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 61, and a portion of the spreadsheet containing the titration data in Figure 62. The endpoint of the titration corresponds to the point on the curve with maximum slope, and this point can be estimated visually in Figure 61. 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 61. 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 62. First derivative of titration data, near the endpoint. (folder 'Chapter 06 Examples', workbook 'Derivs of Titration Data', worksheet 'Derivs')
Figure 62. 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 62 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 62, is
The maximum in ApH/AV indicates the location of the inflection point of the titration (Figure 63).
Figure 63. First derivative of titration data, near the endpoint. (folder 'Chapter 06 Examples', workbook 'Derivs of Titration Data', worksheet 'Derivs')
Figure 63. 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 64) can be used to locate the inflection point more precisely. The second derivative, shown in Figure 65, 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 64. Second derivative of titration data, near the endpoint. (folder 'Chapter 06 Examples', workbook 'Derivs of Titration Data', worksheet 'Derivs')
Figure 64. Second derivative of titration data, near the endpoint. (folder 'Chapter 06 Examples', workbook 'Derivs of Titration Data', worksheet 'Derivs')
Figure 65. Second derivative of titration data, near the endpoint. (folder 'Chapter 06 Examples', workbook 'Derivs of Titration Data', worksheet 'Derivs')
Figure 65. 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 61 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 firstorder approximation is
or, in the notation used in Table 61
tt+i = y> + hy't which, upon rearranging, becomes h
admittedly, an obvious result.
The second derivative can be written as y]  y,+l~y' (68)
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> (6io)
The same result can be obtained from the secondorder Taylor series expansion which is written in Table 61 as
by substituting the backwarddifference formula for F from Table 61. Expressions for higher derivatives or for derivatives using more terms can be obtained in a similar fashion.
Table 61. Some Formulas for Computing Derivatives (For tables with equally spaced entries)
First derivative, using two points:
Forward difference r, ~ ^
Central difference yt =
Responses

Pervinca5 years ago
 Reply

katrin5 years ago
 Reply

Milo4 years ago
 Reply

veronica4 years ago
 Reply

ruth3 years ago
 Reply

berta3 years ago
 Reply

Mungo2 years ago
 Reply

Helen2 years ago
 Reply

luca1 year ago
 Reply

Palmira Folliero1 year ago
 Reply