Method of successive approximations

In most of the problems you will work, the approximation a — x a is valid, and you can solve for [H+] quite simply, as in Example 13.7, where x = 0.012a. Sometimes, though, you will find that the calculated [H+] is greater than 5% of the original concentration of weak acid. In that case, you can solve for x by using either the quadratic formula or the method of successive approximations. 

The method of successive approximations. In the first approximation, you ignored the x in the denominator of the equation. This time, you use the approximate... 

The second answer is physically ridiculous the concentration of H+ cannot be a negative quantity. The first answer is the same one obtained by the method of successive approximations. 

A method of successive approximation (s.a. method) is one that would provide the solution as the limit of an infinite sequence of steps if these steps were carried out exactly. These steps are usually quite simple, and nearly identical, each to the next, so that programming is a relatively easy task. Most methods operate upon a given approximation to obtain a better one, hence they are self-correcting. Because of rounding, at some stage the computed correction will no longer be... 

For differential equations with periodic coefficients, the theorems are the same but the calculation of the characteristic exponents meets with difficulty. Whereas in the preceding case (constant coefficients), the coefficients of the characteristic equation are known, in the present case the characteristic equation contains the unknown solutions. Thus, one finds oneself in a vicious circle to be able to determine the characteristic exponents, one must know the solutions, and in order to know the latter, one must know first these exponents. The only resolution of this difficulty is to proceed by the method of successive approximations.11... 

Iterative methods of successive approximation are in common usage for rather complicated cases of arbitrary domains, variable coefficients, etc. Throughout the entire section, the Dirichlet problem for Poisson s equation is adopted as a model one in the rectangle G = 0 < x < l, a = 1,2 with the boundary P ... 

This can be solved with the quadratic equation, and the result is = 0.0118 moles. We can attempt the method of successive approximations. First, assume that 0.0240. We obtain ... 

Words that can be used as topics in essays 5% rale buffer common ion effect equilibrium expression equivalence point Henderson-Hasselbalch equation heterogeneous equilibria homogeneous equilibria indicator ion product, P Ka Kb Kc Keq KP Ksp Kw law of mass action Le Chatelier s principle limiting reactant method of successive approximation net ionic equation percent dissociation pH P Ka P Kb pOH reaction quotient, Q reciprocal rule rule of multiple equilibria solubility spectator ions strong acid strong base van t Hoff equation weak acid weak base... 

In future, Mulliken and others would attempt to substitute ab initio methods for the earlier methods of successive approximations. With this, quantum chemistry was to become more abstract, more theoretical, than in its initial development. The new approaches seemed to express increasing confidence in a deep level of understanding, the hubris so characteristic of the quest for mathematical certainty. There were dangers in this which Coulson warned might deceive the theoretician. "One is almost tempted to say. .. at last I can almost see a bond. But that will never be, for a bond does not really exist at all it is a most convenient fiction which, as we have seen, is convenient both to experimental and theoretical chemists." 151... 

To evaluate log K", it is necessary to know Ae and, therefore, C,. Yet to know C, we must have a value for a, which depends on a knowledge of Ag. In practice, this impasse is overcome by a method of successive approximations. To begin, we take Ae = Ao and make a first approximation for a" from Equation (20.26). With this value of a", we can calculate a tentative C which can be inserted into Equation... 

For solving equation (1) we use the standard method of successive approximations, putting... 

We are carrying out a method of successive approximations. Each cycle is called an Iteration. 

The method of successive approximations is a good way to deal with difficult equations that do not have simple solutions. For example, Equation 10-11 is not a good approximation when the concentration of the intermediate species of a diprotic acid is not close to F, the formal concentration. This situation arises when Kt and K2 are nearly equal and F is small. Consider a solution of 1.00 X 10 1M HM, the intermediate form of malic acid. 

The method is later extended to time-varying heat inputs on one face with arbitrary boundary conditions on the back face (C7). Citron also has given a simple method of successive approximations for the finite ablating slab (C6) which is shown to converge rapidly for constant heat input. 

I will return to this diagram near the end of the chapter, particularly to amplify the meaning of error removal, which is indicated by dashed horizontal lines in Fig. 7.1. For now, I will illustrate the bootstrapping technique for improving phases, map, and model with an analogy the method of successive approximations for solving a complicated algebraic equation. Most mathematics education emphasizes equations that can be solved analytically for specific variables. Many realistic problems defy such analytic solutions but are amenable to numerical methods. The method of successive approximations has much in common with the iterative process that extracts a protein model from diffraction data. 

When the structure has been approximately determined in some of these ways, the next step is to refine the atomic coordinates until sufficient accuracy has been attained. Many variations of the powerful Fourier method of successive approximation are now available, and for the later stages of refinement the method of least squares can be used with advantage if sufficient computing facilities are available. 

Note that, in more complex problems, especially if the equation to be solved exceeds the abilities of the quadratic formula (cubed values and higher), it may be best to find the equilibrium concentrations by a method of successive approximations. You would select a consistent set of concentration values near where it is guessed the answer will be. Then, the calculations for K are performed and repeated until a sufficiently precise result is reached. 

Because K, depends on concentrations and the product KyKx is concentration independent, Kx must also depend on concentration. This shows that the simple equilibrium calculations usually carried out in first courses in chemistry are approximations. Actually such calculations are often rather poor approximations when applied to solutions of ionic species, where deviations from ideality are quite large. We shall see that calculations using Eq. (47) can present some computational difficulties. Concentrations are needed in order to obtain activity coefficients, but activity coefficients are needed before an equilibrium constant for calculating concentrations can be obtained. Such problems are usually handled by the method of successive approximations, whereby concentrations are initially calculated assuming ideal behavior and these concentrations are used for a first estimate of activity coefficients, which are then used for a better estimate of concentrations, and so forth. A G is calculated with the standard state used to define the activity. If molality-based activity coefficients are used, the relevant equation is... 

In calculations of ionic equilibria, the required activity coefficients depend on ionic strength and, thus, on the equilibrium. As a result, such calculations are usually carried out by the method of successive approximations. Fortunately, convergence to a unique solution is usually achieved very rapidly in these problems. 

This technique is called successive approximation as the digitized value is better approximated with each step of conversion. Compared with the word-at-a-time method the method of successive approximation needs only ld(n) reference voltages but it is slower than the above mentioned technique by a factor of ld(n) (n = number of discrete steps, Id = logarithm with basis 2). 

To make a complete calculation of the phase composition of the test sample, this method can take the experimentally measured value of p or the mass absorption coefficient can be calculated with the help of the method of successive approximations [30,31,39], However, the previously explained method is complex therefore, it is easier to avoid the microabsorption effect by milling the test sample to get particles of about 1 pm [39],... 

Method of Successive Approximations Consider the equation yix) =fix) + X Kix, t)yit) at. In this method a unique solution is obtained in sequence form as follows Substitute in the right-hand member of the equation ydf) for yit). Upon integration there results yiit) =fix) -b A, Kix, t)yoit) dt. Continue in like manner by replacing yo by yi, yi by y, etc. A series of functions yoix), yfx), yfix),. . . are obtained which satisfy the equations... 

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