It should be noted that the applications of thermometric titrimetry discussed on this page are by no means exhaustive. The reader is referred to the bibliography for further reading on the subject.
Potentiometric titrimetry has been the predominant automated titrimetric technique since the 1970s, so it is worthwhile considering the basic differences between it and thermometric titrimetry.
Potentiometrically-sensed titrations rely on a free energy change in the reaction system. Measurement of a free energy dependent term is necessary.
In order for a reaction to be amenable to potentiometric titrimetry, the free energy change must be sufficient for an appropriate sensor to respond with a significant inflection (or "kink") in the curve where sensor response is plotted against the amount of titrant delivered.
However, free energy is just one of three related parameters in describing any chemical reaction:
For any reaction where the free energy is not opposed by the entropy change, the enthalpy change will be significantly greater than the free energy. Thus a titration based on a change in temperature (which permits observation of the enthalpy change) will show a greater inflection than will curves obtained from sensors reacting to free energy changes alone.
In a thermometric titration, titrant is added at a known constant rate to a titrand until the completion of the reaction is indicated by a change in temperature. The endpoint is determined by an inflection in the curve generated by the output of a temperature measuring device.
Consider the titration reaction:
At completion, the reaction produces a molar heat of reaction ΔHr which is shown as a measurable temperature change ΔT. In an ideal system, where no losses or gains of heat due to environmental influences* are involved, the progress of the reaction is observed as a constant increase or decrease of temperature depending respectively on whether ΔHr is negative (indicating an exothermic reaction) or positive (indicating an endothermic reaction).
* among “environmental influences” may be categorized:
• Heat losses or gains from outside the system via the vessel walls and cover
•Differences in the temperature between the titrant and the titrand''
•Evaporative losses from the surface of the rapidly mixed fluid
•Heats of solution when the titrant solvent is mixed with the analyte solvent
•Heat introduced by the mechanical action of stirring(minor influence)
•Heat produced by the thermistor itself (very minor influence)
If the equilibrium for the reaction lies far to the right (i.e. a stoichiometric equilibrium has been achieved), then when all analyte has been reacted by the titrant continuing addition of titrant will be revealed by a sharp break in the temperature/volume curve. Figures 1a and 1b illustrate idealized examples.
The shape of experimentally obtained thermometric titration plots will vary from such idealized examples, and some of the environmental influences listed above may have impacts. Curvature at the endpoint might be observed. This can be due to insensitivity of the sensor or where thermal equilibrium at the endpoint is slow to occur. It can also occur where the reaction between titrant and titrand does not proceed to stoichiometric completion. The determinant of the degree to which a reaction will proceed to completion is the free energy change. If this is favourable, then the reaction will proceed to be completion and be essentially stoichiometric. In this case, the sharpness of the endpoint is dependent on the magnitude of the enthalpy change. If it is unfavourable, the endpoint will be rounded regardless of the magnitude of the enthalpy change. Reactions where non-stoichiometric equilibria are evident can be used to obtain satisfactory results using a thermometric titration approach. If the portions of the titration curve both prior to and after the endpoint are reasonably linear, then the intersection of tangents to these lines will accurately locate the endpoint. This is illustrated in Figure 2.
Consider the reaction for the equation aA + bB = pP which is non-stoichiometric at equilibrium. Let A represent the titrant, and B the titrand. At the beginning of the titration, the titrand B is strongly in excess, and the reaction is pushed towards completion. Under these conditions, for a constant rate of titrant addition the temperature increase is constant and the curve is essentially linear until the endpoint is approached. In a similar manner, when the titrant is in excess past the endpoint, a linear temperature response can also be anticipated. Thus intersection of tangents will reveal the true endpoint.
The most practical sensor for measuring temperature change in titrating solutions has been found to be the thermistor. Thermistors are small solid state devices which exhibit relatively large changes in electrical resistance for small changes in temperature. They are manufactured from sintered mixed metal oxides, with lead wires enabling connection to electrical circuitry. The thermistor is encapsulated in a suitable electrically insulating medium with satisfactory heat transfer characteristics and acceptable chemical resistance. Typically for thermistors used for chemical analysis the encapsulating medium is glass, although thermistors encapsulated in epoxy resin may be used in circumstances where either chemical attack (e.g., by acidic fluoride-containing solutions) or severe mechanical stress is anticipated. The thermistor is supported by suitable electronic circuitry to maximize sensitivity to minute changes in solution temperature. The circuitry in the Metrohm 859 Titrotherm thermometric titration interface moldule is capable of resolving temperature changes as low as 10-5 K.
A critical element in modern automated thermometric titrimetry is the ability to locate the endpoint with a high degree of reproducibility. It is clearly impractical and insufficient for modern demands of accuracy and precision to estimate the inflection by intersection of tangents. This is done conveniently by derivatization of the temperature curve. The second derivative essentially locates the intersection of tangents to the temperature curve immediately pre- and post- the breakpoint.
Thermistors respond quickly to small changes in temperature such as temperature gradients in the mixed titration solution, and thus the signal can exhibit a small amount of noise. Prior to derivatization it is therefore necessary to digitally smooth (or “filter”) the temperature curve in order to obtain sharp, symmetrical second derivative “peaks” which will accurately locate the correct inflection point. This is illustrated in Figure 5. The degree of digital smoothing is optimized for each determination, and is stored as a method parameter for application every time a titration for that particular analysis is run.
Because enthalpy change is a universal characteristic of chemical reactions, thermometric endpoint sensing can be applied to a wide range of titration types, e.g.
Further, since the sensor is not required to interact with the titration solution electrochemically, titrations in non-conducting media can be performed, as can titrations using reactions for which no convenient or cost-effective potentiometric sensor is available.
Thermometric titrations generally demand rapid reaction kinetics in order to obtain sharp reproducible endpoints. Where reaction kinetics are slow, and direct titrations between titrant and titrand are not possible, indirect or back-titrations often can be devised to solve the problem.
Catalytically enhanced endpoints can be used in some instances where the temperature change at the endpoint is very small and endpoints would not be detected satisfactorily by the titration software.
The suitability of a particular chemical reaction as a candidate for a thermometric titration procedure can generally be predicted on the basis of the estimated amount of analyte present in the sample and the enthalpy of the reaction. However, other parameters such as the kinetics of the reaction, the sample matrix itself, heats of dilution and losses of heat to the environment can affect the outcome. A properly designed experimental program is the most reliable way of determining the viability of a thermometric titration approach. Successful applications for thermometric titrations are generally where titrant-titrand reaction kinetics are fast, and chemical equilibria are stoichiometric or nearly so.
Thermometric titration determinations may be recommended where:
A suitable setup for automated thermometric titrimetry comprises the following:
Figure 6 illustrates a modern automated thermometric titration system based on the Metrohm 859 Titrotherm interface module with Thermoprobe sensor, Metrohm 800 Dosino dispensing devices and a computer running the operational software.
Figure 7 is a schematic of the relationship between components in automated thermometric titration system.
A = dosing device
B = thermometric sensor
C = stirring device
D = thermometric titration interface module
E = computer
Applications for thermometric titrimetry are drawn from the major groupings, namely:
Because the sensor does not interact electrically or electrochemically with the solution, electrical conductance of the titrating medium is not a pre-requisite for a determination. Titrations may be carried out in completely non-conducting, non-polar media if required. Further, titrations may be carried out in turbid solutions or even suspensions of solids, and titrations where precipitates are reaction products can be contemplated. The range of possible thermometric titration applications far exceeds the actual experience of this writer, and the reader will be referred to the appropriate literature in some instances.
Mixtures of complex acids can be resolved by thermometric titration with standard NaOH in aqueous solution. In a mixture of nitric, acetic and phosphoric acids used in the fabrication of semi-conductors, three endpoints could be predicted on the basis of the dissociation constants of the acids:
|Endpoint 1||Endpoint 2||Endpoint 3|
|HNO3 (pKa = -1.3)||HOAc (pKa = 4.75)|
|H3PO4 (pKa1 = 2.12)||H3PO4 (pKa2 = 7.21)||H3PO4 (pKa3 = 12.36)|
The key to determine the amount of each acid present in the mixture is the ability to obtain an accurate value for the amount of phosphoric acid present, as revealed by titration of the third proton of H3PO4.
Figure 10 illustrates a titration plot of this mixture, showing 3 sharp endpoints.
The thermometric titrimetric analysis of sodium aluminate liquor (“Bayer liquor”) in the production of alumina from bauxite is accomplished in an automated two titration sequence. This is an adaptation of a classic thermometric titration application (VanDalen and Ward, 1973). In the first titration, tartrate solution is added to an aliquot of liquor to complex aluminate, releasing one mole of hydroxyl for each mole of aluminate present. This is titrated acidimetrically along with “free” hydroxyl present and the carbonate content (as a second endpoint). The second titration is preceded by the automatic addition of fluoride solution. The alumina-tartrate complex is broken in favour of the formation of an aluminium fluoride complex and the concomitant release of three moles of hydroxyl for each mole of aluminium present, which are then titrated acidimetrically. The whole determination can be completed in less than 5 minutes.
Acid leach solutions from some copper mines can contain large quantities of Fe(III) as well as Cu(II). The “free acid” (sulfuric acid) content of these leach solutions is a critical process parameter. While thermometric titrimetry can determine the free acid content with modest amounts of Fe(III), in some solutions the Fe(III) content is so high as to cause serious interference. Complexation with necessarily large amounts of oxalate is undesirable due to the toxicity of the reagent. A thermometric titration was devised by diluting the aliquot with propan-2-ol and titration with standard KOH in propan-2-ol. Most of the metal content precipitated prior to the commencement of the titration, and a clear, sharp endpoint for the sulfuric acid content was obtained.
A recent thermometric titrimetric procedure for the determination of FFA developed by Cameiro et al (2002) has been shown to be particularly amenable to automation. It is fast, highly precise, and results agree very well with those obtained by the official AOAC method. The temperature change for the titration of very weak acids such as oleic acid by 0.1 mol/L KOH in propan-2-ol is too small to yield an accurate endpoint. In this procedure, a small amount of paraformaldehyde as a fine powder is added to the titrand before the titration. At the endpoint, the first excess of hydroxyl ions catalyzes the depolymerization of paraformaldehyde. The reaction is strongly endothermic and yields a sharp inflection. The titration plot is illustrated in Figure 13. The speed of this titration coupled with its precision and accuracy makes it ideal for the analysis of FFA in biodiesel feedstocks and product.
Thermometric iodometric titrations employing thiosulfate as a titrant are also practical, for example in the determination of Cu(II). In this instance, it has been found advantageous to incorporate the potassium iodide reagent with the thiosulfate titrant in such proportions that iodine is released into solution just prior to its reduction by thiosulfate. This minimizes iodine losses during the course of the titration.
An excellent application is the sequential determination of calcium and magnesium. Although calcium reacts exothermically with EDTA (heat of chelation ~-23.4 kJ/mol), magnesium reacts endothermically with a heat of chelation of ~+20.1 kJ/mol. This is illustrated in the titration plot of EDTA with calcium and magnesium in sea water (Figure 18). Following the solution temperature curve, the breakpoint for the calcium content (red-tagged endpoint) is followed by a region of modest temperature rise due to competition between the heats of dilution of the titrant with the solution, and the endothermic reaction of Mg2+ and EDTA. The breakpoint for the consumption of Mg2+ (blue-tagged endpoint) by EDTA is revealed by upswing in temperature caused purely by the heat of dilution.
Direct EDTA titrations with metal ions are possible when reaction kinetics are fast, for example zinc, copper, calcium and magnesium. However, with slower reaction kinetics such as those exhibited by cobalt and nickel, back-titrations are used. Titrations for cobalt and nickel are carried out in an ammoniacal environment; buffered with ammonia:ammonium chloride solution. An excess of EDTA is added, and is back-titrated with Cu(II) solution. It is postulated that the breakpoint is revealed by the difference in reaction enthalpies between the formation of the Cu-EDTA complex, and that for the formation of the Cu-amine complex.
A catalyzed endpoint procedure to determine trace amounts of metal ions in solution (down to approximately 10 mg/L) employs 0.01 mol/L EDTA. This has been applied to the determination of low level Cu(II) in specialized plating baths, and to the determination of total hardness in water. The reaction enthalpies of EDTA with most metal ions are often quite low, and typically titrant concentrations around 1 mol/L are employed with commensurately high amounts of titrand in order to obtain sharp, reproducible endpoints. Using a catalytically indicated endpoint, very low EDTA titrant concentrations can be used. A back-titration is used. An excess of EDTA solution is added. The excess of EDTA is back-titrated with a suitable metal ion such as Mn2+ or Cu2+. At the endpoint, the first excess of metal ion catalyzes a strongly exothermic reaction between a polyhdric phenol (such as resorcinol) and hydrogen peroxide.
The reaction enthalpy for the formation of barium sulfate is a modest −18.8 kJ/mol. This can place a restriction on the lower limit of sulfate in a sample which can be analyzed.
Because 6 mole of fluoride react with one mole of aluminium, the titration is particularly precise, and a coefficient of variance (CV) of 0.03 has been achieved in the analysis of alum.
When aluminium ion (say as aluminium nitrate) is employed as the titrant, fluoride can be determined using the same chemistry. This titration is useful in the determination of fluoride in complex acid mixtures used as etchants in the semi-conductor industry.
Is exothermic. CV’s of under 0.1 have been achieved in test applications. The procedure is suitable for the determination of orthophosphate in fertilizers and other products.
The titration plot illustrated in Figure 19 shows that the endpoint is quite rounded, suggesting that the reaction might not proceed to stoichiometric equilibrium. However, since the regions of the temperature curve immediately before and after the endpoint are quite linear, the second derivative of this curve (representing the intersection of tangents) will accurately locate the endpoint. Indeed, excellent precision can be obtained with this titration, with a CV of less than 0.1.
1. J. M. Bell and C. F. Cowell. J. Am. Chem. Soc. 35, 49-54 (1913)
2. E. VanDalen and L. G. Ward. Thermometric titration determination of hydroxide and alumina in Bayer process solutions. Anal. Chem. 45 (13) 2248-2251, (1973)
3. M. J. D. Carneiro, M. A. Feres Júnior, and O. E. S. Godinho. Determination of the acidity of oils using paraformaldehyde as a thermometric end-point indicator. J. Braz. Chem. Soc. 13 (5) 692-694 (2002)
(All of the following books are out of print, but might be available from academic libraries or through second-hand book sellers such as Amazon.com)
1. G. A. Vaughan (1973) Thermometric and enthalpimetric titrimetry. Van Nostrand Reinhold Company (London) ISBN 0-442-78385 X Library of Congress Catalog Card No. 79-186764
2. L. S. Bark and S. M. Bark (1969). Thermometric titrimetry. International Series of Monographs in Analytical Chemistry Vol 33 Pergamon Press (Oxford) Library of Congress Catalog Card No. 68-57883
3. J. Barthel (1975) Thermometric titrations. John Wiley & Sons, New York. ISBN 0-471-05448-8 Library of Congress Catalog Card No. 75-17503
4. J. K. Grime (1985) Analytical solution calorimetry. John Wiley & Sons, New York. ISBN 0-471-86942-2 Library of Congress Catalog Card No. 84-28424
5. D. J. Eatough, J. J. Christensen, R. M. Izatt (1974) Experiments in thermometric titrimetry and titration calorimetry. Brigham Young University Press, Provo, Utah. ISBN 0-8425-0145-2 Library of Congress Catalog Card 74-13074