Basic Theories

Basic Theory. We consider first an ideal resonance lamp, emitting a non-reversed Doppler-shaped line at a Boltzmann temperature T,. Since most actual lamps operate at low pressures, pressure-broadening is small, as is natural broadening. 

Mitchell and M. W. Zemansky, Resonance Radiation and Excited Atoms , Cambridge University Press, 1971. 

the Doppte profile is a good apinoximation. The energy emitted at any frequency v in the profile, I(o ), is given in the ideal Doppler case by equation (1)/  

In this equation, constant and T,IT where T, is the ambient temperature, equal to that of the absorber in an atomic resonance experiment. u is a reduced frequency variable given by to = 2V In 2(v - vo)/Svd, where o is the frequency of line centre and Aib/cm is the Doppler width, given in turn by 

The absorption coefiScient of the absorber k, at any frequency in the line profile is approximated by equation (2) unless pressure broadening is important ip S 1300 Nm- ). 

This simple experiment illustrates the basic features of the limiting-current method. A particular electrode reaction proceeds at the highest possible rate, indicated by a current plateau. From the limiting current thus recorded the mass-transfer rate and the mass-transfer coefficient at the electrode in question may be determined. 

The term limiting-current density is used to describe the maximum rate at 100% current efficiency, at which a particular electrode reaction can proceed in the steady state. This rate is determined by the composition and transport properties of the electrolytic solution and by the hydrodynamic condition at the electrode surface. 

When a limiting current is encountered, it is almost always caused by the slowness of transport of charged (ionic) or uncharged (molecular) species through the solution.1 These species move toward the appropriate electrodes, where they are consumed in the electrode reaction, or in a reaction coupled with it. Whenever the supply of a dissolved species from the solution to the electrode surface becomes a rate-limiting factor, limiting-current phenomena may be observed. 

Electrode reactions are intrinsically surface reactions, and cause changes in the electrolyte composition near the surface. A thin layer, impoverished in the reacting species, develops at the electrode surface, and the ions or molecules move across this layer by diffusion down the concentration gradients. Ions move also under the influence of the applied electric field that is, they migrate. 

Early investigators assumed that this so-called diffusion layer was stagnant (Nernst-Whitman model), and that the concentration profile of the reacting ion was linear, with the film thickness 5N chosen to give the actual concentration gradient at the electrode. In reality, however, the thin diffusion layer is not stagnant, and the fictitious t5N is always smaller than the real mass-transfer boundary-layer thickness (Fig. 2). However, since the actual concentration profile tapers off gradually to the bulk value of the concentration, the well-defined Nernst diffusion layer thickness has retained a certain convenience in practical calculations. 

The premise behind DFT is that the energy of a molecule can be determined from the electron density instead of a wave function. This theory originated with a theorem by Hoenburg and Kohn that stated this was possible. The original theorem applied only to finding the ground-state electronic energy of a molecule. A practical application of this theory was developed by Kohn and Sham who formulated a method similar in structure to the Hartree-Fock method. 

In this formulation, the electron density is expressed as a linear combination of basis functions similar in mathematical form to HF orbitals. A determinant is then formed from these functions, called Kohn-Sham orbitals. It is the electron density from this determinant of orbitals that is used to compute the energy. This procedure is necessary because Fermion systems can only have electron densities that arise from an antisymmetric wave function. There has been some debate over the interpretation of Kohn-Sham orbitals. It is certain that they are not mathematically equivalent to either HF orbitals or natural orbitals from correlated calculations. However, Kohn-Sham orbitals do describe the behavior of electrons in a molecule, just as the other orbitals mentioned do. DFT orbital eigenvalues do not match the energies obtained from photoelectron spectroscopy experiments as well as HF orbital energies do. The questions still being debated are how to assign similarities and how to physically interpret the differences. 

A density functional is then used to obtain the energy for the electron density. A functional is a function of a function, in this case, the electron density. The exact density functional is not known. Therefore, there is a whole list of different functionals that may have advantages or disadvantages. Some of these 

The advantage of using electron density is that the integrals for Coulomb repulsion need be done only over the electron density, which is a three-dimensional function, thus scaling as N. Furthermore, at least some electron correlation can be included in the calculation. This results in faster calculations than HF calculations (which scale as N ) and computations that are a bit more accurate as well. The better DFT fimctionals give results with an accuracy similar to that of an MP2 calculation. 

Density functionals can be broken down into several classes. The simplest is called the Xa method. This type of calculation includes electron exchange but not correlation. It was introduced by J. C. Slater, who in attempting to make an approximation to Hartree-Fock unwittingly discovered the simplest form of DFT. The Xa method is similar in accuracy to HF and sometimes better. 

The movement of charged particles (ions) in an electric field is called electrophoresis. The basic theory of electrophoresis is related to ionic mobility u, which is also called electrophoretic mobility. When an ion in solution is moving in the direction of a field E, its velocity v depends on three factors the charge z carried by the ion, the frictional coefficient / arising from the resistance of the solution, and the strength of the field E. The quantity E is defined as 

A classical example is the electrophoresis of 0.02 M NaCl. When a current of 1.60 mA was used, the boundary moved 0.020 m in 689 s. The cell is a tube with an inner radius of 0.188 cm. The specific conductance of the solution was k = 1.26 S at 25°C. The electric field strength and the mobility of an ion can then be calculated as follows  

Historically, ionic mobility is measured in terms of conductance because there is a linear relationship between the conductance A and the ionic mobility w in a dilute solution  

Ionic Atmosphere and Mobility Due to electrostatic forces, an ion is always surrounded by many other ions of opposite charge which form an ionic atmosphere. The ionic atmosphere can affect the conductance and mobility of the central ion in 

Viscous Effect Opposing the electrical force that exists between the ion and the field is a frictional viscous drag of the solvent, which, in many cases lowers the conductance and the mobility of the ion. The frictional drag is usually expressed by Stokes law  

As described earlier, Vmax is the maximal rate, and A is the Michaelis constant, which can in practice be viewed as the substrate concentration at half maximal rate. A brief review of the relevant background follows. 

In this scheme, here E, S, [ES] and P are the concentrations of the enzymes, substrates, substrate-bound enzymes, and enzyme products (metabolites), respectively, fci is the association rate constant, and k2 and kj are the dissociation rate constants from [ES] to here E, S, [ES] and P, respectively. 

Consider, once again, a one-dimensional discrete-time map 

Having a complete description of the dynamics of / [0,1] —t [0,1] is equivalent to knowing all orbits X xq) = xq,x, X2,. .. for all initial points xq. An n -order coarse-grained description of the same dynamics may be obtained by (i) partitioning the unit interval into n (not necessarily equal sized) bins, each of which is labeled with a symbol G 0,1. n - 1, and (ii) replacing each point, Xi G X xq), of an orbit by the symbol fti, corresponding to the bin in which that point is situated  

The time evolution of the function f is thus replaced by a sequence of discrete symbols labeling the bins visited by each point of the orbit. Because of the coarse-graining of the phase space, however, detailed knowledge of the actual orbits is generally lost i.e. many different orbits may yield the same symbolic sequence. Different state-space partitionings also generally give rise to different symbolic representations. 

we introduce the shift mapping, 7, which truncates the first orbit symbol and shifts all other symbols one place to the left  

It is easy to check that 7 F — F is continuous under the metric d . 

The reason that a compound ion can be field dissociated can be easily understood from a potential energy diagram as shown in Fig. 2.23. When r is in the same direction as F, the potential energy curve with respect to the center of mass, V(rn) is reduced by the field. Thus the potential barrier width is now finite, and the vibrating particles can dissociate from one another by quantum mechanical tunneling effect. Rigorously speaking, it 

In pulsed-laser field desorption of helium from rhodium surfaces, if the 

Using the field distribution of eq. (2.13), one can show that the peak position of the secondary peak is given by, 

The theoretical bases of QM/MM methods have been covered in detail by many groups [41,90,112,121,122], and so will only briefly be outlined here. 

The QM/MM partitioning is simplest when the border between the QM and MM regions can be considered not to separate covalently bonded atoms and will be considered initially. Following Field et al. [41], the effective Hamiltonian for the whole QM/MM system can be considered as being made up of various terms  

For example, periodic boundary conditions can be applied [124] (although special consideration must be given to the interaction of the QM group with its images), or where only part of a protein can be included, the stochastic boundary method for dynamics can be applied [9,125]. 

The (MM) parameters for the QM atoms can be optimized to reproduce experimental or high level ab initio results for small complexes [126,127]. 

The local self-consistent field (LSCF) or fragment SCF method has been developed for treating large systems [105,134-139], in which the bonds at the QM/MM junction ( frontier bonds ) are described by strictly localized bond orbitals. These frozen localized bond orbitals are taken from calculations on small models, and remain unchanged in the QM/MM calculation. The LSCF method has been applied at the semiempirical level [134-137], and some developments for ab initio calculations have been made [139]. Gao et al. have developed a similar Generalized Hybrid Orbital method for semiempirical QM/MM calculations, in which the semiempirical parameters of atoms at the junction are modified to enhance the transferability of the localized bond orbitals [140]. Recent developments for ab initio QM/MM calculations include the method of Phillip and Friesner [141], who use Boys-localized orbitals in ab initio Hartree-Fock QM/MM calculations. These orbitals are again taken from calculations on small model systems, and kept frozen in QM/MM calculations. 

Consider a solution containing a multiply charged transition metal ion M + in low concentration and a monovalent cation A+ (Na+ or H+) in a significantly higher concentration. When this solution is in contact with a cation exchanger in the A+-form, the following equilibrium is established  

With the law of mass action the equilibrium constant or selectivity coefficient is expressed as  

If we are only dealing with an ion-exchange mechanism, the distribution coefficient DM is given by the ratio of the analyte concentration M + in the stationary and in the mobile phase at equilibrium  

The metal ion concentration is very low compared with the eluent concentration, hence  

It follows that Kma is constant for any given eluent concentration. Since, in good approximation, [RA] is also constant and represents the exchange capacity C of the resin, the distribution coefficient may be expressed as  

Ag AgCl, KC1 salt bridge sample membrane internal solution, AgCl Ag Reference electrode Ion-selective electrode 

The cell potential is the sum of a number of local potential differences generated at the solid/solid, solid/liquid, and liquid/liquid interfaces within the cell and measured between the two reference electrodes. At zero current, the cell potential, Eceu, is given by 

2 (a) Schematic diagram of a potentiometric measuring circuit, (b) Potential 

In ion-selective electrode potentiometry, the cell potential reflects the dependence of the membrane potential on the primary ion activity (concentration). According to the Teorell-Meyer-Sievers (TMS) theory, the sum of 

For an ideally selective electrode, the measured cell potential is described by the Nemst equation  

Steady-state kinetics applies whenever the concentration of the substrate is well above that of the enzyme, so that the rate of change of substrate concentration greatly exceeds the rate of change of the concentration of any enzyme form. The resulting equations for velocity are the ratio of polynomials in reactant concentrations, and if only one substrate concentration is varied, the velocity is usually given by 

Equation (1) is usually inverted for plotting and analysis, since this gives a straight line with slope K V (the reciprocal of VIK) and vertical intercept 1/V (the reciprocal of V) when 1/v is plotted versus 1/A  

In analyzing initial velocity or inhibition patterns, one considers separately the effects on the slopes of such reciprocal plots (which represent effects on VIK) and on the intercepts (which represent effects on V). Similarly, one considers isotope effects on V or VIK separately, and one plots the logarithms of V or VIK versus pH for pH profiles. 

Before polarization is discussed, it is imperative to understand how one measures polarization and to gain a qualitative understanding of how readily or not so readily polarizable a solid is. Consider two metal parallel plates of area A separated by a distance d in vacuum (Fig. 14.1a). Attaching these plates to the simple electric circuit, shown in Fig. 14.1a, and closing the circuit will result in a transient surge of current that rapidly decays to zero. 

Repeating the experiment at different voltages V and plotting Q versus V should yield a straight line, as shown in Fig. 14.2. In other words, the well-known relationship 

Since e is always greater than q, the minimum value for A is 1. By combining Eqs. (14.4) and (14.5), the capacitance of the metal plates separated by the dielectric is 

Thus k is a dimensionless parameter that compares the charge-storing capacity of a material to that of vacuum. 

The foregoing discussion can be summarized as follows when a voltage is applied to a parallel-plate capacitor in vacuum, the capacitor will store charge. In the presence of a dielectric, an additional something happens within that dielectric which allows the capacitor to store more charge. The purpose of this chapter is to explore the nature of this something. First, however, a few more concepts need to be clarified. 

The valencies of main-group elements can be interpreted on the basis of a very simple model of atoms and bonding, as was done originally by Lewis (1916). Such a model is as follows  

Atoms consist of a nucleus, shells of inner or core electrons, and a shell of outer or valence electrons. Except for hydrogen and helium, the number of valence electrons is given by the group number, N. 

Except for hydrogen and helium, the number of electrons in a complete shell is eight. This is the number possessed by inert gas atoms (except He), and constitutes a stable arrangement. This 

Atoms combine in ways that enable them to achieve an even number of electrons, especially a complete shell. (Very few main-group molecules have an odd number of electrons exceptions are NO, NO2, and CIO2.) 

One way atoms achieve complete shells is by the transfer of electrons from those with a smaller number of outer electrons to those with a larger number to form ions, e g. 

As we discussed earlier, the generalized Boltzmann equation leads to a density expansion of the transport coefficients of a dense gas. However, general expressions for transport coefficients of a fluid that are not in the form of an expansion can be derived by another technique, the time correlation function method. This approach has provided a general framework by means of which one can make detailed comparisons between theoretical results, the results of computer-simulated molecular dynamics,and experimental results.  

There are a number of different formulations of the time correlation function method, all of which lead to the same results for the linearized hydrodynamic equations. One way is to generalize the Chapman-Enskog normal solution method so as to apply it to the Liouville equations, and obtain the N-particle distribution function for a system near a local equilibrium state. Expressions for the heat current and pressure tensor for a general fluid system can be obtained, which have the form of the macroscopic linear laws, with explicit expressions for the various transport coefficients. These expressions for the transport coefficients have the form of time integrals of equilibrium correlation functions of microscopic currents, viz., a transport coefficient t is given by 

Here we will study a somewhat simpler nonequilibrium process than we have considered so far, the diffusion of a tagged particle through a gas of particles that are mechanically identical to the tagged one. This diffusion process is called self-diffuswn Although this process cannot be studied in the laboratory, it can be studied on a computer, and we refer the reader to the article of Wood and Erpenbeck in this volume for further details on how these studies are performed.  

To obtain a macroscopic theory for self-diffusion, we define a quantity F(r, t) that is the probability density for finding the tagged particle at the point r at time t. Since the number of tagged partides is conserved, P(r, t) satisfies a conservation law of the form 

The microscopic theory based on the Liouville equation leads to diffusion equation (235) for t with the coefficient of self-diffusion D given by an equation identical to (236) except that in the microscopic theory the angular bracket is taken to be the average over an equilibrium ensemble. To be 

Superconductivity still has not been totally explained. Most of the research done has been experimental rather than theoretical. For example, the formula for critical magnetic field as a function of temperature is an empirical formula based on experimental data rather than theoretical predictions  

The BCS theory, however, developed in 1957 by three physicists, John Bardeen, Leon Cooper, and Robert Schrieffer, does estabhsh a model for the mechanism behind superconductivity. Bardeen, Cooper, and Schrieffer received the Nobel Prize in physics in 1972 for their theory. It was known that the flux quantum was inversely proportional to twice the charge of an electron, and it had also been observed that different isotopes of the same superconducting element had different critical temperatures. Actually, the heavier the isotope, the lower the critical temperature is. The critical temperature, in K, of an isotope with an atomic mass, M, expressed in kg.moT can be predicted by the following equation  

Actually, the more the lattice vibrates, the more the electrons traveling through the lattice will be slowed down by the vibrating atoms in the lattice. Materials with lattices that vibrate easily generally have higher resistivities at room temperature, while materials with lattices that do not vibrate easily generally have lower resistivities at room temperature. However, at very low temperatures, the Cooper pairs can move more easily through the materials with lattices that are more susceptible to vibration. 

In circumstances where the external time-dependent potential is small, it may not be necessary to solve the full time-dependent Kohn-Sham equations. 

Instead perturbation theory may prove sufficient to determine the behavior of the system. We will focus on the linear change of the density, that allows us to calculate, e.g., the optical absorption spectrum. 

Let us assume that for t to the time-dependent potential vtd is zero -i.e. the system is subject only to the nuclear potential, - and furthermore that the system is in its ground-state with ground-state density At to we turn on the perturbation, so that the total external potential now consists of Uext = Clearly will induce a change in the density. 

If the perturbing potential is sufficiently well-behaved (like almost always in physics), we can expand the density in a perturbative series 

The quantity x is the linear density-density response function of the system. In other branches of physics it has other names, e.g., in the context of many-body perturbation theory it is called the reducible polarization function. Unfortunately, the evaluation of x through perturbation theory is a very demanding task. We can, however, make use of TDDFT to simplify this process. 

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R), i.e. there is no effect due to caging of the encounter complex in the common solvation shell. There exist numerous modifications and extensions of this basic theory that not only involve different initial and boundary conditions, but also the inclusion of microscopic structural aspects [31]. Among these are hydrodynamic repulsion at short distances that may be modelled, for example, by a distance-dependent diffiision coefficient... 

Shifts can also be predicted ftom basic theory, using higher levels of computation, if the molecular structure is precisely known [16], The best calculations, on relatively small molecules, vary from observation by little more than the variations in shift caused by changes in solvent. In all cases, it is harder to predict the shifts of less coimnon nuclei, because of the generally greater number of electrons in the atom, and also because fewer shift examples are available. 

In this section, the basic theory of molecular dynamics is presented. Starting from the BO approximation to the nuclear Schrddinger equation, the picture of nuclear dynamics is that of an evolving wavepacket. As this picture may be unusual to readers used to thinking about nuclei as classical particles, a few prototypical examples are shown. 

D. Rumelhart, R. Durbin, R. Golden, Y. Chauvin, Backpropagfition The Basic Theory in Mathematical Perspectives on Neural Networks, P. Smolensky, M. C. Mozer, D. E. Rumelhart (Eds.), Lawrence Earlbaum Assoc, Hillsdale, NJ, 1996, pp. 533-566. 

Presell is the basic theory of tjuaiiHim mechanics, particularly, semi-empirical molecular orbital theory. The authors detail and justify the approximations inherent in the semi-empirical Ham illoTi ian s. Includes useful discussion s of th e appiicaliori s of these methods to specific research problems. 

This experiment provides a nice example of the application of spectroscopy to biochemistry. After presenting the basic theory for the spectroscopic treatment of protein-ligand interactions, a procedure for characterizing the binding of methyl orange to bovine serum albumin is described. 

Color Mixing. The various types of dye powders used to make dye stains are blended to achieve the desired color. Most finishers purchase wood stains premixed to specified colors. In the wood-finishing industry, various shades of brown are the most common. These colors are usually blended from primary colors. Color-matching skills can be acquired only by practice, but the basic theory of color matching is relatively simple and easily understood. The basic theory of color matching can be demonstrated by using the color circle shown in Figure 1 (see Color). 

R. J. Friauf, "Basic Theory of Ionic Transport Processes," in J. Hladik, ed., Phjsics ofPlectroljtes Vol. 1, Academic Press, Inc., New York, 1972. 

Basic Theory of Fiber-Reactive Dye Application. The previously described mechanisms of dyeing for direct dyes apply to the apphcation of reactive dyes in neutral dyebaths. In alkaline solutions important differences are found. The detailed theoretical treatments are described elsewhere (6) but it is important to consider some of the parameters and understand how they influence the apphcation of fiber-reactive dyes. 

Much of the basic theory of reaction kinetics presented in Sec. 7 of this Handbook deals with homogeneous reaclions in batch and continuous equipment, and that material will not be repeated here. Material and energy balances and sizing procedures are developed for batch operations in ideal stirred tanks—during startup, continuation, and shutdown—and for continuous operation in ideal stirred tank batteries and plug flow tubulars and towers. 

All relevant aspects of a machine, including its design, have been discussed but greater emphasis is laid on selection and application. Since this is a reference book the basic theory is assumed to be known to a student or a practising engineer handling such machines and/or technologies. 

These basic components and functions are common to all seals. The form, style, and design vary depending on the service and the manufacturer. The basic theory of its function and purpose nevertheless, remains the same. 

An ESP is a particulate control device that uses electrical forces to move particles entrained within an exhaust stream onto collection surfaces. The basic theory has already been described under dry ESPs, but a brief summary here is included, with... 

Fluid bed reactors became important to the petroleum industry with the development of fluid catalytic cracking (FCC) early in the Second World War. Today FCC is still widely used. The following section surveys the various fluid bed processes and examines the benefits of fluidization. The basic theories of fluidization phenomena are also reviewed. 

The Fundamentals book describes the basic theories and science behind the technical solutions for industrial air technology. Equipment-specific theories will be completed in the Systems and Equipment book. The Applications book will describe technical solutions for specific industrial sectors and technology areas, including design and construction methodology. 

The purpose of this chapter is to present the basic features of air-handling processes and equipment. The aim is to provide a link between the basic theories of air-handling processes, presented in Chapter 4, and the actual equipment covered in the Systems and Equipment book. 

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