Hypothetical systems

Figure III-l depicts a hypothetical system consisting of some liquid that fills a box having a sliding cover the material of the cover is such that the interfacial tension between it and the liquid is zero. If the cover is slid back so as to uncover an amount of surface dJl, the work required to do so will he ydSl. This is reversible work at constant pressure and temperature and thus gives the increase in free energy of the system (see Section XVII-12 for a more detailed discussion of the thermodynamics of surfaces).

The electronic wave functions of the different spin-paired systems are not necessarily linearly independent. Writing out the VB wave function shows that one of them may be expressed as a linear combination of the other two. Nevertheless, each of them is obviously a separate chemical entity, that can he clearly distinguished from the other two. [This is readily checked by considering a hypothetical system containing four isotopic H atoms (H, D, T, and U). The anchors will be HD - - TU, HT - - DU, and HU -I- DT],... 

In an emission spectrum a fixed wavelength is used to excite the molecules, and the intensity of emitted radiation is monitored as a function of wavelength. Although a molecule has only a single excitation spectrum, it has two emission spectra, one for fluorescence and one for phosphorescence. The corresponding emission spectra for the hypothetical system in Figure 10.43 are shown in Figure 10.44. 

We shall see in Sec. 9.10 that sedimentation and diffusion data yield experimental friction factors which may also be described-by the ratio of the experimental f to fQ, the friction factor of a sphere of the same mass-as contours in solvation-ellipticity plots. The two different kinds of contours differ in detailed shape, as illustrated in Fig. 9.4b, so the location at which they cross provides the desired characterization. For the hypothetical system shown in Fig. 9.4b, the axial ratio is about 2.5 and the protein is hydrated to the extent of about 1.0 g water (g polymer)". ... 

Fig. 2. SolubiHty diagram for a hypothetical system. The curves AB and BC represent solution compositions that are in equiHbrium with soHds whose compositions are given by the lines AD and CE. If AD and CE are vertical along the respective axes, the crystals are pure R and S, respectively. Crystallization from any solution whose composition is to the left of the vertical line through point B produces crystals of pure R, whereas solutions to the right of the line produce crystals of pure S. A solution whose composition falls on the line through B produces a soHd mixture that has a composition...

Several features of the hypothetical system in Eigure 2 can be used to illustrate proper selection of crystallizer operating conditions and limitations placed on the operation by system properties. Suppose a saturated solution at temperature is fed to a crystallizer operating at temperature T. Because the feed is saturated, the weight fraction of in the feed is given as shown in Eigure 2. The maximum crystal production rate from such a process depends on the value of and is given by... 

Figure 3-1 shows calculated plots of Eq. (3-16) for hypothetical systems in which kilk2 has the values 1 and 5. It is evident from the example in which k, = 5 2 that the curvature persists well into the reaction and that unambiguous identification of the terminal linear portion may be difficult. The long extrapolation to find eg is also uncertain. The accuracy of this procedure depends upon the ratios kilk2 and... 

Figure 3-1. Plots of Eq. (3-16) for hypothetical systems having k2 = 0.02 s and the indieated values of the ratio ktlk2- In both examples Ca = 0.7 and c% = 0.3.

Clearly Eqs. (3-27) and (3-29) are inapplicable in the special case ki = 2-Figure 3-2 shows the dependence of Ca, Cb, and Cc on time for a hypothetical system for which kdk2 = 4. The rise and then fall of Cb is characteristic substance B is an intermediate in the overall reaction in which A is transformed into C. Figure 3.3 is the same type of plot for a system having kdk2 = 1/4. In this case the concentration of B is at all times less than that in Fig. 3-2 B is a more reactive intermediate in Fig. 3-3. The time at which cb reaches its maximum value is found by setting dc ldt = 0 the result is... 

Figure 3-8 is a plot of Ca, Cb, Cq, and Cd for a hypothetical system of the Scheme X type. An interesting feature is the time delay after the start of the reaction before the final product, D, appears in significant concentrations. This delay in product appearance is called an induction period or lagtime. In order to observe an induction period it is only necessary that the system include several relatively stable intermediates, so that the bulk of the material balance is temporarily stored in these prior forms. An experimental measurement of the induction period requires an arbitrary definition of its length. 

Figure 3-9 shows plots of Eqs. (3-135) and (3-136) for some hypothetical systems. Obviously the equilibrium approximation is poor in the early stages of the reaction, but in the later stages the assumption can be quite good. The preequilibrium assumption, applied to Scheme XIV, amounts to the statement that 2 is negligible relative to ki and i. 

Figure 5-17. Rate-equilibrium relationship according to Eq. (5-69) for a hypothetical system with AG = 20.

The analysis of a k-pH curve in terms of Eq. (6-80) is treated by making approximations that are equivalent to ignoring some of the rate terms in certain pH regions. A common type of system is analyzed as an example the approach can be modified to suit a particular demand. The evaluation of k° and k" can nearly always be accomplished from rate data at very low and high pH, respectively. We are concerned with k, Kf, and /fj. Figures 6-16 and 6-17 are A -pH and log fc-pH plots for a hypothetical system described by Eq. (6-80). The analysis assumes this equation and these types of parameters. 

The Vext operator is equal to Vne for A = 1, for intermediate A values, however, it is assumed that the external potential Vext(A) is adjusted so that the same density is obtained for both A = 1 (the real system) and A = 0 (a hypothetical system with noninteracting electrons). For the A = 0 case the exact solution to the Schrddinger equation is given as a Slater determinant composed of (molecular) orbitals, for which the... 

A detailed explanation of the construction of thermodynamic phase stability diagrams may be found in References 22-25. In this section the basic principles of construction and interpretation for the specific situation of gas-metal equilibria will be addressed using a hypothetical system. 

For the purpose of interpretation, various hypotheses have been built up around the results which have been derived from the two laws of thermodynamics. It must not be forgotten, however, that the deductions of thermodynamics would stand quite firm if the whole hypothetical system collapsed about them. 

The curve in Fig. 2 might represent the relationship between a response Y and a single independent variable X in a hypothetical system, and since we can see the whole curve, we can pick out the highest point or lowest, the maximum or minimum. Use of calculus, however, makes the task of plotting the data or equation unnecessary. If the relationship, that is, the equation for Y as a function of X, is available [Eq. (1)] ... 

The original Hohenberg-Kohn theorem was directly applicable to complete systems [14], The first adaptation of the Hohenberg-Kohn theorem to a part of a system involved special conditions the subsystem considered was a part of a finite and bounded entity regarded as a hypothetical system [21], The boundedness condition, in fact, the presence of a boundary beyond which the hypothetical system did not extend, was a feature not fully compatible with quantum mechanics, where no such boundaries can exist for any system of electron density, such as a molecular electron density. As a consequence of the Heisenberg uncertainty relation, molecular electron densities cannot have boundaries, and in a rigorous sense, no finite volume, however large, can contain a complete molecule. 

The form of the distribution (Eq. 101), as shown in Fig. 7, qualitatively differs from that exhibited by this distribution for the products of homophase copolymerization. This distinction takes place both in real systems (ri < l,r2 < 1), where statistical copolymers are formed and in hypothetical systems (ri l,r2 1), where the formation of multiblock copolymers is expected. Essentially, the composition distribution in the latter systems... 

The copolymer composition produced by these two catalysts can be estimated using the Mayo-Lewis equation [38] and these values of i and r2. Figure 10 depicts the hypothetical comonomer content in the polymer (F2) as a function of the mole fraction of comonomer in the reactor (f2). The good incorporator produces a material with higher F2 as f2 increases. In contrast, the composition from the poor incorporator is relatively flat across a broad range and increases only at very high values of/2. The F2 required to render the copolymer amorphous is comonomer-dependent for 1-octene, this value is near 0.19. In this hypothetical system, the good incorporator produces that composition at f2 = 0.57, at which the poor incorporator incorporates very little comonomer (F2 = 0.01). 

Let us now return to our hypothetical system A-B where we also consider the liquid and where the solid and liquid solutions are both regular (following Pelton and... 

This hypothetical system requires minimal user inputs to determine a data window automatically from reference injections or prior history of the method and coordinate the LC and MS components as well as the LIMS system. This scheme is achievable with a small set of components but requires an industry-wide standard for component flexibility. 

These may be interpreted as the mole fractions of L and H in a hypothetical doubly-occupied system, where Sl = Sh= I - This is a hypothetical system, since in a real system the mole fractions of L and H are given by Eq. (4.5.9). 

The quantity (5) is an average of the direct correlations S. and 5yin a hypothetical system, where the mole fractions of the two configurations are and respectively (see Section 4.5). As such, (S) is bound by and Sj, but y(l, 1) must be larger than unity and is not bound by and In Table 4.4 we see that both the experimental and calculated values of 11 (1,1) are closer to the fumaric rather than the maleic values. One could argue that since the ionized succinic acid would be most of the time in the trans configuration, we should expect that the value of IV for the succinie aeid be closer to the fumaric acid value. The faet that the IV (succinic) is indeed intermediate between IV (maleic) and IV (fumaric) is quite accidental. The value of IV (succinic) is determined by both the negative correlation (5) and the positive correlation y(l, 1). 

The first, 7(0, can be determined from the experimental data for any finite value of C. The second, 7,<0, may be interpreted as the same integral 7(0, but computed for a hypothetical system of independent sites. As we shall see below, this interpretation is somewhat risky and should be avoided. The function 7,.(Q is better viewed as defined in Eq. (5.8.10), with the binding constants determined in Eq. (5.8.8). 

Bochvar, D. A., Gal pem, E. G., Hypothetical Systems Carbododecahedron, s-Icosahedron, and Carbo-s-Icosahedron. [in Russian], Dokl. Akad. Nauk SSSR 1973,209, 610. 

This, by itself, might be considered as a nice accomplishment. However, the instrumentation is at present still very far from the underlying utopistic ideal represented by a hypothetical system with a maximum field in excess of 10 T, high-resolution-grade field homogeneity and stability, and switching times of the order of just a few microseconds. 

Figure 1 Free energies for unreconstructed surface f, and reconstructed surface f, vs slope s. The critical slope, sc, and the slope of the surface at step bunches, St, are given by Sc = ,/ , and St = The thick curve in (a) represents the free energy of a hypothetical system... 

Operational-Scale Estimates Based on the resnlts of a treatabihty stndy at the ANL-East in Chicago, Illinois, the laboratory prepared a cost estimate for an operational-scale Ceramicrete stabilization system. The hypothetical system wonld treat waste in 55-gal batches at a rate of 3 batches per shift. The capital costs for the system were estimated at 2,000,000. This estimate included the cost of eqnipment design and development (D20934H, p. 15). 

The construct of Figure 11.3 is still an ideal, hypothetical system, nevertheless very interesting in one respect it shows that at least in principle, cellular life can be implemented by a very limited number of RNA genes. Those who believe in the RNA world may add that this basic simplicity indicates the predominant importance of RNA in the early stages of life. 

Consider a hypothetical system that has an initial abundance of a short-lived radionuclide Nr. This nuclide is present as a fixed fraction of the parent element and its abundance can be written as the ratio of the radionuclide to a stable reference isotope of the same element, Ns. When the short-lived nuclide has completely decayed to its daughter nuclide, D, we have ... 

When identifying the hypothetical system S we need u. The weighting function found in Example 5.6 is substituted for the input of the hypothetical system. This input does not contain an impulse or a unit step component, and hence we set DC = 0 and US = 0. The response of the hypothetical system equals the "Observed" response. The program is the one used in Example 5.6, only tha data lines are changed as follows ... 

The "weighting function" we found is that of the hypothetical system, therefore it is the absorption curve we were looking for. It is useful to compare it with the "true" input given in Table 5.4. In this special case the input function found and the "true" input are of the same analytical form, so we can compare the parameters of the two functions, as well. In realistic applications, however, we are not interested in the "analytical form" of the input function and rather the table of computed values is of primary interest. 

Consider now a situation where, instead of a measuring instrument, one inserts (Fig. 6.31) into the circuit a source of potential (e.g., an electronically regulated power supply). Here, the total potential difference across the cell must equal (in magnitude) that put out by the source.18 This is, in fact, the law of conservation of energy applied to an electrical circuit, or Kirchhoff s second law The algebraic sum of all potential differences around a closed circuit must be equal to zero. For the simple hypothetical system shown in Fig.6.32, one has... 

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