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Scientist: Examples: Chemistry

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:

Chemical Equilibrium Titration Model

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:

Chemical Equilibrium Parameter Estimates

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.