| Thermometric Titration Theory |
|
| | | | | | | | | | | | | |
|
UPDATE! A monograph giving an outline of the thermometric titration technique and descriptions of a range of applications is now available from Metrohm. Contact your Metrohm dealer, or visit www.metrohm.com to obtain a copy.
The following is a brief introduction to the theoretical basis of thermometric titrimetry.
Heat of Reaction
The heat change in any chemical reaction at constant pressure in a particular system may be expressed as:
D H° = D G° + TD S° (1)
where:
D H° = change in enthalpy
D G° = change in free energy
D S° = change in entropy
T = temperature in Kelvin
For a reaction of the form:
aA + bB + cC « pP + bB (2)
which when complete produces a molar heat of reaction D H manifested by a temperature change D T in the system, then the total heat effect Q Joule is related to D H by:
Q = -npD H (3)
where np is the number of moles of product formed; as well as to the change in temperature D T by:
Q = CsD T (4)
where Cs is the heat capacity of the system. Combining (3) and (4) results in:
D T = -D H np/Cs (5)
From equation (1) it follows that if the value of the entropy term TD S is not negated by the free energy D G, then from equation (5), in an adiabatic system D T will not only be a measure of the total molar heat of reaction, but also the number of moles which have been reacted.
Referring to equation (2); on addition of titrant C to the titrand (A and B), a temperature change will occur as long as molecules of product P are being formed. For every c molecules of titrant C added, p molecules of product P are produced as long as molecules of A are present in the system, and the kinetics of the reaction and the free energy change are such to allow the reaction to proceed immediately to completion. Thus if the change in temperature of the solution is plotted against the volume of titrant added at constant rate, then a change in the slope of the curve will indicate the point at which no more molecules of product are formed, that is, the endpoint of the titration.
This type of titration is known as a thermometric titration.
This is clearly ilustrated in the titration thermogram Figure 1. Here, a titrant of 2M HCl was used to titrate a solution containing 10 mL of 1M NaOH. The change in slope of the direct temperature curve (red) at the endpoint may be observed. The sudden change in reaction rate at the endpoint is shown in the first derivative curve (dark green), and the endpoint is precisely defined by the peak in the second derivative curve.
|
|
 |
| Figure 1. Titration of NaOH with HCl. |
|
Signifigance of equilibrium constant on the endpoint.
The free energy D G of the free reaction may also be related to other reaction parameters:
D G = -RTlnK (6)
where R is the universal gas constant, T is the temperature in Kelvin, and K is the reaction equilibrium constant, which in the context of equation (2) may be defined as:
|
|
 |
where n denotes the appropriate activity coefficient. If the reaction of titrand A with titrant C is not complete near the endpoint, less heat will be evolved or consumed per unit of titrant C added, no matter how high the heat of reaction. This results in a curvature around the equivalence point. The smaller the value of K, the larger is the curvature at the endpoint.
Figure 2 illustrates the effect of dissociation constant on curvature of the thermometric titration temperature curve.
|
|
 |
| Figure 2. Titration of Electroless Copper Plating Solution with HCl. |
|
An aliquot of a bath solution for the electroless copper plating of printed circuit boards was titrated with 2M HCl. The analytes were free hydroxyl ions and ions of a basic copper complex. The hydroxyl ions are fully dissociated, and a sharp first endpoint (as evidenced by the shape and height of the second derivative peak) was obtained. The basic copper complex may be assumed to be weakly dissociated, and a considerable rounding of the temperature curve at the second endpoint is evident. It will also be noted that the second derivative peak relevant to this endpoint is relatively small and somewhat rounded. However, in spite of the rounding, appropriate mathematical treatment permits the true equivalence point to be located, in this case with a precision of 0.2% RSD.
Other parameters affecting the titration curve.
Under ideal conditions, a perfectly linear increase or decrease in temperature could be expected to be observed. However, the titration curve may deviate from linearity for a number of reasons:
1/ Change in heat capacity of the system.
This is due mainly to the increase in the liquid volume resulting from addition of titrant. This may be minimised by utilising a concentrated titrant (about 20-100 times the concentration of the analyte in the sample solution). If weakly dissociated species are being titrated, it may be necessary to reduce the addition rate to avoid overshooting the endpoint. Excessively long titrations may also contribute to unacceptable heat losses from the system. Excessive concentration of titrant may also reduce the titre to such an extent that unacceptable relative errors are observed. In all cases, it should be noted that the molar addition rate of titrant is an important parameter.
2/ The difference in temperature between titrant and titrand.
Although this can be controlled experimentally to control endpoint clarity in some circumstances, this is not usually an option in busy process or quality control situations. For reasons of good analytical practice, the temperature of both the titrant and titrand should be equilibrated with the local ambient temperature.
3/ The heat of dilution.
The dilution of the titrant is usually the largest single contributor to the combined heats of dilution of the titrant and sample. This effect may be observed in Figure 3, a titration thermogram of the determination of sulphate in sea water, using 1.5M barium acetate as titrant. The heat of reaction of Ba2+ with SO42- is quite considerable (D Hf » -26.3 KJ/mol), but the anticipated sharp change in the temperature curve at the endpoint is not observed. However, for titres of only approximately 0.45 mL, an RSD of approximately 0.9% was obtained.
|
|
 |
| Figure3. Titration of Sulphate in Sea Water with 1.5M Ba(OAc)2. |
|
4/ The heat of mixing.
This occurs when the titrant and titrand are in different solvents.
5/ The heat of stirring.
Part of the mechanical action of the stirrer can be expected to be transformed into thermal energy, however, this is rarely sufficient to interfere with endpoint determinations.
6/ The Joule effect of the temperature sensor.
The thermistors used as temperature sensors are semi-conducting resistance elements whose resistance varies as a function of temperature. They often possess a high resistance to the passage of a small but significant current. The energy produced is dissipated as thermal energy. However, in the volume of fluid normally utilised in thermometric titrimetry, the consequent contribution to the temperature of the system is minimal.
Titration of multi-component systems.
Figure 2 illustrates one example of titration of more than one analyte in a sample, namely free hydroxyl ions, and copper in the form of a basic complex. Thermometric titrimetry offers a number of examples of determination of analytes in multi-component systems. The essential criteria are that firstly, that the titration of one component should be virtually complete before that of the "next" component commences. The second criteria in respect of thermometric titrimetry is that the heats of reaction should be sufficiently different to produce a clear inflection between them.
In respect of the first criterion, a "rule of thumb" in potentiometric titrimetry (eg, Kortüm et al. (1961)) is that the ratio of the two equilibrium constants K1/K2 should not be less than 103, ie pK2 - pK1 should not be less than 3. In thermometric titrimetry, pKa differences of 2 or lower have been successfully exploited (Vaughan, 1973).
The discrimination of hydroxyl and carbonate in industrial aluminate solutions using potentiometric titrimetry can cause problems, where the aluminate is complexed with a ligand such as gluconate. It has been reported by Connop (1996) that it is necessary to use the Gran's plot method to satisfactorily discriminate between the two species. There is no such problem with thermometric titrimetry.
Figure 4 illustrates the thermometric titration of hydroxyl and carbonate in an industrial aluminate solution. The heat of reaction for hydrogen ions with hydroxyl ions is approximately -56.2 KJ/mol, with that for hydrogen ions and carbonate approximately -14.8 KJ/mol (pKa = 10.3).
|
|
 |
| Figure 4. The Titration of Hydroxyl and Carbonate in an Industrial Aluminate Solution. |
|
The ability to discriminate between different species is not restricted to acid/base titrations with thermometric titrimetry. For example, calcium and magnesium may be determined in admixture using EDTA as a titrant.
Figure 5 illustrates the sequential determination of calcium in magnesium in sea water using tetrasodium EDTA as titrant.
|
|
 |
| Figure 5. Sequential Titration of Calcium and Magnesium by Thermometric Titration with 1M Tetrasodium EDTA. |
|
The chelation of calcium with EDTA is exothermic, resulting in a heat of reaction of approximately -24 KJ/mol. In contrast, magnesium reacts endothermically with EDTA (D Hr » +21 KJ/mol). Precision (1 standard deviation) was 0.003 mL for the calcium titration and 0.005 mL for the magnesium titration.
REFERENCES:
Vaughan, G. A. (1973). Thermometric and enthalpimetric titrimetry. Van Nostrand Reinhold Company Ltd. (London).
Bark, L. S., and Bark S. M. (1969). Thermometric titrimetry. Pergamon Press (Oxford) International Series of Monographs in Analytical Chemistry Vol 33.
Kortüm, G., et al. (1961). Pure Appl. Chem. 1, 224.
Barthel, J. (1975) Thermometric titrations. Wiley-Interscience (New York).
Connop, W. L. (1996). A new procedure for the determination of alumina, caustic and carbonate in Bayer liquors. Proc. 4th Int. Alumina Qual. Workshop, Darwin, 2-7 June 1996, 321-330
|
|
| | | | | | | | | | | | | |
|
| This website and its contents are the copyright of Multitrator Pty Ltd (2003). All rights reserved. "Multitrator" and "Lab Rat" (words and image) are registered trade marks of Multitrator Pty Ltd. |
|
| | | | | | | | | | | | | |
|
|
|
