Simultaneous determination of equilibrium constants and enthalpy changes by titration calorimetry: Methods, instruments, and uncertainties
Section snippets
Comparison of calorimeters for determination of equilibrium constants
Fig. 1 shows simulated calorimetric titration curves for reactions with differing equilibrium constants. The calorimetric method for determination of equilibrium constants depends on the reaction being significantly incomplete at the equivalence point. Quantitatively, the degree of completion at the equivalence point determines how “rounded” the titration curve is and is related to the product of Kf and CR. The equilibrium constant for the reaction titrant (T) + reactant (R) = product (P) with 1:1
Partial fill versus overfill
The choice to operate with a partially filled or overfilled reaction vessel (see Table 1) may be dictated by the calorimeter design, but some calorimeters can be operated in either mode. In overfill mode, both the reaction vessel and a narrow-bore fill tube, which may also contain the stirrer shaft, are completely filled with reactant solution. As titrant enters the solution, reactant solution overflows through the fill tube. The advantage of an overfilled reaction vessel is the absence of a
Analysis of calorimetric titration data for equilibrium constants and enthalpy changes
Generalized computer programs are available to calculate species distributions for the following categories of reactions:
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Complexation: M + xL = MLx, where M is a metal ion or macromolecule and x = an integer with units of moles of ligand (L) per mole of M. Note that Kf values can be determined only for those species that are present to a significant extent at some time during the titration.
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Sorption: A + xL = ALx, where A is a sorbent and x = moles ligand (L) per unit mass of A. Various sorption isotherms
Modeling the system
Once the data have been collected (e.g., heat, concentrations), a model that describes the chemistry of the system must be chosen. Such a model is composed of the balanced chemical reactions, associated equilibrium constants, and mass and heat balance equations [57]. In general, the model iswhere nj,i is the number of moles of the jth reaction product at point i in time (see Table 1 for definitions of symbols). In addition,where ∏ indicates the equilibrium
Differential computation versus integral computation of Kf and ΔH
In the integral method, data in the form of qc,i versus t are fit to the model, and in the differential method, data in the form Δqc,i versus t are fit to the model (see Table 1 for definitions of symbols). Data collected by either continuous or incremental titration and by any calorimetric method can be analyzed by either method. However, because the primary measurements are different, the effects of random and systematic errors differ among the titration and calorimetric methods. In
Estimation of uncertainties in Kf and ΔH
Uncertainties in estimated Kf and ΔH values can be qualitatively assessed with the help of Fig. 1, which shows curves for simulated titrations with different KfCR values. Note first that uncertainties in Kf and ΔH depend on the value of KfCR chosen for the titration. The most accurate values of ΔH will be obtained from early points in a titration with KfCR > 50 because the reaction is near completion and ΔH ≈ Δqi/nT,i. Under these conditions, ΔH is nearly independent of Kf. Obtaining accurate Kf
Systematic errors
After Kf and ΔH values have been computed, the results must be examined for the presence of significant systematic errors by plotting the residuals, [qc,i(measured) − qc,i(calculated)] or [Δqc,i(measured) − Δqc,i(calculated)], versus i (see Table 1 for definitions of symbols). Points should be randomly distributed around zero. A nonrandom distribution or offset from zero is indicative of systematic error [74].
A nonrandom distribution of points in a plot of the residuals between the model and
Random errors
The effects of random errors on the uncertainties in determined values of equilibrium constants and enthalpy changes can be determined by adding random values representing relative errors to simulated heat data for a 1:1 reaction. Fig. 6 illustrates the results from such a study where the magnitude of the relative errors, 1, 5, 10, and 20%, was determined as a percentage of the first data point and established as the standard deviation of the data. This procedure is similar to that described by
Random plus systematic errors
Fig. 7 shows the effects of 10% random error plus systematic errors in corrections for baseline, heats of dilution, and secondary reactions. The combination of random and systematic error causes serious problems in determination of Kf values, particularly at larger Kf values (i.e., where KfCR > 500). The effects of combined systematic and random errors on ΔH are similar to the effects of systematic error only, with random error actually tending to diminish the effects of systematic error,
Optimal conditions to minimize uncertainties
Choosing conditions with 50 < KfCR < 500 is crucial to obtaining accurate results. The optimum is near KfCR = 100 but also depends on the value of ΔH. Values of KfCR much smaller than this can lead to large errors in ΔH, and values much larger than this can lead to large errors in Kf. The next condition that must be optimized is the number and size of injections of titrant (or, in the case of continuous titration, the titrant delivery rate). Too few injections will not define the curve sufficiently
Conclusions
General equations that allow comparisons of calorimeters and optimizing conditions for simultaneous determinations of equilibrium constants and enthalpy changes by titration calorimetry have been derived. Different methods for titration and for computation of Kf and ΔH values were compared. Continuous titration is advantageous over incremental titration both in rapidity of experiments and in definition of data near equivalence points, but continuous titrations should be done only in temperature
Acknowledgment
L.D.H. thanks Brigham Young University for continuing support of his research.
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