Solution curves

Figure A3.8.3 Quantum activation free energy curves calculated for the model A-H-A proton transfer reaction described 45. The frill line is for the classical limit of the proton transfer solute in isolation, while the other curves are for different fully quantized cases. The rigid curves were calculated by keeping the A-A distance fixed. An important feature here is the direct effect of the solvent activation process on both the solvated rigid and flexible solute curves. Another feature is the effect of a fluctuating A-A distance which both lowers the activation free energy and reduces the influence of the solvent. The latter feature enliances the rate by a factor of 20 over the rigid case.

Iron, cobalt, and nickel catalyze this reaction. The rate depends on temperature and sodium concentration. At —33.5°C, 0.251 kg sodium is soluble in 1 kg ammonia. Concentrated solutions of sodium in ammonia separate into two Hquid phases when cooled below the consolute temperature of —41.6°C. The compositions of the phases depend on the temperature. At the peak of the conjugate solutions curve, the composition is 4.15 atom % sodium. The density decreases with increasing concentration of sodium. Thus, in the two-phase region the dilute bottom phase, low in sodium concentration, has a deep-blue color the light top phase, high in sodium concentration, has a metallic bronze appearance (9—13). 

Plox X (= 0.0215) and Y (= 0.0035) to get equilibrium curve, which accounts for this effect of heat of solution Curve B, (Figure 9-77). 

Figure 1. Oxidation of 2-deoxy-D-glucose (6 10 4M) with sodium metaperiodate (6.6 10 SM) in 0.1N H2SO4 (curve A) and in unbuffered solution (curve...

Strong acid with a weak base. The titration of a strong acid with a moderately weak base (K sslO-5) may be illustrated by the neutralisation of dilute sulphuric acid by dilute ammonia solution [curves 1 and 3, Fig. 13.2(a)]. The first branch of the graph reflects the disappearance of the hydrogen ions during the neutralisation, but after the end point has been reached the graph becomes almost horizontal, since the excess aqueous ammonia is not appreciably ionised in the presence of ammonium sulphate. 

Fig. 4. Swelling ratio of PAAm hydrogels containing varying amount of AAc units as a function of CuCl2 concentration in the outer solution. Curves are numbered with respect to increasing ionic group content 0 (0), 36.5 (J), 107 (2), 145 (3), and 212 mM (4). From Rieka and Tanaka [101]...

As we have seen before, exact differentials correspond to the total differential of a state function, while inexact differentials are associated with quantities that are not state functions, but are path-dependent. Caratheodory proved a purely mathematical theorem, with no reference to physical systems, that establishes the condition for the existence of an integrating denominator for differential expressions of the form of equation (2.44). Called the Caratheodory theorem, it asserts that an integrating denominator exists for Pfaffian differentials, Sq, when there exist final states specified by ( V, ... x )j that are inaccessible from some initial state (.vj,.... v )in by a path for which Sq = 0. Such paths are called solution curves of the differential expression The connection from the purely mathematical realm to thermodynamic systems is established by recognizing that we can express the differential expressions for heat transfer during a reversible thermodynamic process, 6qrey as Pfaffian differentials of the form given by equation (2.44). Then, solution curves (for which Sqrev = 0) correspond to reversible adiabatic processes in which no heat is absorbed or released. 

According to the Caratheodory theorem, the existence of an integrating denominator that creates an exact differential (state function) out of any inexact differential is tied to the existence of points (specified by the values of their x, s) that cannot be reached from a given point by an adiabatic path (a solution curve), Caratheodory showed that, based upon the earlier statements of the Second Law, such states exist for the flow of heat in a reversible process, so that the theorem becomes applicable to this physical process. This conclusion, which is still another way of stating the Second Law, is known as the Caratheodory principle. It can be stated as... 

We distinguish the solution curves from solution surfaces in Figure 2.12. The curved solid path marked 6qKV = 0 is a solution curve, while the solution surfaces are designated by S, Sj, and S3. Each surface corresponds to a different value for the constant entropy. From equation (2.38)... 

Presumably all points on the same surface can be connected by some solution curve (reversible adiabatic process). Flowever, states on surface S2, for example, cannot be connected to states on either Si or S3 by any reversible adiabatic path. Rather, if they can be connected, it must be through irreversible adiabatic paths for which dS 0. We represent two such paths in Figure 2.12 by dashed lines. 

A1.5c Differential Equations, Solution Curves, and Solution Surfaces... 

Within each solution surface are numerous subsets of points that also satisfy the differential equation bQ = dF = 0. These subsets are referred to as solution curves of the Pfaffian. The curve z — 0, y + y2 = 25.00 is one of the solution curves for our particular solution surface with radius = 5.00. Others would include x = 0, y2 + z2 — 25.00, and r — 0,. v2 + r2 = 25.00. Solution curves on the same solution surface can intersect. For example, our first two solution curves intersect at two points (5, 0, 0) and (-5, 0. 0). However, solution curves on one surface cannot be solution curves for another surface since the surfaces do not intersect. That two solution surfaces to an exact Pfaffian differential equation cannot intersect and that solution curves for one surface cannot be solution curves for another have important consequences as we see in our discussion of the Caratheodory formulation of the Second Law of Thermodynamics. 

When the Pfaffian expression is inexact but integrable, then an integrating factor A exists such that AbQ = d5, where dS is an exact differential and the solution surfaces are S = constant. While solution surfaces do not exist for the inexact differential 8Q, solution curves do exist. The solution curves to dS = 0 will also be solution curves to bQ = 0. Since solution curves for dS on one surface do not intersect those on another surface, a solution curve for 8Q — 0 that lies on one surface cannot intersect another solution curve for bQ = 0 that lies on a different surface. 

Thus, exact or integrable Pfaffians lead to non-intersecting solution surfaces, which requires that solution curves that lie on different solution surfaces cannot intersect. For a given point p. there will be numerous other points in very close proximity to p that cannot be connected to p by a solution curve to the Pfaffian differential equation. No such condition exists for non-integrable Pfaffians, and, in general, one can construct a solution curve from one point to any other point in space. (However, the process might not be a trivial exercise.)... 

Pfaffian differentials 22-3. 608-11 in Second law of thermodynamics 63-7 three or more variables 67 two variables 64-6 solution curves and surfaces 610-11 in three dimensions 609 in two dimensions 611 phase equilibria criteria for 231-7... 

The cleavage mechanism can be clarified by cyclic voltammetries as shown in Figure 5. In aprotic solution (curves a) steps (l) and (2) correspond to the successive electron transfers leading finally to the dianion. On the other hand, in protic solution (curve c), step (2) has disappeared while step (l) has grown and then obviously corresponds to an ECE process. Anyhow, and whatever the medium, step (3) is identified as that in which the produced olefin (here 1,1-diphenylethylene) is reduced in all cases. 

Fig. 11. Nonlinear scattering spectra for chloroform solution of nitro-azobenzene dendrimer (curve a) and neat chloroform solution (curve b). Inset is absorption spectrum of sample...

Fig. 10 a UV-Vis DRS spectra of TS-1 (curve 1, full line), immediately after contact with H2O2/H2O solution (curve , dotted line), after time elapse of 24h (curve 3, dashed line) and after subsequent H2O dosage (curve 4, scattered squares), b as for a for the XANES spectra, c as for a for the -weighted, phase imcorrected, FT of the EXAFS spectra. Spectra 2-4 of b and c have been reordered at liquid nitrogen temperatime. Adapted from [49] with permission. Copyright (2004) by ACS... 

FIGURE 14.6 Influence of surface-active ions [N(C4H9)4]+ (curve 2) and I (curve 3) on the polarization curve for hydrogen evolution at a mercury electrode in acidic solutions (curve 1 is for the base electrolyte). 

The electrochemical response of analytes at the CNT-modified electrodes is influenced by the surfactants which are used as dispersants. CNT-modified electrodes using cationic surfactant CTAB as a dispersant showed an improved catalytic effect for negatively charged small molecular analytes, such as potassium ferricyanide and ascorbic acid, whereas anionic surfactants such as SDS showed a better catalytic activity for a positively charged analyte such as dopamine. This effect, which is ascribed mainly to the electrostatic interactions, is also observed for the electrochemical response of a negatively charged macromolecule such as DNA on the CNT (surfactant)-modified electrodes (see Fig. 15.12). An oxidation peak current near +1.0 V was observed only at the CNT/CTAB-modified electrode in the DNA solution (curve (ii) in Fig. 15.12a). The differential pulse voltammetry of DNA at the CNT/CTAB-modified electrode also showed a sharp peak current, which is due to the oxidation of the adenine residue in DNA (curve (ii) in Fig. 15.12b). The different effects of surfactants for CNTs to promote the electron transfer of DNA are in agreement with the electrostatic interactions... 

In the event of technological solutions, curves of cyanide destruction have horizontal sections which occurrence is caused by various speed of degradation of free cyanide and cyanide, combined in complex with metal. 

Figure 13. Plot of current against time, (a), (b), and (c) Current-time transients on Pt(l 11) in 0.5M H2SO4, when CO was endorsed at potentials of 0.08.0.3, and 0.5 V(RHE) of curve 3 of (d), respectively. When CO was ptesent in the solution, curves I and 2 were observed. The sweep rate was 50 mV s". (From Ref. 24.)...

The shape of the solution curve depends strongly on the values of the physical parameter C. the damping coefficient. Let us now look at the various possibilities. 

The curve with B/RT= 2.00 is the one with the highest value of B/RT in a one phase system. It is called the critical solution curve. We can determine the thermodynamic significance of B from the following analysis. 

Figure 21.3. Concentration distribution of solute in solution at sedimentation equilibrium. Curve A represents ideal behavior of a monodisperse solute curve B represents nonideality and curve C represents a polydisperse system.

Fig. 5 Voltammetric pattern obtained with 10-= M a-K4SiWi2O40 in DMF-1-0.1 M LiCl04 solution - Curve 1 in the absence of acid Curve 2 with 2 10 M perchloric acid curve 3 with 5 10 M HCIO4. The pattern is restricted to the evolution of the first wave observed in the absence of acid. Sweep rate 100 mV s GC electrode surface area 0.07 cm (taken from Ref 33c).

Fig. 7. Solvent weight fraction in P(MAA-g-EG) complexing gels vs swelling pH in distilled water (curve /) or in 0.1 M NaCI solution (curve 2) at 37 °C. Gels prepared with PEG of M = 200 and PEG MAA = 60 40...

Fig. 8. Nuclear Overhauser enhancement of PEG-ethylene protons vs irradiation time in P(MAA-j -EG) gels exhibiting complexation. Proton enhancements of graft copolymer with PEG M = 400 in D20 (curve 1), graft copolymer in NaOD solution (curve 2), and polymer mixture with PEG M = 1000 in D20 (curve i). The PEG concentration was 0.01 wt%, PMAA concentration was 0.09 wt%, copolymer concentration was 0.1 wt%, and temperature was 21 °C...

While in the first step the deviation from the exact solution stems only from approximating the solution curve by its tangent line, in further steps we calculate the slope at the current approximation y1 instead of the unknown true value (), thereby introducing additional errors. The solution of (5.2) is given by (5.3), and the total error Ei = y(t ) - y for this simple equation is... 

Typical results are shown in Fig. 44. The spectral threshold of the proper photoconductivity and the photo-emf of PAC is situated at 520 nm. The spectral response for the photo emf of PAC itself is shown by curve 1. After PAC has been immersed in an ethanol solution of methylene blue and dried its spectral response is represented by curves 2 and 2. The photo-response appears in the range of the absorption maximum of the dye at 680 nm characteristic of the monomolecular form in the dilute initial solution (curve 3). The observed enhancement of the second maximum at 620 nm in comparison to the solution spectrum is obviously connected with the presence of dye dimers. The shift of the maximum photoresponse to the longer wavelength by 15 nm relatively to the solution is usually the case for the adsorbed state. The sign of the charge carriers both in the proper and sensitized spectra ranges is positive. As seen in Fig. 44 the adsorption of the dye also markedly changes the proper photosensitivity of the PAC. When the monomolecular form of the adsorbed dye dominates, the... 

Curves 4 and 4 in Fig. 5.6 show an example of the current-potential relation obtained for an irreversible electrode process. For a reversible electrode process, the reduction wave appears at the same potential as the oxidation wave, giving an oxidation-reduction wave if both Ox and Red exist in the solution (curves 1, 2 and 3 in Fig. 5.6). For an irreversible process, however, the reduction wave (curve 4) is clearly separated from the oxidation wave (curve 4 ), although the limiting currents for the two waves are the same as those in the reversible process. The cur-rent-potential relation for the irreversible reduction process can be expressed by... 

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