Equation Symbolism

Up to this point, we have only investigated stoichiometric lattices. Let us now examine non-stoichiometric lattices in light of our S5mibolism. Consider the semi-conductor, Ge. The defect reactions associated with the formation of p-type and n-t3rpe lattices are  

The excess charges shown are spread over the entire lattice, as stated before. 

Whether you realize it or not, we have already developed our own symbolism for defects and defect reactions based on the Plane Net. It might be well to compare our system to those of other authors, who have also considered the same problem in the past. It was probably Rees (1930) who wrote the first monograph on defects in solids. Rees used Dm 

Although the results are equal as far as utility is concerned, we shall continue to use our symbolism, for reasons which will become clear later. 

The following compares defect symbolism, as used by prior Authors. Note that our S5ntnbollsm most resembles that of Kroeger, but not in all aspects. 

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A balanced chemical equation symbolizes both the qualitative and the quantitative changes that take place in a chemical reaction. The stoichiometric coefficients tell us the relative numbers of moles of reactants and products taking part in the reaction. 

In the above equations, symbol a represents the area of biofilm per unit volume of bulk liquid. The diffusion rates in and out of the biofilm are driven by concentration differences between the bulk liquid (0) and the outer biofilm segment (1). 

Modern mathematical software, such as Mathematica, allows us to compute symbolically the mean square deviation of this approximation from the exact acceleration, integrated over the feasible region, differentiate the resulting expression symbolically with respect to the parameters a and b, set the results to zero and solve the equations symbolically, and simplify the whole lot to find the following remarkably simple expressions... 

Students will write a balanced chemical equation and explain the meaning of the equation symbols. 

Bases that produce hydroxide ions in solution turn phenolphthalein red. Calcium and magnesium produce hydroxide ions in water. The following equations symbolize the reactions ... 

Figure 1. Liquid-liquid equilibria (LLE) for nitrogen + ethane, new EOS , TBS equation symbols, exp. data [12, 13].

Figure 17.15 Effect of polymerization temperature on catalytic activity. Line, calculated by kinetic equation symbols, polymerization results...

In equations, symbols are used to show the physical states of the reactants and products. Reactants and products can exist as sohds, liquids, and gases. When they are dissolved in water, they are said to he aqueous. It is important to show the physical states of a reaction s reactants and products in an equation because the physical states provide clues about how the reaction occurs. Some basic symbols used in equations are shown in Table 10-1. 

Use Maple to solve the following equations symbolically (use the solve command). 

One of the advantages of FEMLAB is that it differentiates the equations symbolically, including any terms that you add to the problem, and this enhances convergence. The parametric solver in FEMLAB is possible because the Newton-Raphson method is being used. Consider Eq. (F.l) when it depends upon a parameter ... 

Mathematica has a powerful capability to carry out symbolic mathematics on algebraic expressions and can solve equations symbolically. In addition to the arithmetic operations, the principal Mathematica statements for manipulating algebraic expressions are Expand ], Factor ], Simplify ], Together ], and Apart ]. The Expand statement multiplies factors and powers out to give an expanded form of the expression. The following input and output illustrate this action In l] =Clear a,x]... 

Herbert Spencer has defined a law of Nature as a proposition stating that a certain uniformity has been observed in the relations between certain phenomena. In this sense a law of Nature expresses a mathematical relation between the phenomena under consideration. Every physical law, therefore, can be represented in the form of a mathematical equation. One of the chief objects of scientific investigation is to find out how one thing depends on another, and to express this relationship in the form of a mathematical equation—symbolic or otherwise—is the experimenter s ideal goal.1... 

Symbols—Formulae— Equations.—Symbols are conventional abbreviations of the names of the elements, whose purpose it is to introduce simplicity and exactness into descriptions of chemical actions. They consist of the initial letter of the Latin name of the element, to which is usually added one of the other letters. If there be more than two elements whose names begin with the same letter, the single-letter symbol is reserved for the commonest element. Thus, we have nine elements whose names begin with C of these the commonest is Carbon, whose symbol is C tho others have double-letter symbols, as Chlorin, Cl Cobalt, Co Copper, Cu (Cuprum), etc. 

In the above equation, symbols have the following meaning hi is the weight of the t-th vertex depicted by loop, the first product gives the... 

For the formation of the differentials and the solution of the sets of equations, symbolic computational programs can be used in favor, such as DERIVE(r) or Mathematica . 

Figure 3.2-7 Fitting of the propane/activated carbon data with the Sips equation (symbol -data line ...

This equation gives the criticality condition. In many cases of practical interest, the lattice configuration will exhibit several degrees of symmetry. As a result, various elements of the determinant will be repeated, and this in turn can lead to high orders of degeneracy in the determinantal (characteristic) equation. Thus the complicated equation symbolized by (10.295) will in actual practice reduce to considerably simpler forms. This point is well demonstrated in the example which follows. 

Besides identifying what substances are the reactants and products of a chemical reaction and how many atoms, molecules, and formula units are involved, chemical equations can provide additional information if the writer so chooses. Symbols may be used to provide some qualitative information about the substances involved or about the reaction in general. However, such symbols are usually optional. Symbols giving specific information about the reactants or products are placed in parentheses immediately to the right of the chemical symbol or formula in the equation. Symbols giving information about the reaction in general are usually placed above the arrow. 

The first two equations symbolize heat balances for th zones at the flue gas side. The balances for the coil... 

Figure 12.2. Cathode kinetics fitted by different equations. Symbols represent the measurement by Carbonio et al. (1987). The dotted line represents the fitting results using kinetic law by Chan and Savinell (1991).

MATLAB can solve ordinary differential equations symbolically with or without boundary conditions or initial value parameters. The dsolve command is used for this purpose. Within dsolve, the letters D(/ are used to indicate a derivative, where i is the order of differentiation andj is the dependent variable. D implicitly specifies a first-order derivative, D2 signifies a second-order derivative, and... 

Many illustrations have three parts what you can see with your eyes (macroscopic) what the molecules are doing (molecular) and how chemists represent the process with equations (symbolic). 

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