Simultaneous determination of equilibrium constants and enthalpy changes by titration calorimetry: Methods, instruments, and uncertainties

doi:10.1016/j.ab.2010.11.002

Analytical Biochemistry

Volume 409, Issue 2, 15 February 2011, Pages 220-229

https://doi.org/10.1016/j.ab.2010.11.002 Get rights and content

Abstract

Calorimetric methods have been used to determine equilibrium constants since 1937, but no comprehensive review of the various calorimeters and methods has been done previously. This article reports methods for quantitative comparison of the capabilities of calorimeters for simultaneous determination of equilibrium constants and enthalpy changes, for determining optimal experimental conditions, and for assessing the effects of systematic and random errors on the accuracy and precision of equilibrium constants and enthalpy changes determined by this method.

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 K_f and C_R. 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:

•
Complexation: M + xL = ML_x, 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 K_f values can be determined only for those species that are present to a significant extent at some time during the titration.
•
Sorption: A + xL = AL_x, 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 is $q_{c, i} = \sum_{j} Δ H_{j} n_{j, i},$ where n_j_,_i is the number of moles of the j^th reaction product at point i in time (see Table 1 for definitions of symbols). In addition, $K_{j} = \prod (C_{T, i}) (C_{R, i}) γ_{j, i},$ where ∏ indicates the equilibrium

Differential computation versus integral computation of K_f and ΔH

In the integral method, data in the form of q_c_,_i versus t are fit to the model, and in the differential method, data in the form Δq_c_,_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 K_f and ΔH

Uncertainties in estimated K_f and ΔH values can be qualitatively assessed with the help of Fig. 1, which shows curves for simulated titrations with different K_fC_R values. Note first that uncertainties in K_f and ΔH depend on the value of K_fC_R chosen for the titration. The most accurate values of ΔH will be obtained from early points in a titration with K_fC_R > 50 because the reaction is near completion and ΔH ≈ Δqi/nT,i. Under these conditions, ΔH is nearly independent of K_f. Obtaining accurate K_f

Systematic errors

After K_f and ΔH values have been computed, the results must be examined for the presence of significant systematic errors by plotting the residuals, [q_c_,_i(measured) − q_c_,_i(calculated)] or [Δq_c_,_i(measured) − Δq_c_,_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 K_f values, particularly at larger K_f values (i.e., where K_fC_R > 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 < K_fC_R < 500 is crucial to obtaining accurate results. The optimum is near K_fC_R = 100 but also depends on the value of ΔH. Values of K_fC_R much smaller than this can lead to large errors in ΔH, and values much larger than this can lead to large errors in K_f. 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 K_f 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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Simultaneous determination of equilibrium constants and enthalpy changes by titration calorimetry: Methods, instruments, and uncertainties

Abstract

Section snippets

Comparison of calorimeters for determination of equilibrium constants

Partial fill versus overfill

Analysis of calorimetric titration data for equilibrium constants and enthalpy changes

Modeling the system

Differential computation versus integral computation of Kf and ΔH

Estimation of uncertainties in Kf and ΔH

Systematic errors

Random errors

Random plus systematic errors

Optimal conditions to minimize uncertainties

Conclusions

Acknowledgment

Biophys. Chem.

J. Chem. Thermodyn.

Thermochim. Acta

Thermochim. Acta

Thermochim. Acta

Thermochim. Acta

J. Biochem. Biophys. Methods

Methods Enzymol.

Curr. Opin. Struct. Biol.

Anal. Biochem.

Biophys. Chem.

Thermochim. Acta

Inorg. Chim. Acta

Methods Cell Biol.

Anal. Biochem.

Methods Enzymol.

Methods Cell Biol.

Anal. Biochem.

Anal. Biochem.

Methods Enzymol.

Biophys. Chem.

Anal. Biochem.

Anal. Biochem.

Anal. Biochem.

Anal. Biochem.

Methods Cell Biol.

Biophys. J.

Thermochim. Acta

Anal. Biochem.

Methods

J. Mol. Biol.

Biophys. J.

Methods Enzymol. A

Anal. Biochem.

Stupid statistics

Methods Cell Biol.

Equations to obtain, for equilibrium reactions, free-energy, heat, and entropy changes from two calorimetric measurements,

Proc. Natl. Acad. Sci. USA

Entropy titration: a calorimetric method for the determination of ΔG (K), ΔH, and ΔS

J. Chem. Soc., Chem. Commun.

New precision thermometric titration calorimeter

Rev. Sci. Instrum.

Entropy titration: a calorimetric method for the determination of ΔG, ΔH, and ΔS from a single thermometric titration

J. Phys. Chem.

Thermodynamics of proton dissociation in dilute aqueous solution: V. An entropy titration study of adenosine, pentoses, hexoses, and related compounds

J. Am. Chem. Soc.

Computer evaluation of entropy titration data: calorimetric determination of log βi, ΔHi , ΔSi for the silver(I) and copper(II)–pyridine systems

J. Phys. Chem.

Calorimetric investigation of adsorption of aromatic compounds by Linde Molecular Sieve 13X

Anal. Chem.

Binding of the Triton X series of nonionic surfactants to bovine serum albumin

Biochemistry

Theoretical evaluation of entropy titration method for calorimetric determination of equilibrium constants in aqueous solution

Anal. Chem.

Differential computation versus integral computation of K_f and ΔH

Estimation of uncertainties in K_f and ΔH