Second D e r i v a t i v e Curves and End-Point Determination Titration end-mint titration curves. first derivative curves. and second derivative curves . determination usine..reeular ,. is commonly treated in yuantmtivc analbsia and rnstrumental annlviis texts. The treatment 01 the f ~ r s t\\o t methods is generally adequate, hut that vf the wcmd deriratwe m e t h d . in whrh a plot o:the secot~dderivative a w n s t volume heeumps zero a t the end-point, is in many eases misleading or incorrect. The problem arises in the intended simplificatian produced by making all titrant volume increments equal. Where some err in this case is in giving the heading of a data column (column Dala from a Tllration of Fe(ll) by Ge(lV) a ~
I
~
II
ill
~
v
IV
vi
VII
Vlll
ix
Average V 10. .- l d
V(ml)
E(mV)
7.20
505
7.30
511
7.40
AE
AV (First)
6
0.10
60
7.25
8
0.10
80
7.35
519
1
11 7.50 7.65 7.75 7.85
AUAV
Derivative Plot
530
49
0.15
327
7.575
341
0.70
3410
7.70
108
0.10
1080
7.80
579 920
X Average V
A( AB AV)
AV (Second)
A2ElAP
for - -2ndDerivative Plot
+20
0.10
+ZOO
7.30
+30 110 +217
0.10 7.45 0.125
+300
7.40
+1736
7.513
+3083
0.125
+24.664
7.638
-2330
0.10
-23,300
7.75
1028
p~ ~ a t ham a an acid-bane titration could be d a s well. ande number of texts use p ~ d a f whwr a illustrating end-point determination methcds. in several in~tancesin this table. significantfigwe rules have not been adhered to in ader tar the origin of various numbers to be more owlous. VII in the table) as AZEIAVZor d2EldVZwhen it is only A(AE/AV); that is, the authors have failed to divide the change in first derivative by the change in aveiage volume. I t is, when increments are equal, proportional to the second derivative and will produce thesame end-point when plotted, but i t isequal to the second derivativeonly when theadded increments are unity. The student, however, who carries out a titration using unequol volume increments and examines for guidance a textbook table with the final and plotted column headed AZEIAV20rd2EldVZwhen i t is actually A(AE/AV) has cause t o he perplexed. This can he remedied by dividing by the change in average volume in each case and adding the necessary column or columns to the textbook tables and using correct column headings. (For meximum clarity columns equivalent t o columns I-X of the tahle should he included. Then, with equal or with unequal increments, the method of correct data treatment is immediately obvious.) The classic text' by Kolthoff, Sandell, Meehan, and Bruckenstein gives an excellent and detailed treatment of the second derivative method, including a discussion of the necessity of small titrant increments and the desirability of equal ones.
1 Kolthoff, I. M., Sandell, E. B., Meehan, E. J., and Bruckenstein, S., "Quantitative Chemical Analysis," 4th ed., The Macmillan Co., New York, 1969, p. 946.
Presbyterian College Clinton, South Carolina 29325
26 / Journal of Chemical Education
K. N. C a r t e r R. B. Huff
