Models to analyze acid-base and other titration data employ the usual theory of equilibrium [1], In general, an equation describing the measured response in terms of the initial concentration of analyte (c"), concentration of titrant (cb), initial volume of analyte (V^), and volume of titrant added (Vb) is used as the model.

Weak Acids Consider a potentiometric titrations of a weak monobasic acid with a strong base; that is,

The exact equation [3] for the concentration of hydrogen ion during the titration is

where

Ka = the dissociation constant for the weak acid, Kw = the ion product [H+][OH-], = initial concentration of analyte, cb = concentration of titrant, V° = initial volume of analyte, Vb = volume of titrant added, / = fraction of analyte titrated = Vbcb/V^c% a = c°afl( 1 + rf) + ^ — ca Ka (1 — /)/(l + rf) + Km r = c°Jcb.

The third-order polynomial in [H+] is used along with the relation converting [H+] to pH:

where yH+ = the activity coefficient of hydrogen ion.

Equations (5.2) and (5.3) can be used in the model subroutine to compute a value of pH after each increment of titrant added. Solution of the third-order polynomial by an appropriate numerical method, such as Newton-Raphson (Section 3.B.3) is required.

Weak Bases A second-order equation in [H+] obtained by neglecting Kw (eq. (5.4)) was shown by Barry and Meites [4] to give excellent results for the titration of acetate ion with hydrochloric acid, using 3 M KCl to keep activity coefficients constant. This equation has the advantage that it can be solved by the quadratic method.

where

Ka = the dissociation constant for the conjugate acid of the weak base, c° = initial concentration of analyte, ca = concentration of titrant,

Vb = initial volume of analyte,

Va = volume of titrant added,

/ = fraction of analyte titrated = VacaIVt,Cb,

Ignoring the negative root, the quadratic solution to eq. (5.2) is

A nonlinear regression program to analyze data from titrations of weak bases with strong acids requires eqs. (5.3) and (5.5) and definitions of a and ¡3 in the subroutine used to calculate pH as a function of Va. A total of four regression parameters are possible:

B(3) = yH+ an effective activity coefficient for hydrogen ion.

A subroutine for this calculation of pH as a function of Va suitable for use in a nonlinear regression analysis is listed in Box 5.1.

When using this model, it is important to keep the ionic strength relatively large and constant, to keep yH+ constant throughout the titration. This is best accomplished by using an inert 1:1 electrolyte such as NaCl at concentrations of 1-3 M.

Although this model contains four parameters, in few situations do we need to find all four of them in the analysis. We found that using this model with two or three parameters greatly improved convergence properties compared to four-parameter fits when using a Marquardt-Levenberg algorithm.

The most important parameter to be found in the titration is the analyte concentration The best way to find it by nonlinear regression is discussed later. We suppose that determination of ca is also desirable, because then we can use an unstandardized reagent with a concentration known only to two significant figures. We can envision the following analytical situations.

Case 1 The identity of the weak base is known, and pKa and yH+ are known or can be found by a standard titration where both ca and c° are known.

a = Ka + (Vgcg - Vaca)!{Vl + Va), j3 = VaKacJ(V°h + Va).

BASIC SUBROUTINE FOR COMPUTING pH (YCALO IN TITRATION OF WEAK BASE WITH A STRONG ACID

5 REM "YCALC subroutine ** FITS titration of weak base with strong acid

20 ALPHA=(B(1)*VBASE-A(2,I%)*B(2))/(A(2,I%)+VBASE) + B(0) 30 BETA=(-A(2,I%)*B(0)*B(2))/(A(2,I%)+VBASE) 40 H=(-ALPHA+SQR(ALPHAA2-4*BETA))/2 50 YCALC = -LOG(B(3)*H)/LOG(10) Documentation:

line 20 - computes a from Vb°, Va = A(2,I%), and values of parameters B(0), B(l), and B(2).

line 30 - computes (3

line 40 - computes [H+] from a and |3 using the quadratic formula line 50 - computes pH from [H+] and B(3)._

Box 5.1 BASIC subroutine for computing pH (YCALC) in titration of a weak base with a strong acid.

Finding pKa and -yH+ in this situation can be done experimentally by titrating a solution of the analyte of known concentration with a standardized titrant. Reliable values of pKa and yH+ can be found by fitting these standard data to the model with ca and fixed. These values of pKa and yH+ can then be used in subsequent titrations of real samples.

volume of acid (mL)

Figure 5.1 Simulated titration curve of a weak base (0.0100 M) with a strong acid (0.0100 M) with yni = 0.90, pKa = 3 for conjugate acid of the weak base, and an error of ±0.1% of a pH unit. Circles are simulated data, line is the best fit with model in Box 5.1.

Figure 5.1 Simulated titration curve of a weak base (0.0100 M) with a strong acid (0.0100 M) with yni = 0.90, pKa = 3 for conjugate acid of the weak base, and an error of ±0.1% of a pH unit. Circles are simulated data, line is the best fit with model in Box 5.1.

Figure 5.S Simulated titration curve of a weak base (0.0100 M) with a strong acid (0.0100 M) with yH+ = 0.90, pKa = 5 for conjugate acid of the weak base, and an error of ±0.1% of a pH unit. Circles are simulated data, line is the best fit with model in Box 5.1.

volume of acid (mL)

Figure 5.S Simulated titration curve of a weak base (0.0100 M) with a strong acid (0.0100 M) with yH+ = 0.90, pKa = 5 for conjugate acid of the weak base, and an error of ±0.1% of a pH unit. Circles are simulated data, line is the best fit with model in Box 5.1.

We have followed the preceding scenario by using simulated data for titrations of 10 mL of 0.010 M weak base with 0.010 M strong acid. Data were simulated for various values of pKa with the equations in Box 5.1, using yH+ = 0-90. Absolute noise of 0.1% of a pH unit was added by using the subroutine in Box 3.1 in Chapter 3.

The shapes of titration curves computed with the above parameters for pKa values of 3, 5, and 7 are given in Figures 5.1, 5.2, and 5.3, respectively. Data sets were computed with 40 points equally spaced on the Va axis for extent of titration of 60% to 140%. This is the pKa for the conjugate acid of the weak base. Thus, if X is the weak base, the values of pKa pertain to the reaction:

volume of acid (mL)

Figure 5.3 Simulated titration curve of a weak base (0.0100 M) with a strong acid (0.0100 M) with yH+ = 0.90, pKa = 7 for conjugate acid of the weak base, and an error of ±0.1% of a pH unit. Circles are simulated data, line is the best fit with model in Box 5.1.

Since the basicity constant pKh = 14lpKw the larger the pKa, the stronger is the base X . Recall that the endpoint of a potentiometric titration is located at the inflection point of a sigmoid-shaped titration curve. For the data with pKa = 3, the base is so weak that the shape of the titration curve is not sigmoid at all (Figure 5.1). This is expected because the pKh is 11. The titration curve does not show any inflection point and remains in the lower pH region of the coordinate system.

When the pKa is increased to 5, a rather poorly defined inflection point appears (Figure 5.2). At pKa = 7, the base is strong enough for a reasonable inflection point (Figure 5.3), and the endpoint could be readily located by conventional derivative techniques [1].

Following the case 1 scenario, we determined pKa and yH+ by nonlinear regression onto the model in Box 5.1, assuming that we know ca and c° to an accuracy of ±0.2% in the experiment on standards. Then, in a separate set of analyses meant to emulate analysis of sample data, we used the values of pKa and found in the first analysis in a new regression to obtain ca and Results show (Table 5.1) that the errors in Ka and yH+ increase as pKa increases; that is, as the base gets stronger.

The error in c° does not become significant, however, until we reach pKa > 7. The error in c° appears to be most sensitive to the error in yH+ under these conditions. We shall see later that much better results are obtained by using three- or four-parameter fits.

Case 2 The identity of the weak base is unknown but yH+ can be found by a titration of a known base in which the ionic strength and electrolyte used is the same as in our analytical titration.

In this case, we assume that we can find yH+ by a one-parameter fit of the standard data, where pKa for the standard known base, and ca and c° are fixed at their known values. To find in the unknown samples, we now have the option of a three-parameter fit with yH+ fixed at its known value or a four-parameter fit. We have examined both situations for our test data. The results of the three-parameter fit were used as initial guesses in the four-parameter fits.

True pKa |
Ka found |
-yH+ found |
eg, found* |
Error in c" |

3 |
1.001 x lO"3 |
0.904 |
0.01001 |
+0.1% |

5 |
1.002 x lO'5 |
0.878 |
0.01007 |
+0.7% |

7 |
1.010 x 10-7 |
0.940 |
0.00848 |
-15% |

" True values are -yH+ = 0.90, c„ = c°h = 0.0100 M. b Initial guess of c" had +30% error.

" True values are -yH+ = 0.90, c„ = c°h = 0.0100 M. b Initial guess of c" had +30% error.

The results of this analysis show that the approach works quite well for nearly all the data sets. Errors of <0.1% were found for all data with pKa < 7, using either three- or four-parameter fits (Table 5.2). Only for a pKa value of 9, that is, for the strongest base considered, was an error of 0.5% encountered. This latter data set required a two-parameter fit, because trouble with convergence of the Marquardt algorithm was encountered when three or four parameters were used.

Examination of Tables 5.1 and 5.2 suggests that an accurate value of -yH+ is important for the success of the regression analysis. We also find, at least with the Marquardt-Levenberg algorithm, that the method works best for weaker bases. Convergence problems are encountered when pKa > 7. However, this does not pose a serious problem to the analyst. Conventional endpoint detection methods can be used in these cases, which are characterized by a clearly visible inflection point in the titration curve (Figure 5.3). It also appears that the approach taken in case 2 will give better results than in case 1. That is, even if the pKa is known, the three- and four-parameter fits give better results than two-parameter fits.

Thus, we see that, as long as an accurate value of yH+ can be obtained, excellent results can be expected for determination of the concentrations of very weak bases by titration with a strong acid. These titration-regression analyses can provide accurate values of analyte concentrations without standardization of the titrant. The concentration of the titrant need be known to only about two significant figures.

In practice, fitting the pH vs. Vb data by nonlinear regression can yield highly accurate and precise analyte concentrations and equilibrium constants. Typically, accuracy and precision within two parts per thousand can be achieved, even in titrations of weak bases such as acetate (pKb = 9.3), which give poorly defined potentiometric endpoint breaks [4, 5].

True |
No. |
TH+ |
Error | |||

pKa |
parameters" |
Ka found |
found |
ca, found |
cü, found |
in c' |

3 |
3 |
1.000 x 10-3 |
Fixed |
0.01000 |
0.010001 |
+0.01% |

3" |
1.016 x 10-3 |
Fixed |
0.01007 |
0.010001 |
+0.01% | |

4 |
1.000 x 10 3 |
0.900 |
0.01000 |
0.010001 |
+0.01% | |

5 |
3 |
9.995 X 10'6 |
Fixed |
0.009996 |
0.09996 |
-0.04% |

4 |
9.996 X lO 6 |
0.899 |
0.009994 |
0.09993 |
-0.07% | |

7 |
3 |
1.000 x 10-7 |
Fixed |
0.009999 |
0.099998 |
-0.01% |

4 |
Matrix inversion error | |||||

9 |
2C |
1.29 X 10"' |
Fixed |
Fixed |
0.01005 |
+0.5% |

" Number of parameters not fixed; true values are yH+ = 0.90, c„ = c° = 0.0100 M. Initial guesses had ±30% error unless otherwise noted. b Initial guess of c° had +90% error.

c Only two-parameter fits would converge for this set of data.

" Number of parameters not fixed; true values are yH+ = 0.90, c„ = c° = 0.0100 M. Initial guesses had ±30% error unless otherwise noted. b Initial guess of c° had +90% error.

c Only two-parameter fits would converge for this set of data.

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