Equations and Symbols

It is very difficult to describe coherence transfer using the vector model. To understand it we will need to expand our theoretical picture to include product operators. Product operators are a shorthand notation that describes the spin state of a population of spins by dividing it into symbolic components called operators. You might wonder why you would trade in a nice pictorial system for a bunch of equations and symbols. The best reason I can give is that the vector model is useless for describing most of the interesting NMR experiments, and product operators offer a bridge between the familiar vectors and the more formal and... 

Equation and Symbolic answer types so that the results of a self-derivation can be entered to check for correctness, feedback, and assistance... 

End-of-chapter summary tables of important equations and symbols used in these equations... 

Molar concentrations are used so frequently that a symbolic notation is often used to simplify its expression in equations and writing. The use of square brackets around a species indicates that we are referring to that species molar concentration. Thus, [Na ] is read as the molar concentration of sodium ions. ... 

The frequency-dependent coefficients in this equation are given separate names and symbols to facilitate discussion. Remember it is these coefficients that determine the behavior of the system the trigonometric functions merely describe the oscillations. The following can be said of the coefficient of the cosine term ... 

See nomenclature for definition of symbols and units. The units presented are English engineering units, unless a conversion is required. The friction factor is the only experimental variable that must be determined by reference to the above equations and it is represented by Figure 2-3. Note that this may sometimes be referred to as the Fanning formula, and may be modified to )held a fric-... 

This expression shows that the maximum possible useful work (i.e., reversible work) that can be obtained from any process occurring at constant temperature and pressure is a function of the initial and final states only and is independent of the path. The combination of properties U + PV - TS or H - TS occurs so frequently in thermodynamic analysis that it is given a special name and symbol, F, the free energy (sometimes called the Gibbs Free Energy). Using this definition, Equation 2-143 is written... 

We use primes to denote the quantities after collision, while unprimed symbols label those before the collision taking another time average of the equation, and using/(1)/(1) /a)/tt), which is valid for first order deviations from equilibrium, we obtain... 

Students often ask, What is enthalpy The answer is simple. Enthalpy is a mathematical function defined in terms of fundamental thermodynamic properties as H = U+pV. This combination occurs frequently in thermodynamic equations and it is convenient to write it as a single symbol. We will show later that it does have the useful property that in a constant pressure process in which only pressure-volume work is involved, the change in enthalpy AH is equal to the heat q that flows in or out of a system during a thermodynamic process. This equality is convenient since it provides a way to calculate q. Heat flow is not a state function and is often not easy to calculate. In the next chapter, we will make calculations that demonstrate this path dependence. On the other hand, since H is a function of extensive state variables it must also be an extensive state variable, and dH = 0. As a result, AH is the same regardless of the path or series of steps followed in getting from the initial to final state and... 

In this expression, the n are the amounts of each substance in the chemical equation and the symbol X (sigma) means a sum. The first sum is the total standard enthalpy of formation of the products. The second sum is the similar total for the reactants. 

Consider the examples of some of the forms of chemical equations (and related representations) met in school and college (i.e. middle and senior high school) science and chemistiy classes that are shown in Table 4.1. For the purposes of this chapter half-equations (Example 11) and symbolic representations of processes such as ionisation (Example 10) will be included under the generic heading of chemical equations . Table 4.1 does not include examples of chemical reactions and reaction schemes that include stmctural formulae, as are commonly nsed in organic chemistiy. 

Discuss the observed changes using submicroscopic and symbolic representations. Students deduce the ionic equation for the chemical reaction. 

When interpreting the chemical equation for the reaction between aqueous sodium hydroxide and dilute nitric acid, 20% of students appeared to hold the view that Na+ and NO3" ions (submicroscopic and symbolic representations) had reacted in aqueous solution to produce aqueous sodium nitrate. It was not apparent to these students that the net chemical reaction had only involved removal of H+ and OH in aqueous solution to produce molecules of H2O. 

This section began with a class discussion about the importance of water softening and the different factors that influence water hardness. As an example of everyday situation, the efficiency of dishwasher Finish salt was presented. A set of short chemical experiments entitled Testing the dishwasher Finish salt was carried out as a wet laboratory task in groups of students (macro). Later on teachers explained one of those chemical experiments by the use of an animation and also by its 2D presentation with models then students in groups tried to write 2D representations for other chemical experiments (submicro). Students also tried to write down word and symbolic equations and to select the appropriate energy diagrams (symbolic). The results of students work were discussed and corrected when necessary. 

Students ability to connect observations at the macroscopic level with their descriptions using the submicro and symbolic types of representation improved as a consequence of the LON teaching approach. Teachers attributed the improvement to the consistent use of all three types of representation and to the use of visible models as a tool for bridging the gap between macroscopic observations and symbolic notations of chemical equations. 

The superscript appearing above and in related symbols below will serve as a reminder that the particle dimensions have been assumed small compared to the wavelength of the light. Owing to failure to meet this condition it is often necessary to apply a correction (see Sec. 2d) to the observed intensity ie in order to obtain the intensity applicable in this equation and subsequent relationships. 

In the following, definitions of essential analytical terms are compiled, if possible on the basis of international agreements. Attached are sparse references and cross-references. The symbols, being used here, means —> see also (cross-references to terms, additional references as well as paragraphs, chapters, equations and figures of this book), and X is a warning notice. 

Note that the rate of formation of A is rA, as defined in section 1.4 for a reactant, this is a negative quantity. The rate of disappearance of A is (-rA), a positive quantity. It is this quantity that is used subsequently in balance equations and rate laws for a reactant. For a product, the rate of formation, a positive quantity, is used. The symbol rA may be used generically in the text to stand for rate of reaction of A where the sign is irrelevant and correspondingly for any other substance, whether reactant or product. 

Point defect populations profoundly affect both the physical and chemical properties of materials. In order to describe these consequences a simple and self-consistent set of symbols is required. The most widely employed system is the Kroger-Vink notation. Using this formalism, it is possible to incorporate defect formation into chemical equations and hence use the powerful methods of chemical thermodynamics to treat defect equilibria. 

We sometimes call the equilibrium constant in Equation (6.33) a basicity constant, and symbolize it as Kb. 

In these equations, HA symbolizes a weak acid and A symbolizes the anion of the weak acid. The calculations are beyond our scope. However, we can correlate the value of the equilibrium constant for a weak acid ionization, Ka, with the position of the titration curve. The weaker the acid, the smaller the IQ and the higher the level of the initial steady increase. Figure 5.2 shows a family of curves representing several acids at a concentration of 0.10 M titrated with a strong base. The curves for HC1 and acetic acid (represented as HAc) are shown, as well as two curves for two acids even weaker than acetic acid. (The IQ s are indicated.)... 

This definition of c is the same as that given in Section 2.4.2 with m = 1 and different units. A variation in the units and symbols used is quite common in the literature and one should be wary of this. Combining Equations (5.65) and (5.72) we obtain an expression related to the overlap parameter ... 

Solutions for the integration of ODE s, such as the ones given in equations (3.76), are not always readily available. For non-specialists it is difficult to determine if there is an explicit solution at all. The symbolic toolbox (which is not contained in the standard Matlab) provides very convenient means to integrate systems of differential equations and also to test whether there is an explicit solution. As an example the reaction 2A —-—> B ... 

Enzyme kinetics Michaelis constant, symbol iCm maximum velocity of an enzyme catalysed reaction, Vm DC inhibitor constant, symbol X Michaelis-Menten equation and graph in the absence and the presence of inhibitors. Lineweaver-Burke and Eadie-Hofstee plots. 

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