## Calculating First and Second Derivatives

A pH titration (measured volumes of a base solution are added to a solution of an acid and the pH measured after each addition) is shown in Figure 6-1, and a portion of the spreadsheet containing the titration data in Figure 6-2. The endpoint of the titration corresponds to the point on the curve with maximum slope, and this point can be estimated visually in Figure 6-1. The first and second derivatives of the data are commonly used to determine the inflection point of the curve mathematically.

14.0

12.0

10.0

Volume of 0.1000 M NaOH

Figure 6-1. Chart of titration data, (folder 'Chapter 06 Examples', workbook 'Derivs of Titration Data', worksheet 'Derivs')

 A B c D E F | 2 V/lllL PH AV ApH V(avye) ¿pHAV 22 1.90 4.981 0.100 0.229 1.850 2.29 [ 23 1.95 5.157 0.050 0.175 1.925 3.52 24 2.00 5.389 0.050 0.232 1 975 4.64 ; 25 2.05 5.928 0.050 0.539 2.D25 10.78 ; 26 2.08 7.900 0.030 1.972 2.065 65.73 27 2.10 9.115 0.020 1.215 2.090 60.75 28 2.15 9.604 0.050 0.489 2.125 9.78 29 2.20 9.856 0.050 0.252 2.175 5.04 30 2.30 10 125 0.100 0.269 2.250 2.69

Figure 6-2. First derivative of titration data, near the endpoint. (folder 'Chapter 06 Examples', workbook 'Derivs of Titration Data', worksheet 'Derivs')

Figure 6-2. First derivative of titration data, near the endpoint. (folder 'Chapter 06 Examples', workbook 'Derivs of Titration Data', worksheet 'Derivs')

Columns A through F of the spreadsheet shown in Figure 6-2 are used to calculate the first derivative, ApH/A V. Since the derivative has been calculated over the finite volume AV = Fi+i - V„ the most suitable volume to use when plotting the ApH/A V values, as shown in column E of Figure 6-2, is

The maximum in ApH/AV indicates the location of the inflection point of the titration (Figure 6-3).

Figure 6-3. First derivative of titration data, near the endpoint. (folder 'Chapter 06 Examples', workbook 'Derivs of Titration Data', worksheet 'Derivs')

Figure 6-3. First derivative of titration data, near the endpoint. (folder 'Chapter 06 Examples', workbook 'Derivs of Titration Data', worksheet 'Derivs')

The maximum in the first derivative curve must still be estimated visually. The second derivative, A( ApH/A V)/A V, calculated by means of columns E through J of the spreadsheet (shown in Figure 6-4) can be used to locate the inflection point more precisely. The second derivative, shown in Figure 6-5, passes through zero at the inflection point. Linear interpolation can be used to calculate the point at which the second derivative is zero.

 E F G H I ! J j 2 U(iivge) ApH ÀV AV AUpH) V(iivge) Ai Ajilli AU 22 1.850 2.29 0 1 GO 0.57 1 800 5.7 ! 23 1.925 3.52 0.075 1.23 1.888 16.4 24 1975 4 64 0.050 1.12 1 950 22.4 25 2.025 10.78 0.050 6.14 2.000 122.8 26 2.065 65.73 0.040 54.95 2.045 1373.8 27 2.090 60.75 0.025 -4.98 2.078 -199.3 23 2.125 9 78 0.035 -50.97 2.108 -1456.3 29 2.175 5.04 0.05D -4.74 2 150 -94.8 30 2.250 2.G9 0.075 -2.35 2.213 -31.3

Figure 6-4. Second derivative of titration data, near the endpoint. (folder 'Chapter 06 Examples', workbook 'Derivs of Titration Data', worksheet 'Derivs')

Figure 6-4. Second derivative of titration data, near the endpoint. (folder 'Chapter 06 Examples', workbook 'Derivs of Titration Data', worksheet 'Derivs')

Figure 6-5. Second derivative of titration data, near the endpoint. (folder 'Chapter 06 Examples', workbook 'Derivs of Titration Data', worksheet 'Derivs')

Figure 6-5. Second derivative of titration data, near the endpoint. (folder 'Chapter 06 Examples', workbook 'Derivs of Titration Data', worksheet 'Derivs')

There are other equations for numerical differentiation that use three or more points instead of two points to calculate the derivative. Since these equations usually require equal intervals between points, they are of less generality. Again, their main advantage is that they minimize the effect of "noise." Table 6-1 lists equations for the first, second and third derivatives, for data from a table at equally spaced interval h.

These difference formulas can be derived from Taylor series. Recall from Chapter 4 that the first-order approximation is

or, in the notation used in Table 6-1

tt+i = y> + hy't which, upon rearranging, becomes h

The second derivative can be written as y] - y,+l~y' (6-8)

When each of the y terms is expanded according to the preceding expression fory, the expression for the second derivative becomes

• = (y,+2 - yM) ■1 h - Cv/+i -yt)ih h or yl = y>«-2y"+y> (6-io)

The same result can be obtained from the second-order Taylor series expansion which is written in Table 6-1 as

by substituting the backward-difference formula for F from Table 6-1. Expressions for higher derivatives or for derivatives using more terms can be obtained in a similar fashion.

Table 6-1. Some Formulas for Computing Derivatives (For tables with equally spaced entries)

First derivative, using two points:

Forward difference r, ~ ^

Central difference yt =

+1 -3