Mathematical expression

Most DKRs of this type are achieved with lithium carbanions possessing O [136], N [137], S [138] and Se [139] atoms at the a-position. Enantioenrichment with lithiated N,N-diisopropyl-o-ethylbenzamide arises from DKR in a similar way to the a-hetero-substituted carbanions [140]. 

1) reactions of Sj and S, with rate constants fejj and kg, and stereoinversion of the substrate proceed by first or pseudo-first-order kinetics  

Sjj reacts faster than S, and hence, Pjj is the prevaihng enantiomeric product  

Using these four conditions, the rate of consumption of Sjj and S5 can be expressed as 

The mathematical treatment of the two equations gives the concentration of each component, Sg, Sjj, Pg and P,j, as a function of time with the parameters kjj, kg and 

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The mathematical formulation of a typical molecular mechanics force field, which is also called the potential energy function (PEF), is shown in Eq. (18). Do not wony yet about the necessary mathematical expressions - they will be explained in detail in the following sections ... 

In making certain mathematical approximations to the Schrodinger equation, we can equate derived terms directly to experiment and replace dilTiciilL-to-calculate mathematical expressions with experimental values. In other situation s, we introduce a parameter for a mathematical expression and derive values for that parameter by fitting the results of globally calculated results to experiment. Quantum chemistry has developed two groups of researchers ... 

Spray Correlations. One of the most important aspects of spray characterization is the development of meaningful correlations between spray parameters and atomizer performance. The parameters can be presented as mathematical expressions that involve Hquid properties, physical dimensions of the atomizer, as well as operating and ambient conditions that are likely to affect the nature of the dispersion. Empirical correlations provide useful information for designing and assessing the performance of atomizers. Dimensional analysis has been widely used to determine nondimensional parameters that are useful in describing sprays. The most common variables affecting spray characteristics include a characteristic dimension of atomizer, d Hquid density, Pjj Hquid dynamic viscosity, ]ljj, surface tension. O pressure, AP Hquid velocity, V gas density, p and gas velocity, V. ... 

Several requirements must be met in developing a stmcture. Not only must elementary analysis and other physical measurements be consistent, but limitations of stmctural organic chemistry and stereochemistry must also be satisfied. Mathematical expressions have been developed to test the consistency of any given set of parameters used to describe the molecular stmcture of coal and analyses of this type have been reported (4,6,19,20,29,30). 

In the broadest sense, thermodynamics is concerned with mathematical relationships that describe equiUbrium conditions as well as transformations of energy from one form to another. Many chemical properties and parameters of engineering significance have origins in the mathematical expressions of the first and second laws and accompanying definitions. Particularly important are those fundamental equations which connect thermodynamic state functions to real-world, measurable properties such as pressure, volume, temperature, and heat capacity (1 3) (see also Thermodynamic properties). 

The common theme in the evolution of methods for property and parameter prediction is the development of equations, either theoretical or empirical, containing quantities that can be calculated from theoretical considerations or experimental data. Mathematical expressions for correlating thermodynamic data may take several forms. 

Here a suitable equation of state is required to provide a mathematical expression for the mixture molar volume, V. For some equations of state, it is better to use a form of equation 28 in which the integral is volume expHcit (3). Note also that for an ideal gas — Z — 1, and 0 = 1. 

An important benefit of QSAR methods is that no enzyme stmcture is required. However, a series of stmcturaHy related inhibitors of known stmcture and known inhibitory activity is required. The limitations of the method involve the difficulties in describing something as compHcated as a stmcture-function relationship in a single mathematical expression. 

By contrast, a numerical computer program for solving such integration problems would depend on approximating the mathematical expression by a series of algebraic equations over expHcit integration limits. 

Essential Features of Optimization Problems The solution of optimization problems involves the use of various tools of mathematics. Consequently, the formulation of an optimization problem requires the use of mathematical expressions. From a practical viewpoint, it is important to mesh properly the problem statement with the anticipated solution technique. Every optimization problem contains three essential categories ... 

Develop via mathematical expressions a valid process or equipment model that relates the input-output variables of the process and associated coefficients. Include both equality and inequality constraints. Use well-known physical principles (mass balances, energy balances), empirical relations, implicit concepts, and external restrictions. Identify the independent and dependent variables (number of degrees of freedom). 

We can derive the same inference from Tables 30.2, 30.4 and 30.5, specifying current ratings for different cross-sections. The current-carrying capacity varies with the cross-section not in a linear but in an inconsistent way depending upon the cross-section and the number of conductors used in parallel. It is not possible to define accurately the current rating of a conductor through a mathematical expression. This can be established only by laboratory tests. 

Mixing of product and feed (backmixing) in laboratory continuous flow reactors can only be avoided at very high length-to-diameter (aspect) ratios. This was observed by Bodenstein and Wohlgast (1908). Besides noticing this, the authors also derived the mathematical expression for reaction rate for the case of complete mixing. 

A minimum basis set for molecules containing C, H, O, and N would consist of 2s, 2p, 2py, and 2p oibitals for each C, N, and O and a 1 j orbital for each hydrogen. The basis sets are mathematical expressions describing the properties of the atomic orbitals. 

The mathematical expressions for the force Aelds are derived from classical-mechanical potenAal energy funcAons. The energy required to stretch a bond or to bend a bond angle increases as the square of the distortion. For bond stretching,... 

The phase rule is a mathematical expression that describes the behavior of chemical systems in equilibrium. A chemical system is any combination of chemical substances. The substances exist as gas, liquid, or solid phases. The phase rule applies only to systems, called heterogeneous systems, in which two or more distinct phases are in equilibrium. A system cannot contain more than one gas phase, but can contain any number of liquid and solid phases. An alloy of copper and nickel, for example, contains two solid phases. The rule makes possible the simple correlation of very large quantities of physical data and limited prediction of the behavior of chemical systems. It is used particularly in alloy preparation, in chemical engineering, and in geology. 

The rate equation involves a mathematical expression describing the rate of progress of the reaction. To predict the size of the reactor required in achieving a given degree of conversion of reactants and a fixed output of the product, the following information is required ... 

The following details mathematical expressions for instantaneous (point or local) or overall (integral) selectivity in series and parallel reactions at constant density and isotliermal conditions. An instantaneous selectivity is defined as the ratio of the rate of formation of one product relative to the rate of formation of another product at any point in the system. The overall selectivity is the ratio of the amount of one product formed to the amount of some other product formed in the same period of time. 

In general, the optimum conditions cannot be precisely attained in real reactors. Therefore, the selection of the reactor type is made to approximate the optimum conditions as closely as possible. For this purpose, mathematical models of the process in several different types of reactors are derived. The optimum condition for selected parameters (e.g., temperature profile) is then compared with those obtained from the mathematical expressions for different reactors. Consequently, selection is based on the reactor type that most closely approaches the optimum. 

Many attempts have been made to obtain mathematical expressions which describe the time dependence of the strength of plastics. Since for many plastics a plot of stress, a, against the logarithm of time to failure, //, is approximately a straight line, one of the most common expressions used is of the form... 

It is important to derive the mathematical expression relating y, and xy on the practical-feasibility line. For a given y,, the values of xj can be obtained by evaluating x that is in equilibrium with y, then subtracting Cy. i.e.,... 

For computer applications, it is useful to have the friction factor in a mathematical expression (empirical). 

Langmuir equations The mathematical expressions that describe vapor adsorption equilibria. 

Mixture rule A mathematical expression applying to workers simultaneously exposed to chemicals that act on the same... 

How are the electrons distributed in an atom You might recall from your general chemistry course that, according to the quantum mechanical model, the behavior of a specific electron in an atom can be described by a mathematical expression called a wave equation—the same sort of expression used to describe the motion of waves in a fluid. The solution to a wave equation is called a wave function, or orbital, and is denoted by the Greek letter psi, i/y. 

Wave equation (Section 1.2) A mathematical expression that defines the behavior of an electron in an atom. 

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