Acid-Base Titration Curves
Theory
The most important characteristics of a neutralization titration can be summarized in the titration curve (usually pH as a function of volume Vtitr of the titrant). The titration curve can be calculated theoretically, whereupon conclusion can be drawn from it for the feasibility and the expected accuracy of a titration, and the selection of the proper protolytic indicator.
The most common approach for the calculation of titration curves is based on approximations depending on the relative strength of the acid and base, the concentration levels, and the actual region of the titration curve relatively to the equivalence point.
Unfortunately, there is no explicit function of the type [H^{+}] = F(V_{titr}), that can be used for the straightforward calculation of the titration curve, but an implicit function of the type F([H^{+}], V_{titr}) = 0 can be readily obtained. This function is exact and valid over the entire titration curve for any acid-base system and at any concentration level. The only problem is that due to the complexity of this function a numerical procedure is required to calculate [H^{+}] at a given volume of titrant V_{titr}, a task that can be easily accomplished by a computer.
Derivation of the implicit function F([H^{+}], V_{titr}) = 0
The mass- and charge-balance of the chemical system must be used to derive the exact function F([H^{+}], V_{titr}) = 0. It is also, useful to consider the base as the corresponding conjugate acid, treating thus all species involved as acids or derived from acids. Therefore, the species that can be involved in the general case of titration of a n-protic acid (H_{n}A) with an m-protic base (B) titration (or vice versa) are:
The equilibria between the species of the n-protic acid are:
The equilibria between the species of the m-protic base, treated as the corresponding conjugate acid, are:
The charge balance is described by the following equation:
which can be written in the implicit form:
C_{B} and C_{A} are the actual analytical concentrations of the base and the acid at any point of the titration. b_{i} and a_{i} are the (molar) fractions of base ions with charge +i and of the acid ions with charge -i (e.g. b_{1} = [HB^{+}]/C_{B}, b_{2} = [H_{2}B^{2+}]/C_{B}, ..., a_{1} = [H_{n-1}A-]/C_{A}, a_{2} = [H_{n-2}A^{2-}]/C_{A}). These fractions are functions of [H^{+}] and depend also on the dissociation constants K_{a,1}, K_{a,2}, ... K_{b,1}, K_{b,2}, ... (see applet "Distribution Diagrams of Polyprotic Acids").
When a volume V_{A} of acid of molarity M_{A} is titrated with a base of molarity M_{B} and a volume V_{titr} of the titrant (base) has been added, we have:
When a volume V_{B} of base of molarity M_{B} is titrated with an acid of molarity M_{A} and a volume V_{titr} of the titrant (acid) has been added, we have:
The implicit function F([H^{+}], V_{titr})=0 can be solved (i.e. [H^{+}] can be calculated for a given V_{titr} value) by applying a numerical technique, such as the Newton-Raphson iterative method. However, it is important to realize that the implicit function F([H^{+}], V_{titr})=0 can easily acquire the form V_{titr} = f([H^{+}]). The latter function can be used indirectly for drawing the titration curves, i.e. by calculating V_{titr} for a given value of [H^{+}] (or pH).
Applet
With this applet we can obtain the titration curves of some of the more common acids and bases at any combination. The user can select the acid (among 26 real acids) and the base (among 22 real bases), by using the small move forward (>>) or backward (<<) buttons, next to the corresponding light red (for acids) and light blue (for bases) windows. The K_{a} and K_{b} values are displayed along the name of the selected reagents. Among the acids, NH_{4}Cl has been included as a weak cationic acid, whereas among the bases, CH_{3}COONa, NaCN, and Na_{2}CO_{3} have been included as anionic bases.
The concentrations and the volumes of the titrant and the titrated solution (titrand) can be freely selected between certain limits (click on "Solutions" radiobutton), set mostly for programming convenience. The plot area can also redesigned (click on "Graph" radiobutton).
Many titration curves can be displayed on the graph area (simply, do not click the CLEAR button). It is of interest to compare the titration plots of a group of structurally similar acids (e.g. CH_{3}COOH, CH_{2}ClCOOH, CHCl_{2}COOH, and CCl_{3}COOH, or of diprotic acids e.g. oxalic through adipic acid) with a strong base or NH_{3}.
It is also of interest to examine cases of titration curves with multiple equivalence points (e.g. oxalic acid or citric acid with ethylenediamine) and see which deflection would better serve for the location of the titration end-point.
It should be stressed that we consider all titrations taking place under constant ionic strength (e.g. in the presence of large excess of an inert salt), otherwise the gradually increasing ionic strength would require much more complicated calculations.
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