A simple case

Reactions in solution often have a contribution to the overall rate from reaction of the solvent with one or other of the reactants e.g., hydrolyses of esters in aqueous solution often have a contribution from a water rate. 

A typical base hydrolysis may have two contributing reactions  

Experimental rate constants gathered for experiments over a range of [OH ] should determine whether there is a water rate or not, and allow k and/or k2 to be found. 

If there is no water rate, then kobs values will be independent of [OH-]. If kobs does depend on [OH-], then a graph of kobs versus l/[OH ] should be linear, with slope = k2 and intercept = k.  

The method of analysis implies that kobs is found at various [OH-]. But there is a problem - OH is used up during each experiment. How can this be overcome  

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Cementation, the removal of a metal ion from solution by reduction of the metal with a more electropositive material, is also known as reductive precipitation. For a simple case the following can be used ... 

A simple case of heat conduction is a plate of finite thickness but infinite in other directions. If the temperature is constant around the plate, the material is assumed to have a constant thermal conductivity. In this case the linear temperature distribution and the heat flow through the plate is easy to determine from Fourier s law (Eq. (4.154)). 

Protons are equivalent to one another and have the same chemical shift when they are in equivalent environments. Often it is an easy matter to decide, simply by inspection, when protons are equivalent or not. In more difficult cases, mentally replacing a proton in a molecule by a test group can help. We ll illustrate the procedure for a simple case—the protons of propane. To see if they have the same chemical shift, replace one of the methyl protons at C-1 by chlorine, then do the same thing for a proton at C-3. Both replacements give the same molecule, 1-chloropropane. Therefore the methyl protons at C-1 are equivalent to those at C-3. 

The mechanism for the Wenker aziridine synthesis is very straightforward. As depicted by conversion 2—>3, the transformation is a simple case of intramolecular Sn2 displacement process, in which the sulfate ester is the leaving group. 

One way to separate two cations in water solution is to add an anion that precipitates only one of the cations. This approach is known as selective precipitation. To see how it works, consider a simple case, a solution containing Mg2+ and Na+ ions. Referring back to Table 16.1 (p. 433), you can see that Mg2+ forms a couple of insoluble compounds MgC03 (K = 6.8 X 10-6) and Mg(OH)2 (K = 6 X 10-12). In contrast, all of the common compounds of sodium are soluble, including the carbonate and hydroxide. It follows that you could readily separate Mg2+ from Na+ by adding either C032- or OH- ions to the solution. In either case, Mg2+ will precipitate while Na+ remains in solution. 

In Sections 10.11-10.16 it is shown how the change in pH during acid-base titrations may be calculated, and how the titration curves thus obtained can be used (a) to ascertain the most suitable indicator to be used in a given titration, and (b) to determine the titration error. Similar procedures may be carried out for oxidation-reduction titrations. Consider first a simple case which involves only change in ionic charge, and is theoretically independent of the hydrogen-ion concentration. A suitable example, for purposes of illustration, is the titration of 100 mL of 0.1M iron(II) with 0.1M cerium(IV) in the presence of dilute sulphuric acid ... 

For this calculation, let us assume a simple case in which a suitable monochromatic beam of intensity /x falls perpendicularly upon unit area (1 square centimeter) of a monolayer of cobalt atoms resting upon a transparent substrate. The first term in Equation 4-9, A I, is given by... 

This qualitative discussion will be illustrated by a simple case. Consider the determination of iron in the five samples of Table 7-1, the analytical line being iron Ka, and all the samples being assumed uniform. Figure 7-2 gives some of the spectra significant for Table 7-1. 

The analytical chemist often wishes to express the counting error relative to the amount present of the element sought in a simple case we have... 

To illustrate such a calculation, Balder (Bl) considered a simple case wherein he assumed that the (symmetric) excess Gibbs energy of the ternary system is given by a two-suffix Margules expansion ... 

Therefore, the development of an open system can be described by a set of nonlinear equations that usually have solutions in equilibrium at infinity. In some cases, the solutions change their states greatly before and after the specific values of physical parameters these phenomena are called bifurcations. Figure 1 shows a simple case of bifurcation. For example, the following nonlinear differential equation is considered,... 

Consider some particular cases, which are of interest to the understanding of the general flow properties in a heated capillary. First, we examine a simple case when the Peclet number is much less than unity. Since Zl < 1, the assumption Pcl 1 corresponds to the condition Pcl l C 1. In this case Eq. (9.62) reduces to... 

Before proposing a general formulation, we illustrate a DIFF with a simple case, in which we consider bone collagen as the body component, and the diet as containingjust two components, protein and non-protein. In the notation, B stands for body 5 values, and D for diet 8 values suffixes distinguish the particular component specified. Thus Bcolla stands for the collagen 8 C, Dp and Dn for the dietary protein and non-protein 6 C values. 

Assiune that we already know the relevant values of d,j and flj(). A simple case, often implicit in past diet reconstructions, is to assume that all d,j = -l-5%o, where j corresponds to collagen, and that all fy() = 1. If we consider the diet to be partitioned into n components, it is specified by 2n - 1 values (the n 8s of Dt and the n - 1 values of F,) and these can only be evaluated from... 

Let s start off with a simple case where we only have two stereocenters. Consider the two compounds below ... 

Let a simple case of heat transfer by convection be considered. Let it be assumed that the surface of a solid at temperature is in contact with a flowing fluid at temperature T2 (Tx > T2) as indicated in Figure 3.17. Let dQJdt represent the rate ofheat transfer by convection from the solid over a finite area of solid-fluid contact, S. In such a situation it is found in many cases that dQJdt is proportional to S (Tx - T2), so that... 

A simple case is the diffusion of a single type of ion in a solution containing a sufficient excess of an indifferent electrolyte (see page 116), which then occurs in the same way as in the case of a non-electrolyte. Isotope (tracer) diffusion has the same character, where a concentration gradient of the radioactive isotope of an ion, present in a much lower concentration, is formed in a solution with a much larger, constant salt concentration. 

Electrochemical reactions are driven by the potential difference at the solid liquid interface, which is established by the electrochemical double layer composed, in a simple case, of water and two types of counter ions. Thus, provided the electrochemical interface is preserved upon emersion and transfer, one always has to deal with a complex coadsorption experiment. In contrast to the solid/vacuum interface, where for instance metal adsorption can be studied by evaporating a metal onto the surface, electrochemical metal deposition is always a coadsorption of metal ions, counter ions, and probably water dipols, which together cause the potential difference at the surface. This complex situation has to be taken into account when interpreting XPS data of emersed electrode surfaces in terms of chemical shifts or binding energies. 

Once these nuclides deposit on the ocean floor they are likely to be subjected to particle mixing processes. In the following we discuss attenuation due to a simple case of mixing, in which the sedimentary particles are mixed to a constant depth, L, from the sediment-water interface [72,73]. For such a case the temporal variation in the standing crop (atoms/cm2) C, of the radionuclide in the mixed layer is given by ... 

The band structure that appears as a consequence of the periodic potential provides a logical explanation of the different conductivities of electrons in solids. It is a simple case of how the energy bands are structured and arranged with respect to the Fermi level. In general, for any solid there is a set of energy bands, each separated from the next by an energy gap. The top of this set of bands (the valence band) intersects the Fermi level and will be either full of electrons, partially filled, or empty. 

The following example illustrates a simple case of optimal operation of a multistage CSTR to minimize the total volume. We continue to assume a constant-density system with isothermal operation. 

Here we derive Avrami s theorem for a simple case [4]. Consider an area A that is partially covered by N circles each of area a, where a A. The circles overlap so that the area that is actually covered is smaller than the extended area Na. Show that the probability that a particular point in not covered by any circle is ... 

The conditions for mechanical equilibrium can now be applied to a simple case of great practical importance. Let us consider the interfaces that occur when a liquid phase is brought into equilibrium with a solid surface in a gaseous... 

H2, N2, or CO dissociates on a surface, we need to take two orbitals of the molecule into account, the highest occupied and the lowest unoccupied molecular orbital (the HOMO and LUMO of the so-called frontier orbital concept). Let us take a simple case to start with the molecule A2 with occupied bonding level a and unoccupied anti-bonding level a. We use jellium as the substrate metal and discuss the chemisorption of A2 in the resonant level model. What happens is that the two levels broaden because of the rather weak interaction with the free electron cloud of the metal. 

As a simple case, reconcile a single data set for the stream flows as follows ... 

Analogous to simple carbocyclic aryl halides, 5-halopyrimidines readily take part in Pd-catalyzed olefinations under standard Heck conditions. In a simple case, Yamanaka et aL synthesized ethyl 2,4-dimethyl-5-pyrimidineacrylate (102) via the Heck reaction of 5-iodo-2,4-dimethylpyrimidine and ethyl acrylate [70]. 

There are, however, other components where the process of degradation is clearer and the rate of change easier to deduce, particularly when that process is mechanical rather than chemical. This would be so in a simple case where surface wear proceeded at a linear rate. Discussion of these problems will explain why there is still a shortage of reliable data and also why much durability testing falls short of the breadth and quality necessary to make reliable predictions. 

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