Titration Curve Calculus

Refinement of the use of AF/ApH and its reciprocal as buffer and sharpness indices (as mentioned above) is achieved by obtaining the derivative of the titration curve, dF/dpH. As will be seen, this derivative is a function of the several da/dpH values, which in turn are simple functions of the a values themselves. 

Equations 8-8 and 8-9 can be transformed (daj/2.303aj = loga ) to express the variation of log a with pH 

For diprotic acids the derivatives of a values with respect to pH can be shown to be 

In each of these cases, the da/dpH is proportional to the a in question times a factor which includes all the other a values, each multiplied by the number of protons gained (a minus sign for protons lost). The factor containing the other a values represents the slope of the log a - pH curve, i.e., the EquiligrapH. Thus, the slope of the log [ 03 ] vs. pH curve will be +2 when H2CO3 predominates (a, 1, aj and a2 being almost zero), +1 when HC03 predominates (a, 1, the other as w 0) and 0 when CO predominates (a2 1, the other as 0). 

We can generalize from these considerations that one can write the value for da/dpH for any a in an N-protic acid system as 2.303 times the product of two factors (1) the a in question and (2) the sum of all the other alphas in the system and the number of protons gained in going from the species corresponding to the a in question to that of each of the other as. Thus, to illustrate with da2/dpH for H4Y (EDTA) 

---

An expression for instantaneous buffer capacity, jS, can be derived using calculus. Essentially, /S is the reciprocal of the slope of the titration curve at any point. Starting with the Henderson-Hasselbalch equation ... 

The end point can be taken as the inflection point of the titration curve. With a sigmoid-shaped titration curve, the inflection point is the steepest part of the titration curve where the pH change with volume is a maximum. This can be estimated visually from the plot or by using calculus to find the first and. second derivatives of the titration curve. The first derivative, ApH/AK gives the slope of the titration curve. It goes from nearly zero far before the end point to a maximum at the end point back to zero far beyond the end point. We can differentiate a second time to locate the maximum of the first derivative, since the slope of the first derivative... 

With points on the titration curve closely spaced, obtaining slopes (ApH/AVg) is a numerical differentiation that is a reasonable approximation of the calculus operation of differentiation. Compare the numerical and calculus approaches in titration curves using several different acid concentrations. 

Because values of the critical variable can be listed in very small increments without the customary tediousness, the spreadsheet format is very well suited for performing reasonably accurate calculus operations by numerical differentiation and integration. Useful examples of the former include titration curve slopes, dV /dpH and dpH/dV, which give us important titration curve parameters, namely, the buffer and sharpness indices. Another area of great interest, chemical kinetics, represents an additional topic where numerical differentiation and integration are of great use. 

---