This example illustrates the use of the model for triprotic acid equilibrium to determine the values of the pKa's of the acid. We will work with data from the titration with base of a solution containing about 0.1 molar phosphoric acid. Since the phosphoric acid concentration, ATOT, is not known exactly it will be treated as a fitted parameter along with the acid pKa's. The total amount of proton initially present, HTOT0, is 3 times the total phosphate present, since we start with H3PO4. Each mole of OH added has the effect of correspondingly reducing HTOT, the net proton balance. This leads to the following Scientist model for the titration:

The final equation is the proton mass balance. The existence of variables defined in terms of PH on both sides of the equation indicates that this is an implicit equation for PH. Thus for a given value of OHADD, Scientist will iteratively seek the value for PH that causes the proton mass balance to be satisfied after all the other species concentrations are calculated. We therefore indicate that PH is to lie in the range of 0 to 14 using the constraint term following the last equation.

As is typical for titration experiments, we will use a data set with a large number of points, in this case 61 points that span the interval from OHADD = 0 to OHADD = 0.300 in equal increments of 0.005. Thus each iteration of the model parameters will involve the solution of the implicit system of equations above some 61 times. We give Scientist initial estimates and ranges for the parameters as follows:

After a brief calculation, Scientist reports success in fitting the model. We calculate Statistics Report for the fit as follows.

Micromath Scientist Statistics Report | ||||

Input Information | ||||

Model: | pK Determination.eqn | |||

Data Set: | pK Determination | |||

Parameter Set: | Parameter Set 3 | |||

Report Options | ||||

Descriptive Statistics | N | |||

Goodness-of-Fit Statistics | Y | |||

Confidence Intervals | Y | |||

Variance-Covariance Matrix | Y | |||

Correlation Matrix | Y | |||

Rigorous Limits | N | |||

Residual Analysis | Y | |||

Confidence Interval | 95% | 50 | ||

Goodness-of-Fit Statistics | ||||

Data Column Name: | OHADD | |||

Weighted | Unweighted | |||

Sum of squared observations: | 1.8453 | 1.8453 | ||

Sum of squared deviations: | 3.2736E-006 | 3.2736E-006 | ||

Standard deviation of data: | 0.00023965 | 0.00023965 | ||

R-squared: | 0.99999 | 0.99999 | ||

Coefficient of determination: | 0.99999 | 0.99999 | ||

Correlation: | 1 | 1 | ||

Data Set Name: | pK Determination | |||

Weighted | Unweighted | |||

Sum of squared observations: | 1.8453 | 1.8453 | ||

Sum of squared deviations: | 3.2736E-006 | 3.2736E-006 | ||

Standard deviation of data: | 0.00023965 | 0.00023965 | ||

R-squared: | 0.99999 | 0.99999 | ||

Coefficient of determination: | 0.99999 | 0.99999 | ||

Correlation: | 1 | 1 | ||

Model Selection Criterion: | 11.749 | 11.749 | ||

Confidence Intervals | ||||

Parameter Name: | PKH3A | |||

Estimated Value: | 2.1459 | |||

Standard Deviation: | 0.0013264 | |||

95% Range (Univariate): | 2.1432 | 2.1485 | ||

95% Range (Support Plane): | 2.1417 | 2.1501 | ||

Parameter Name: | PKH2A | |||

Estimated Value: | 7.197 | |||

Standard Deviation: | 0.0022613 | |||

95% Range (Univariate): | 7.1925 | 7.2016 | ||

95% Range (Support Plane): | 7.1898 | 7.2042 | ||

Parameter Name: | PKHA | |||

Estimated Value: | 12.344 | |||

Standard Deviation: | 0.0029078 | |||

95% Range (Univariate): | 12.338 | 12.35 | ||

95% Range (Support Plane): | 12.335 | 12.353 | ||

Parameter Name: | ATOT | |||

Estimated Value: | 0.097024 | |||

Standard Deviation: | 5.426E-005 | |||

95% Range (Univariate): | 0.096915 | 0.097132 | ||

95% Range (Support Plane): | 0.096851 | 0.097196 | ||

Variance-Covariance Matrix | ||||

1.7592E-006 | ||||

1.0788E-006 | 5.1134E-006 | |||

1.5902E-006 | 4.9092E-006 | 8.4551E-006 | ||

3.2078E-008 | 9.9037E-008 | 1.4594E-007 | ||

Correlation Matrix | ||||

1 | ||||

0.3597 | 1 | |||

0.4123 | 0.74661 | 1 | ||

0.44573 | 0.80717 | 0.92501 | ||

Residual Analysis | ||||

Expected Value: | The following are normalized parameters with an expected value of 0.0. Values are in units of standard deviations from the expected value. | |||

Serial Correlation: | -0.87966 | is probably not significant. | ||

Skewness | -13.231 | indicates the likelihood of a few large negative residuals having an unduly large effect on the fit. | ||

Kurtosis | 6.8116 | is probably not significant. | ||

Weighting Factor: | 0 | |||

Heteroscedacticity: | 1.3441 | |||

Optimal Weighting Factor: | 1.3441 |

As might be expected, parameter values can be very well determined as a result of the number and precision of the data points available in titration studies.