The Method of Substitution

Anions obtained formally by loss of a hydron from a parent hydride (see Table 5.2 p. 99 for a list of parent hydride names) are conveniently named by the methods of substitutive nomenclature. 

Carbon is, of course, unique in the number of hydrides it forms, but the elements in the proximity of carbon in the Periodic Table have a similar, if more restricted, propensity to form hydrides. Silicon, germanium, boron and phosphorus are obvious examples. For hydrides of these elements, and especially for their organic derivatives, the methods of substitutive nomenclature can be applied to obtain suitable names. 

An area of current development is the nomenclature of organometallic compounds. Organometallic compounds of Main Group elements can, to a first approximation, be considered to be derivatives of hydrides, and the methods of substitutive nomenclature can be applied. Even then, the accessibility of different oxidation states, as with phosphorus(iii) and phosphorus(v), introduces complications. Transition metal organometallic compounds are even more difficult to treat, and the development of a unified, self-consistent and accepted and applied nomenclature is not easy. Witness the different ways (k, t and italicised symbols) for denoting donor atoms in ligands. 

This same expression can be derived for single-pan balances that use the method of substitution of weights. The fact that the lever arms are unequal and that there is a constant load on the balance does not alter the final expression. 

The location of the metal cation (or cations) in alcoholates has been investigated by the method of substitutive methylation for the sodium derivatives of methyl a-D-glucopyranoside, amylose, and cellulose. The method assumes that methylation by either dimethyl sulfate or methyl iodide occurs only at an anionic oxygen atom. Lenziu methylated the mono-sodium alcoholates (prepared in boiling, butanolic sodium hydroxide by the... 

In the second case, if the species mobilities differ greatly, the dimensionality of the system of kinetic equations decreases [103], Let all the components be divided into two groups of species a slow (5) and a rapid (r) one. This yields three types of pair functions. For the rapid species the condition of the equilibrium distribution can be considered as satisfied. Then, for the pair functions of types sr and rr instead of the kinetic Eqs. (32) algebraic relations in Appendix A apply, whose dimensionality can be lowered using the method of substitution variables according to Appendix B. In this case the kinetic Eqs (31) for the local concentrations and Eq. (32) for the pair functions type ss do not change. A similar situation remains in passing to the one-dimensional discrete and point-like models. 

The method of substitution is not limited to two equations and is not limited to linear equations. In the following example, we treat a nonlinear system of two equations ... 

Solve the simultaneous equations by hand, using the method of substitution ... 

In Chapter 3, we discussed the use of the method of substitution and the method of elimination to solve the linear inhomogeneous set of two simultaneous equations ... 

To solve for numerical values of two variables, two equations are required, and they must be solved simultaneously, and similarly for more variables. We presented several methods for solving simultaneous equations. First was the method of substitution, which is not limited to linear equations, but which is not practical for more than two or three equations. We then presented several methods which can apply to sets of linear inhomogeneous equations. Cramer s method is a method... 

II. To solve the normal equations. We can determine the values of x, y, z, from this set of simultaneous equations (2) by any method we please, determinants ( 179), cross-multiplication, indeterminate multipliers, or by the method of substitution.1 The last method is adopted here. Solve the first normal equation for x, thus... 

The method of substitution is not limited to two equations and is not limited to linear equations. 

In the method of substitution a change of variables is performed. The integrand function is expressed in terms of the new independent variable, which then becomes the variable of integration. The limits of integration must be expressed in terms of the new variable. 

The p integral is done by the method of substitution, letting u = 2ap and du = 4apdp. We obtain... 

The integral in this equation can be performed by the method of substitution. We let w = v, so that the integral becomes... 

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