The Pointer Function

The use of successive approximations can be avoided altogether by means of the POINTER function, which will locate all real roots of any equation within a selected interval. The use of this function has the added advantage of providing a global overlay on the graphical representation of the problem under consideration. 

The principle of the POINTER function can be stated very simply If a problem can be reduced to an algebraic expression of one independent variable (pH, pM, x, etc.), called F(x), which has one or more real solutions (roots) in a given range of interest for the variable, these can be found very simply and accurately without sacrificing any rigor. 

This will be applied to chemical problems throughout this text, but it is being introduced here to emphasize the point that its value is not restricted to the types of analytical chemistry, or even to chemistry in general. Examples in physics, engineering, and mathematics are equally useful. 

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Use the Pointer function approach to solving the following equations. = lOx -. ... 

Even in those cases where the solution does not reduce to the pH value of the principal intersection, making this value increasingly uncertain, one can see from the EquiligrapH and the PBE both the source and extent of the error. It must be remembered, however, that the PBE itself can guide us to the rigorously correct solutions for such problems, regardless of their complexity. This can be done with the spreadsheet, and a novel way to use the PBE called the POINTER function. 

As we saw in Chapter 1, even very complex algebraic expressions can be solved by a simple graphical method involving the POINTER function. Application of this idea to acid-base as well as many other analytical calculations is far from complicated. This is especially true since for pH, pM, etc., no matter how complex the expression may look, we know that any single aqueous solution can have only one pH value. Hence, the expression has a single, unique solution. Further, when the problem involves pH calculations, the range of values to be examined is obviously between 0 and 14 in almost every case. 

The value of P will vary with pH but will be uniquely ZERO at the pH of the solution. While this could be represented by the intersection of the P function with pH axis, it can be conveniently transformed to the log of its absolute (one cannot obtain the log of a negative number) value, or LOG( ABSP). This is seen in Figure 4.4 by the minimum in the pointer function which points to pH = 8.0, the answer. Pointer functions obtained in the same manner have been included in Figures 4.2 and 4.3. 

Every term in Equation 5-5 can be expressed in terms of [NH3], the only variable in the a expressions. The solution to this problem can be found simply by using the pointer function, P, which here is... 

This represents a fairly formidable equation which can be cut to size , however, with the help of the Pointer function. The spreadsheet is... 

From the resulting spreadsheet. Figure 6.6 was obtained which gives us S as a function of total [CT], and a pointer enabling us to obtain the water solubility of PbCl2. According to the pointer function, a saturated solution has a [CT] of 0.045 M at which S = 0.025 M. 

This results in a quadratic equation which readily yields the result of X = 0.1366 and D = 107.7. The equation can also be solved by using the Pointer function after transposing one side to the other and the entire expression, F(x), to get log( abs(F(x)). 

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