Search Results

This chapter derives formulas for gross sections of potential scattering, and shows their generalizations to nuclear reactions. A projection technique is introduced, which allows one to separate in the T-matrix the part which varies smoothly in energy from that for resonances. For the latter, general expressions are deduced which involve partial and total widths. Non-hermitean Hamiltonians for the optical model are obtained by applying suitable energy averages. The doorway mechanism is explained by which the intermediate structure in the resonances can be understood: it employs widths for decays to the continuum and to the compound states. The statistical theory is addressed, for which on the basis of general assumptions, analytic formulas for the fluctuating cross section are derived, which are in accord with N. Bohr's hypothesis of the independence of entrance and exit channels. The Weisskopf-Ewing relations and the Hauser-Feshbach theory are described, which involve branching ratios and formation probabilities. A critique of the statistical model is included.

Chemical kinetics is the aspect of chemistry that deals with the speed or rate of chemical reactions and the mechanisms by which they occur. The rate of a chemical reaction is a measure of how fast the reaction occurs, and it is defined as the change in the amount or concentration of a reactant or product per unit time. The mechanism of a reaction is the series of steps or processes through which it occurs. Most experimental techniques for determining reaction rates involve measuring of the rate of disappearance of a reactant, or the rate of appearance of a product. For a reaction in which the reactant Y is converted to some products: Rate = Concentration of Y at time t2 −Concentration of Y at time t1/t2 −t1 Rate = Δ [Y]/ Δt where [Y] indicates the molar concentration of the reactant of interest, and Δ refers to a change in the given amount. Rate for a reactant, by this definition, is a negative number. For a product, it is positive. The value of the rate at a particular time is known as the instantaneous rate and will be different from the average rate. Its value can be obtained from the plot of concentration (mol/L) vs. time (s) as the slope of a line tangent to the curve at a given point. Consider the following kinetic data for the decomposition of N2O5 to gaseous NO2 and O2 at 40°C (see table 16-3). A plot of [N2O5] vs. time is shown in figure 16-2. From this curve, the instantaneous rate of reaction at any time t can be obtained from the slope of the tangent to the curve. This corresponds to the value of Δ [N2O5]/ Δt for the tangent at a given instant. The instantaneous rate at the beginning of the reaction (t =0) is known as the initial rate.

In most undergraduate chemistry classes, students are taught to consider reactions in which cations and anions dissolved in water are depicted as isolated ions. For example, the magnesium ion is depicted as Mg2+, or at best Mg2+(aq). For anions, these descriptions may be adequate (if not accurate). However, for cations, these abbreviations almost always fail to describe the critical chemical attributes of the dissolved species. A much more meaningful description of Mg2+ dissolved in water is [Mg(H2O)6]2+, because Mg2+ in water does not behave like a bare Mg2+ ion, nor do the waters coordinated to the Mg2+ behave anything like water molecules in the bulk fluid. In many respects, the [Mg(H2O)6]2+ ion acts like a dissolved molecular species. In this chapter, we discuss the simple solvation of anions and cations as a prelude to exploring more complex reactions of soluble oxide precursors called hydrolysis products. The two key classes of water–oxide reactions introduced here are acid–base and ligand exchange. First, consider how simple anions modify the structure and properties of water. As discussed in Chapter 3, water is a dynamic and highly fluxional “oxide” containing transient rings and clusters based on tetrahedral oxygen anions held together by linear hydrogen bonds. Simple halide ions can insert into this structure by occupying sites that would normally be occupied by other water molecules because they have radii (ranging from 0.13 to 0.22 nm in the series from F− to I−) that are comparable to that of the O2− ion (0.14 nm). Such substitution is clearly seen in the structures of ionic clathrate hydrates, where the anion can replace one and sometimes even two water molecules. Larger anions can also replace water molecules within clathrate hydrate cages. For example, carboxylate hydrate structures incorporate the carboxylate group within the water framework whereas the hydrophobic hydrocarbon “tails” occupy a cavity within the water framework, as in methane hydrate (see Chapter 3). Water molecules form hydrogen bonds to dissolved halide ions just as they can to other water molecules, as designated by OH−Y−.

Although dissolution reactions involving water can etch and decompose oxides, truly catastrophic failures of oxide structures usually involve fractures and mechanical failures. Geologists and geochemists have long recognized that water and ice both play key roles in promoting the fracture and crumbling of rock (see Chapter 17). Freezing and thawing create stresses that amplify the rate at which water attacks metal–oxygen bonds at the crack tip. The interplay between water and stressed oxides also leads to common failures in man-made objects, ranging from the growth of cracks from flaws in windshields to the rupture of optical fibers in communication systems. In this chapter, we outline how mechanical deformations change the reactivity of metal–oxygen bonds with respect to water and other chemicals, and how reactions on strained model compounds have been used to predict time to failure as a function of applied stress. The basic phenomenon of stress corrosion cracking is illustrated in Figure 16.1. Cracks can propagate through oxide materials at extremely fast rates, as anyone who has dropped a wine glass on the floor can attest. High-speed photography reveals that when glass shatters, cracks can spread at speeds of hundreds of meters per second, or half the speed of sound in the glass. At the other end of the spectrum, cracks in glass can grow from preexisting flaws so slowly that only a few chemical bonds are broken at the crack tip per hour. Because mechanical failures are associated with cracking, it is critical for design engineers to understand the factors that control crack growth rates for this enormous range of crack velocities (a factor of 1012). In addition, because it is difficult to measure crack velocities slower than 10−8 m/second, it is often necessary to make major extrapolations from measured data to predict the long-term reliability of glass and ceramic objects. Will an optical fiber under stress fail in 1 year or 10 years? Answering this question can require accurate extrapolations down to crack growth rates as low as 10−10 m/second.