Ebbe Still and Rolf Sara
Abo Akademi Abo, Finland
I
Calculation of Potentiometric Titration Curves The masfer variable concept
Potentiometric titration curves have been discussed in many studies since the beginning of this century. A commonly used method in the chemistry curriculum for the calculation of the shape of the titration curve is suggested by Kolthoff (1) in his analytical textbooks. According to this method approximate formulas are used for the calculation of the p H (or p M ) of the solution. Practical experience has, however, revealed that correct approximations are difficult for inexperienced students to make; exact methods are to he preferred provided calculators are at hand. Kolthoff's method can be applied to oompleximetric titrations as well. The difficulty caused by the fact that one or more of the species in the main reaction are involved in side reactions can be avoided by using conditional constants (2). The influence of these side reactions can easily be calculated by means of rr-coefficients. A graph plotting p M or pM' as a function of the added amount of complexing agents will illustrate this type of titration. An alternative type of graph illustrating the course of a titration is the so-called logarithmic diagram with a master variable (3,4). A logarithmic diagram represents a graphical method to solve equilibrium prohlems and such a diagram has become quite popular when dealing with rather complicated systems. Logarithmic diagrams can easily be constructed by choosing a suitable independent variable, the master variable. The master variable is here defined as by Sillhinref. (4). This means, for instance, in acid-base titrations that p H is chosen as the master variahle. A given p H will determine the acidbase ratio in each acid-base pair and thus the consumption of titrant. In complex formation equilibria the ratio of various species to any other is determined by the free ligand concentration [Y] or p Y = -log [Y], or alternatively by the free metal ion concentration [MI @M = -log [MI). When the ligand and the metal ions are involved in various side reactions i t is best to use the concept of conditional constants and rr-coefficients (2) and replace pY with pY', or p M with pM'. Logarithmic diagrams and other graphical methods are of considerable help in finding appropriate experimental conditions as these methods immediately tell us the predominating and negligible species. In the last decade the introduction of digital computers bas vigorously supported the use of algebraic methods when solving equilibrium problems. The Swedish (SillBn) and the Australian (Perrin) schools have developed very general programs which require a high-speed computer with a sufficiently large computer storage. These programs can he used to solve fairly large prohlems with almost any combination of chemical species. The development of programmable pocket calculators combined with a rapid reduction in prices have made these calculators available to practically every student. This means that the possibilities for the students to solve various calculating prohlems have increased drastically. The limited memory capacity has, however, made it necessary to run smaller programs, tailor-made for one's own use. The need for compact algorithms is thus obvious. Our aim in the present paper is to give compact algorithms for the calculation of some potentiometric titration curves. I t will he shown that the master variahle concept, i.e. the 348 / Journal of Chemical Education
choice of the same independent variable as in logarithmic diagrams, is an advantage in the calculation of the titration curve. In this way the equation will usually he linear in the dependent variable and iterations are thus avoided. This is in contrast to the general approach of solving equilibrium prohlems where iterative procedures are always required. The Titration of Several Acids With a Strong Base As an example the titration of one of several weak acids HA; (i = 1.2.. . . . .) with a strone base will he considered. If Vo cm"
of theacids of themncentrat~onsC H , ~are titrated with V cm' of a stn)
[email protected]. NaOH. of the concentratmn Cn. -. the 101lowing conditions must hold (for convenience, all signs of charge are omitted). The law of mass action states
-
-
H+Ai-HA;
KHA.=-IHAil IHllAiJ
(1)
The solution must be electrically neutral, meaning that [Na]
+ [HI = [OH] + [Al] + [An]+ . . .
(2)
The concentration of the sodium ion is [Nal = -CB
vo + v
(3)
The expression for the total concentration of the acids HA; is [Ail
+ [HA;] = vo + v CHA.
which, when combined with eqn. (I), gives
Substitution of (3) and (5) into (2) yields
- ..
.
.
Equation (6) is linear in V, hut not in [HI. This means that the choice of p H as the independent variahle will give a linear equation in the unknown V, whereas a choice of Vas the independent variable will lead to an eauation of third or higher degree in [HI. The calculation of the titration curve can he approached in the following way: The input statements are the values of Vo and Ce, the ionic product of water K , = [HJIOH],and the concentrations and the stability constants of the acid. For a given pH-value the calculator finds the corresponding volume value. A neeative value of V means that the chosen DH has htering been less than the i n i t i i p ~ - v a l u eof the the DH-valuewith certain steos the whole titration curve can easily he calculated. The same stratew can be used for a ~roeranunahledesk- to^ calculator equipperwith a plotter. he ti6ation can he started from an estimated pH-value (e.g., assuming all acids heing strong ones) for the initial solution. The corresponding Vvalue is calculated and the pH-value is then increased by a
solution.^^
Complexlmetrlc Titratlons
The course of a com~leximetrictitration can be recorded by measuring the chanie in either the metal ion mncentmtion by an electrode of the first order or the ligand ion concentratcon by an indirect electrode. The formercase will he considered first. Assume that a Lewis acid M of concentration CMis titrated with a Lewis base Y of concentration Cu to form a 1:lcomplex MY according to the reaction M+Y-MY KMY=-[MY1 (8) [MI[Yl The mass balance equations will give [MI + [MY] = Fiowe 1. Onehundred cubic centimeters of a solution containino three weak a c m n he cancennal~ms 3 X 10-'molldm3. 5 X 10Pmoildml, am 4 X 10 ' moildm3 re~pect~veiy s rlwsted worn 0 1 moildm3 NaOH The denvatwe d V l o p i or 8. is also ploned as a functionof V.
+
[Y] [MY] =-
CM
(9)
vo+ v CY
(10)
vo+ v
A combination of eqns. (8)-(10) will give
The latter expression is derived to save program steps in small calculators. If M, S , and/or MS are involved in side reactions, u-coefficients and conditional constants ran he used to simdifv the eauilihriurn calculations. The conditional constants &
