the change of potential. In this study the SPAIRS version of SNIFTIRS was performed by stepping the potential from a reference potential in the anodic direction, allowing sufficient time at each potential to reach stationary conditions. 

Following the Bolland-Gee Scheme 2 for non-inhibited oxidation occurring under stationary conditions, the following equation is frequently valid. 

Normally we expect /ta to be 1 but, as was shown in the first chapter, fi can now take on values in excess of unity. Determination of fi in this way is certainly possible, though in fact it is customary to measure I vs. E plots under stationary conditions, for reasons that will become apparent later. 

Catalytic performances in ethylene ammoxidation as function of reaction temperature of the different catalysts are compiled in Table 2. Data were collected under stationary conditions after a transition period of one hour. All catalysts are active and selective toward acetonitrile. Wherein, Cr-Cl catalyst exhibits the higher ethylene conversion and the higher acetonitrile selectivity. Chromium with highly oxidation state (VI) seems to play a key role in the ammoxidation reaction as confirmed by TPR and DRS spectroscopy results. This idea is strongly supported by the difference between catalytic behaviour of Cr03 and Cr203 supported on ZSM-5. Nevertheless, Cr(III) oxide seems to... 

Procedure Cholinesterase activity in analyzed tissue or the matrix (biotest with immobilized AChE) is determined in the incubation media [consisting of substrate ATCh - 34 mmol maleate buffer 0.1 M, pH = 6.0- 6.5 ml sodium citrate 0.1 M - 0.5 ml CuS045H20 0.03M -1.0 ml distilled H20 (or inhibitor in variant with toxin analyzed) -1.0 ml potassium ferricyanide 0.005 M -1 ml.] Volume of incubation media in one test - 400 mcl. As a blank (control sample), a treatment of the exposure without the substrate is used. If inhibitory effects of allelochemical (or any toxin) are analyzed, before the substrate addition the sample was preliminary exposed to allelochemical inhibitor. Two methods for the AChE-biotests may be recommended (i) in microcells ( stationary conditions ) and (ii) in flowing columns-reactors ( dynamic conditions ). 

Under stationary conditions for long chains and at sufficiently high quinone concentrations when 2 q[Q] > (2ktv j)1/2... 

These conditions are incorporated into the stationary condition (2.29) using Lagrange multipliers... 

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. 

Many ion-transfer reactions involve two or more steps. Often one of these steps proceeds more slowly than the others, and if the reaction proceeds under stationary conditions, this step determines the overall rate. We will elaborate this concept of a rate-determining step further. For this purpose consider a reaction taking place according to the general scheme ... 

Under stationary conditions dc /dt = 0, and an ordinary differential equation results with Eq. (10.5) as boundary conditions, which can be solved explicitly by standard techniques. The resulting expression for the current density is ... 

On application of an overpotential rj we have under stationary conditions ... 

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. 

We consider the investigation of two consecutive electron-transfer reactions with a ring-disc electrode under stationary conditions. A species A reacts in two steps on the disk electrode first to an intermediate B which reacts further to the product C. The intermediate is transported to the ring, where the potential has been chosen such that it reants bank to A. The overall scheme is ... 

Recently, a new (and now commercially available) methodology was reported for measuring in-situ high pressure NMR spectra up to 50 bar under stationary conditions. The instrument uses a modified sapphire NMR tube, and gas saturation of the sample solution and exact pressure control is guaranteed throughout the overall measurement, even at variable temperatures. For this purpose, a special gas cycling system is positioned outside the magnet in the routine NMR laboratory [51]. 

Fig. 44.4 N MR-spectroscopic monitoring of the enantioselective hydrogenation of (Z)-N-acetylamino methyl cinnamate using [Rh((R,R)-DI PAM P) (COD)]BF4 as catalyst under stationary conditions (reaction conditions 0.01 mmol Rh-complex,...

In continuous reactor systems, all reactants are continuously fed to the reactor, and the products are continuously withdrawn. Typical continuous reactors are stirred tanks (either single or in cascades) and plug flow tubes. Continuous reactors are characterized by stationary conditions in that both heat generation and composition profiles remain constant during operation (provided that operating conditions remain unchanged ). 

Low inventory Stationary condition Stationary condition Agitation provides safety tool Streams may be diluted to slow reaction Agitation provides safety tool Controllable addition rate Agitation provides safety tool Large exotherm controllable... 

Critical steady-state temperature (CSST) the highest ambient temperature at which self-heating of a material as handled (in a package, container, tank, etc.) does not result in a runaway but remains in a stationary condition (see Self-Accelerating Decomposition Temperature). 

Stationary conditions conditions that are characterized by constant concentrations and temperatures as a function of time (i.e., the time derivatives are zero). 

A gravity sewer pipe with a diameter D=0.5 m and a slope s=0.003 m m-1 is flowing half full under stationary conditions, i.e., the DO concentration is constant and equal to about 0.3 g02 m-3. The pipe is made of concrete, and the roughness is 1.0 mm. The sewer is an interceptor and serves a separate sewered catchment. The wastewater originates from domestic sources and has a temperature of T= 15°C. The characteristics of the wastewater are approximately as depicted in Figure 3.10, i.e., the potential process rates for the aerobic transformations are relatively high. Only aerobic processes in the water phase are considered in the example. 

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