9 Calorimetric Measurements: Guidelines and ...

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9 Calorimetric Measurements: Guidelines and Applications

9.1 General Considerations

Selecting the appropriate calorimeter for a given task – out of the large variety of available instruments – is a major problem for the uninitiated. Even those who possess some experience in calorimetry may find it difficult to make the right decision when buying or building a new or different instrument. In any case, the calorimeter is expected to yield the desired information in a state that is as accurate and distortion-free as possible. The decision should not be too difficult in view of the apparent triviality of this requirement. Nevertheless, in this chapter, we shall discuss the considerations involved in greater detail in order to prevent waste of time and money resulting from the purchase or construction of an unsuitable instrument. Experience shows that erroneous decisions in this field stem mostly from the following two causes: i) The experimenter has little experience with the possibilities or limitations of calorimetric procedures and is consequently unable to decide whether calorimetry provides a correct and useful approach to the problem at hand. ii) Calorimetric nomenclature and the description of calorimeters are generally inadequate and ambiguous. From the data published by the manufacturers, it is often difficult to identify the advantages and drawbacks of the respective instrument or its different features with regard to other calorimeters. Of course, the calorimeter manufacturers should not be blamed for the existing confusion of terms concerning calorimetric procedures. Yet one is left with the impression that commercial interests are incompatible with a frank disclosure of the disadvantages of a given measuring principle for the solution of a specific problem in this field. These matters will be considered here. To begin with, the physical principles of calorimetry that were treated in detail in the first part of this book will be Calorimetry: Fundamentals, Instrumentation and Applications, First Edition. Stefan M. Sarge, G€ unther W. H. H€ohne, and Wolfgang Hemminger. Ó 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

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reexamined in terms of a critical evaluation of the respective method. Next, readers will be given clear-cut guidelines for judging the suitability of a specific instrument for the particular measuring problem at hand. This necessitates a precise, unambiguous description of the calorimeter and its properties. A calorimeter can be described in terms of the measuring principle and the mode of operation; in addition, essential aspects of its design may be pertinent in some cases. On the basis of the general remarks contained in the respective chapters of this book, the user will obtain an idea of the positive and negative aspects of the instrument in question, including its possible error sources. We still lack the well-defined terms necessary for judging the suitability of a given calorimeter for a particular task – for example, whether a specific thermal event can be detected and resolved at all. The following proposals might contribute toward a standardization of the terminology. Many concepts originate from general metrology, in which their usefulness has been established. 9.1.1 Sensitivity (DX/Q or DX/DW)

The term “sensitivity” when applied to a measuring instrument is used in a variety of ways, which leads to frequent misunderstandings in its interpretation. Here we shall define this term in order to describe the relationship between cause and effect. The cause is a heat flow rate change DW or an exchanged heat Q in the calorimeter. The effect is the output signal DX, which varies according to the measuring principle used. The sensitivity 1/K (K, calibration factor) of a calorimeter thus equals DX/DW or DX/Q. Accordingly, sensitivity data are rendered in terms such as grams of a given substance per joule or volts per watt. For commercial instruments, it is often not possible to judge the sensitivity, it is hidden in internal electronics and computer software, and the given output quantity DX is already transformed into heat Q or heat flow rate W units. Because of this, the sensitivity is often defined as the lowest possible quantity of heat or heat flow rate detectable with that calorimeter. This quantity is closely connected with the noise of the instrument. 9.1.2 Noise (dQ or dW)

This term refers to the random fluctuations of the output signal. In a calorimeter, it designates the random changes of the output signal of the measuring system without a sample (i.e., empty). Noise limits the resolution of the calorimeter with regard to the measured heat flow rate (or heat) regardless of whether it actually originates from the measured quantity itself (e.g., temperature difference) or from the processing of the primary signal (e.g., in an electronic amplifier). Because the user is mainly interested in the smallest


Φ δΦ t

Heat flow rate Short-term noise Time

Figure 9.1 On the definition of “short-term noise” of the output signal of a heat flow calorimeter without sample.

heat or heat flow rate that can be detected by a given calorimeter, the causes of the noise of the output signal need not be considered. More appropriately, the noise of a given calorimeter is rendered in units of heat or heat flow rate with reference to the measured signal proper (e.g., 2 mJ or 10 mW). A distinction should be made here between the short-term noise (noise proper) and the long-term noise. The former term designates the “rapid” random fluctuations of the output signal (Figure 9.1). The term long-term noise (Figure 9.2) refers to the overall fluctuations of the baseline over a longer time interval (in comparison with the measuring time) or temperature range (in scanning instruments). This concept consists of, for example, the repeatability of the baseline over a number of repeated measurements of the same kind. But the baseline “drift” during the measurement also limits the resolution of the calorimeter and should be assigned to the category of long-term noise. Long-term noise is more troublesome than short-term noise because the latter can be eliminated with relative ease by averaging; the middle of the measured curve is broadened by the short-term noise, which is easily found in

Φ dΦ1 dΦ2 δΦ t Δt ·····

Heat flow rate Baseline drift Long-term noise Long-term noise with subtracted baseline drift Time Time of measurement Averaged baseline

Figure 9.2 On the definition of “long-term noise” and “baseline drift” of the output signal of a calorimeter without sample.

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most cases. Elimination of the long-term noise by averaging is more time consuming because it necessitates a large number of protracted measurements. A small long-term noise (or baseline stability) therefore represents an essential quality criterion of a calorimeter. This is especially important in the study of processes extending over long time intervals or large temperature ranges (as is the case with biological phenomena or slow chemical reactions). 9.1.3 Linearity (Xout ¼ K Xin) and Linearity Error (dK/K)

The concept of linearity refers to the proportionality between two quantities. It can be represented geometrically by a straight line in the Xout Xin coordinate system. In calorimetry, this term applies first and foremost to the proportionality between the quantity of interest: heat Q (or heat flow rate W) of the sample and the measured output signal, namely, Qout ¼ K Qin (or Wout ¼ K Win). Strict linearity cannot be achieved in practice. The linearity error dK/K measures the maximum deviation of the actual functional relationship from the (ideal) straight line; it can be easily found from the graphic presentation of Qout ¼ K Qin or Wout ¼ K Win (the calibration function) (Figure 9.3). However, a good linearity also means that the calibration factor K does not depend on sample parameters (e.g., the size, mass, thermal conductivity of the sample) or on temperature in the case of scanning calorimeters. If the calibration factor depends on further parameters (e.g., sample mass, temperature, heat evolved), it is recommended that K be presented graphically as a function of the respective parameter. A good linearity exists when K is constant (horizontal line). The linearity error dK/K equals the maximum deviation from the horizontal line. A good linearity of the apparatus is a precondition for the existence of an apparatus function (see Section 6.3), which makes it essential for a reconstruction of the true heat flow rate function Win(t) from the measured curve Wout. 9.1.4 Apparatus Function (fapp(t))

The apparatus function, also called Green’s or response function, is obtained by dividing the function values of the measured function (Wout(t)) traced after a pulselike heat production in the sample (see Section 6.3.4) by the area under this function (normalization) (Figure 9.4). The apparatus function measures the degree to which the heat exchange function of the sample has been “smeared” by the calorimeter; it serves also for estimating the time constant of the instrument. Variation of the sample and test parameters usually yields different apparatus functions. But if the differences obtained are small in comparison with the required accuracy of measurement, a single apparatus function exists. The existence of a single apparatus function is a precondition for a desmearing of the measured curve (see Section 6.3) and any kinetic evaluation.


Ideal case :Φ = K·X Φ Heat flow rate X Respective output quantity K Calibration factor

Ideal case: K is independent on parameters K Calibration factor X Respective output quantity

Real case: K depends on parameters, for example, the temperature, that is, Φ = K(T)·X Φ Heat flow rate X Respective output quantity T Temperature

Linearity deviation of K relative to X, that is, K depends on the output signal X K Calibration factor X Respective output quantity δK Linearity error of K relative to the respective quantity X Linearity deviation of K relative to T, that is, K depends on the temperature K Calibration factor T Temperature δK Linearity error of K relative to the temperature T Figure 9.3 On the definition of “linearity” of a calorimeter.

9.1.5 Accuracy and Total Error ({Qmeasured --- Qtrue}/Qtrue)

The concept of accuracy refers to the agreement between a measured (mean) value and the corresponding true value of a quantity. The value indicated as accuracy is 100 minus the amount of the difference between the two magnitudes in relation to

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True event: “heat pulse” of weight Q at t1 inside the calorimeter Φin True heat flow rate t Time

Measured function (“response function”): Φout(t) Measured heat flow rate function t Time Q Heat released at t1: Q =∫Φoutdt

Apparatus function (normalized “response function”): f(t') Apparatus function f(t’) = Φout (t)/Q with t’ = t — t1 Φ(t) Measured heat flow rate at time t ′ Figure 9.4 On the definition of “apparatus function” of a calorimeter.

the true value (given in percent). The term “error” (absolute error and relative error, respectively) includes the sign of the deviation Qmeasured Qtrue. The total error is the sum of the (estimated) systematic error and the random one. Usually, not the above-defined accuracy value is given, but rather the relative error; what is said is “this value is accurate to 5%,” instead of “this value is measured with an accuracy of 95%” (or “with an inaccuracy of 5%”). High accuracy means a small error. Accuracy has much to do with the ability to calibrate and the calibration of the instrument. In an exactly calibrated instrument (i.e., one free from systematic errors), the accuracy depends on the repeatability (see below) alone; otherwise, the accuracy is smaller. In a given instrument, accuracy also depends on the kind of measurement. Thus, rapid phase transitions (in the presence of a low short-term noise and a high long-term noise) can be measured with a high degree of accuracy. Conversely, with the same conditions, measurements of slow processes can be highly inaccurate. Obviously, this term alone cannot characterize an instrument in the absence of further information on the nature of the involved process. Because

9.2 Guidelines to Calorimetric Experiments

accuracy includes the systematic errors, it can merely be estimated based on experience or on the results of round robin experiments. Possibly it can be found out by means of a standard. 9.1.6 Repeatability and Random Uncertainty (DQ/Q)

The term “repeatability” refers to the agreement between the results of a number of measurements of the same quantity performed by the same method, the same observer, and the same instrument in the presence of random fluctuations (i.e., errors). Here the random uncertainty, a relative or absolute plus/minus semirange of an interval around the mean value, is usually indicated. A measure is the standard deviation or a certain multiple of the standard deviation. A high repeatability means a small random uncertainty (i.e., small random errors). In most cases, the repeatability is higher than the accuracy because random errors can occur independent of systematic ones. Similar to the accuracy, the repeatability can vary from one process to another in a given instrument. Further details on the nature of the examined process must therefore be available in order to use the repeatability of an instrument as an efficient characteristic. The repeatability of the measured signal of an empty instrument (baseline) is equivalent to the long-term noise with an eliminated baseline drift (Figure 9.2). Concerning the evaluation of a specific calorimeter, the characteristic “repeatability” is more appropriate than “reproducibility.” Reproducibility refers to the agreement between results of measurements of a specific quantity when the individual measurements are performed by different persons using various methods and different instruments. Reproducibility depends on the various random errors and the systematic ones. For that reason, in general, reproducibility is poorer than repeatability, for example, a value is repeatable to 5% but reproducible to 9%.

Conclusion

To sum up, it is obvious that the noise, linearity, and apparatus function provide the main criteria to quantify the quality of a calorimeter. Regrettably enough, these data are often missing from the manufacturers’ prospectuses. Such terms as sensitivity, accuracy, and repeatability are of little significance in the absence of additional data.


Obtaining reliable experimental results needs some preparation stages: the problem to be solved must be formulated, the proper instrument must be picked

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out, the experimental conditions must be chosen, the measurements must be executed properly, the results must be evaluated, and last but not least an uncertainty estimation must be performed. These ambitious demands measure up to scientific problems. For calorimetric routine measurements (e.g., for quality control), many of the steps can then be skipped. These guidelines may help to avoid failures in calorimetric practice. 9.2.1 Definition of the Problem to be Investigated

The first step toward a decision as to the suitability of a given calorimeter invariably consists of a thorough analysis of the problem at hand. If the process under study is known or expected to involve an exchange of heat, calorimetry promises to be the chosen approach. But calorimetric procedures may also provide valuable information toward an understanding of processes where the enthalpy remains constant (i.e., there is no exchange of heat), but one of the derivatives of enthalpy with regard to temperature (e.g., the first derivative – the heat capacity) undergoes a change during the process. The following questions must be answered in order to select a suitable calorimeter. i) Nature of the Process under Study Is the study aimed at the heat flow proper or at the process that causes it? Is there a phase transition? (first, second, or higher order?) Is the process reversible or irreversible? Is a chemical reaction involved? What reaction products are expected to appear? Is a physical reaction involved? Does a change (anomaly) of specific heat capacity occur? Are metastable states to be expected (activation problems)? Are the processes involved slow or rapid? Does the process take place at a given temperature or within a certain temperature range? Are any kinetic parameters sought? ii) Boundary Conditions Is a constant pressure, volume, or temperature required? In what range? With what uncertainty? Are isothermal or adiabatic operating conditions required? How strictly must they be maintained? Is a scanning operation required? At what heating rates? Do the boundary conditions have to be varied? If so, according to what program? iii) Accuracy Requirements What is the magnitude of the expected thermal effect? What is the time interval or temperature range of the process? What is the measured quantity and how accurately does it have to be measured?


What is to be exactly measured: just the overall heat or the heat flow rate as a function of time (or temperature)? How exact must the heat flow rate time (or heat flow rate temperature) correlation be? Is any desmearing of the measured results required for further evaluations (e.g., for kinetic analysis of rapid reactions)? iv) Nature and Composition of the Sample What is the available or required mass, size (volume), and form (e.g., liquid, powder) of the substance to be tested? Do the substance, the intermediate products, or the reaction products possess any properties that require particular consideration (e.g., chemically aggressive, explosive, inert)? Is the thermal conductivity of the sample low or high? What is the vapor pressure? How can a defined heat transfer between sample and measuring system be ensured? v) Further Requirements Are any simultaneous studies of other effects required (e.g., heat flow rate– weight change or heat flow rate–light emission)? The answers to these questions should be laid out in the form of a list in order to provide a reliable basis for further considerations. 9.2.2 Selection of the Proper Calorimeter 9.2.2.1 Calorimeter Requirements The requirements with regard to a calorimeter can be derived on the basis of the above analysis of the measuring problem. The necessary operating conditions have to be defined first: an isothermal, isoperibol, adiabatic, or a scanning calorimeter? What temperature range? What heating rate? Any other boundary conditions: a constant pressure, constant volume, gas flow rate, and so on? The requirements concerning the maximum admissible noise (both short- and long-term noise) can likewise be derived from the requirements for accuracy. In general, the noise level determines the smallest amount of the thermal effect that can be detected by the calorimeter. If a heat flow rate of 1 mW is to be measured with an accuracy of 99% (i.e., with an error of 1%), the noise (peak to peak) must not exceed 10 mW; otherwise, this requirement cannot be fulfilled. It is assumed in this connection that a suitable accuracy has been ensured by calibration. The requirements concerning the linearity of the calorimeter and the demands for the existence of a single apparatus function and a definite time constant stem from the accuracy requirements with regard to the thermal effect–time (or thermal effect–temperature) relationship. Slow processes that take much longer than a multiple of the time constant of the calorimeter require no desmearing of the measured curve. Rapid processes, on the other hand, have to be desmeared before

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being analyzed. This requires a single apparatus function, which in turn necessitates a good linearity of the calorimeter unless the work can be restricted to a very narrow range of sample masses, thermal effects, and temperatures. The time constant limits the accuracy of the desmearing of the measured curve with regard to time (or temperature). To bring the desmeared measured curve to an accuracy much larger than that corresponding to half of the time constant, one has to make a thorough analysis of the possible errors – a task that usually necessitates a very extensive test program and a theoretical mastery of the behavior of the calorimeter (see Section 6.3.5). Slow processes always require a low level of long-term noise (high baseline stability). But if only rapid processes are to be measured, poor baseline stability is not necessarily a disadvantage (good baseline stability usually represents a rather costly quality feature). The most difficult decision concerns the choice of the measuring principle. This task requires a thorough understanding of the various measuring principles with all their advantages and drawbacks in order to select the approach that best fits the requirements of the particular problem – a difficult undertaking indeed in view of the complexity and intricate interrelationships of the requirements involved. The procedure of choice consists of first eliminating the measuring principles that are obviously unsuitable and concentrating then on the remaining ones, thinking them over again together with additional characteristics of the instrument. All these considerations yield a list of requirements, that is, data on noise, linearity, apparatus function, operating conditions, and possible measuring principles. 9.2.2.2 Selection of the Calorimeter Usually the first choice is between a homemade calorimeter and a commercial instrument. A homemade calorimeter would be justified either for simple tasks requiring only a modest performance in a minimal layout or in order to meet very special requirements for which no commercial calorimeter exists on the market. Every commercial instrument is the result of an unavoidable compromise between diverse requirements for the sake of greater applicability. On the other hand, a do-it-yourself construction faces major difficulties that must not be underestimated. If the problem can be solved by a commercial calorimeter, let there be no hesitation in buying one. The considerable saving of time will undoubtedly cover any cost difference. The next stage consists in the preparation of a list of calorimeters in such a way that the possible measuring principle and the operating conditions serve as selection criteria in terms of the list of requirements. Chapter 7 contains all the different calorimeters treated in this book and many commercially available calorimeters. The construction of a do-it-yourself calorimeter necessitates a study of the vast original literature in this field in order to avoid mistakes and problems that have happened previously. On the other hand, a commercial calorimeter must be accompanied by the documents necessary for the testing of such technical data as


noise, linearity, and apparatus function. Regrettably enough, the manufacturers’ literature is usually insufficient for this purpose because it often lacks a number of essential data. We recommend that trial measurements be carried out at the manufacturer’s laboratory. The results of these enquiries are usually sufficient for a correct choice. 9.2.3 Testing of the Calorimeter

Section 9.2.2 was devoted to an examination of calorimeters in general for the purpose of selecting a suitable instrument. Here we shall deal with the testing of a particular calorimeter for determining the accuracy of measurement and checking the operation of the device. 9.2.3.1 Calibration At least two measuring values have to be the subject of calibration: on the one hand, the extent to which the displayed or recorded temperature corresponds to the actual temperature of the substance being tested (especially in scanning calorimeters); on the other hand, the calibration factor for heat or heat flow rate must be determined or verified. The measured temperature is checked in a variety of ways depending on the particular type of calorimeter, and the same applies to the information on temperature fluctuations. Heat flows are invariably associated with a temperature gradient whose magnitude must be taken into account in order to be able to analyze the accuracy of temperature measurement. If the heat flows are approximately known – by means of model calculations, for example (see Sections 4.1 and 7.9.2.3) – the temperature field can be calculated if the geometry and thermal conductivity of the measuring device are known (see Section 4.1). If this is not the case, the only remaining testing procedure consists of measurements of temperature at different parts of the measuring device. One must bear in mind in this connection that the placing of temperature probes naturally affects the device, influencing the heat flow paths and consequently the temperature field, hence the need for temperature sensors of the smallest possible size (e.g., thermocouples with very thin wires). This is particularly true if the temperature probe is placed at or inside the sample. If the calorimeter is not of the scanning type, a previously calibrated thermometer would permit an easy check of the temperature that is set or displayed by the instrument. In scanning calorimeters, the calibration of the temperature scale involves a somewhat more laborious procedure because a thermometer usually cannot be used for this purpose because there is not enough room for it. Such instruments are calibrated by means of reference substances, that is, highly pure substances that undergo phase transitions at definite temperatures. The test runs should be carried out with different substances, melting at different temperatures, using a wide variety of heating rates because the temperature correction may depend not only on

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the respective temperature but also on the heating rate. The thermal conductivity (and mass) of the sample may also be involved, so the testing should not be restricted to metals. In any case, to be suitable for temperature calibration, a substance must be available in a highly pure form; its transition temperature must be precisely known, and there must be little or no tendency to superheating or supercooling in the course of the phase transition (see Section 3.2.2). The International Union of Pure and Applied Chemistry (IUPAC) has published two papers on the calibration of calorimeters that contain guidelines and recommended reference materials for the calibration (i) of calorimeters designed for use in the microwatt range under essentially isothermal conditions and (ii) for differential scanning calorimeters (DSCs). Wads€ o and Goldberg (2001) recommended test and calibration materials for different thermodynamic processes (e.g., dilution, mixing, sorption). Certified and non-certified materials as recommended by Della Gatta et al. (2006) are given in Table 9.1. The values for the metals are consensus values based on published literature and calculated by weighted averaging, the values for the organic materials are taken from the respective certificate. A number of heat flow rate calibration materials have also been suggested for the calibration of the heat capacity scale of calorimeters. Table 9.2 describes these materials and the applicable temperature ranges. The determination or checking of the calibration factor usually takes place through the release of a definite amount of heat in an electric heater (resistor). Although many instruments have built-in calibration resistors, the test is best made by placing an additional very small resistor at or in the sample in order to release the heat exactly at the spot where it is generated during the measurement. It is not self-evident that the calibration factor determined by means of exothermic effects is the same as for endothermic effects. In many instruments, the calibration factor varies in accordance with the site of emergence of the heat that is to be measured; the built-in resistor yields systematic errors and has only a limited applicability in such calorimeters. Calibrations of this kind can only be used for exothermic effects, of course, and for heat addition at a rate similar to the heat flow occurring in the course of the reaction under investigation. For an endothermic calibration, the choice is restricted to the Peltier effect (see Chapter 7) on the one hand and endothermic phase transitions on the other. The advantage of the latter approach lies in the fact that the heat is consumed at the same spot where it is needed for the measurement (namely, in the sample); moreover, in this case, there is no further intervention (by means of electric leads and so on) in the measuring system, but there is, of course, a difference in thermal conductivity between the calibration substance and the sample. Calibration by endothermic heat effects is the usual procedure in scanning calorimeters. However, one has to take into consideration the inaccuracy of the tabulated values as well as weighing errors, purity data, chemical stability, and so on and the disturbance of steady-state conditions. Hence, in many instruments, the calibration factor depends also on the magnitude of the thermal effect as well as the nature and mass of the substance used (level of filling,

Hg Ga In Sn Bi Cd Pb Zn Sb Al Ag Au Cu Ni Co C13H10O3 C12H10 C10H8 C14H10O2 C8H9NO C7H6O2 C14H12O2

Mercurya) Galliumb) Indiuma,b,c) Tina,b,c) Bismutha,b) Cadmium Leadc) Zincc) Antimony Aluminumc) Silver Gold Copper Nickel Cobalt Phenyl salicylatec) Biphenylc) Naphthalenec) Benzilc) Acetanilidec) Benzoic acidc) Diphenylacetic acidc) 200.59 69.723 114.818 118.710 208.98038 112.411 207.2 65.39 121.760 26.981538 107.8682 196.96655 63.546 58.6934 58.933200 214.21 154.21 128.17 210.23 135.17 122.12 212.25

M/g mol1 234.315 302.9146 429.748 505.078 544.552 0.001 594.219 0.001 600.612 0.001 692.677 903.778 0.001 933.473 1234.93 1337.33 1357.77 1728 1 1768 3 314.94 342.08 353.38 368.00 387.49 395.50 420.34

Tfus/K

Dfush/J g1 80.07 0.13 28.62 0.04 60.38 0.15 53.18 0.12 55.25 0.68 23.08 0.11 108.09 0.43 162.55 4.91 399.87 1.33 104.61 2.09 64.58 1.54 203.44 4.36 290.36 6.41 272.44 6.26 89.5 0.4 120.6 0.8 147.6 0.6 110.6 0.5 161.2 0.6 147.2 0.3 146.8 0.6

Tfus/ C 38.8344 29.7646 156.5985 231.928 271.402 321.069 327.462 419.527 630.628 660.323 961.78 1064.18 1084.62 1455 1495 41.79 68.93 80.23 94.85 114.34 122.35 147.19

a) Available from National Institute of Standards and Technology (United States) as a certified reference material. b) Available from Physikalisch-Technische Bundesanstalt (Germany) as a certified reference material. c) Available from Laboratory of the Government Chemist (United Kingdom) as a certified reference material.

Symbol/formula

Element/substance

5583 9 3287 5 7168 18 11114 25 6211 77 4782 22 7068 28 19792 598 10789 36 11284 225 12720 304 12928 277 17042 376 16056 369 19180 80 18600 100 18920 80 23260 100 21790 80 17980 40 31160 130

DfusHm/J mol1

Table 9.1 Recommended values of temperatures and enthalpies of fusion of metallic and organic certified and non-certified reference materials (Della Gatta et al., 2006).

9.2 Guidelines to Calorimetric Experiments 241

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9 Calorimetric Measurements: Guidelines and Applications Table 9.2

Heat flow rate calibration materials (Della Gatta et al., 2006).

Substance a-Alumina Copper Benzoic acid Polystyrenea) Molybdenuma) a)

Symbol or formula

Temperature range (K)

a-Al2O3 Cu C7H6O2 (C8H8)n Mo

70---2250 14---320 115---350 115---350 273.15---2400

a) Available from National Institute of Standards and Technology (USA) as a certified reference material.

heat transfer, and thermal conductivity); in scanning calorimeters, the heating rate may also be a factor. These parameters should therefore be varied systematically in calibration measurements in order to cover their effects as thoroughly as possible (Sarge et al., 1997). The test measurements described above reveal the repeatability and an estimation of the accuracy of a calorimeter and their possible dependence on such parameters as sample mass, magnitude of the effect, heating rate, temperature, and so on, at least in the case of rapid heat exchange processes. The linearity of the calorimeter (see Figure 9.3) is also tested on the basis of these results. 9.2.3.2 Other Testing Now the noise level of the instrument remains to be determined. For a measurement of the short-term noise, it is sufficient to operate the empty calorimeter at various temperatures (or heating rates) in the most sensitive measuring range and evaluate the fluctuations of the recorded measurement over a period of about 1 min (see Figure 9.1). The long-term noise (see Figure 9.2) is determined by averaging the fluctuations of several measurements (in the most sensitive measuring range) over a period of 10–20 h (in nonscanning calorimeters) or by measuring a number of baselines across the entire temperature range (in scanning calorimeters). For testing the resolution limits (i.e., the smallest effects that are still clearly detectable) inside the sample, a small heat pulse (e.g., by melting a specific quantity of substance) or a constant heat flow rate on the order of twice the magnitude of the long-term noise is generated. Under such conditions, it is checked to see whether the effect can clearly be distinguished from the baseline measured without extra heat production. One must bear in mind that switching on a constant heat flow in the calorimeter under steady-state conditions causes a baseline step change that in general is easy to recognize. However, the testing of the influence of the long-term noise on the resolution comprises a clear identification of a superimposed small but permanent heat flow. For this reason, the heat flow must be switched on before the introduction of the heat source in the calorimeter (in the case of an electric heat source); alternatively, one can use a longlived radioactive sample, which by its very nature generates heat at a constant, unchanging rate.


Finally, the apparatus function (see Section 6.3) should be known because it yields the apparatus-caused smearing of the thermal effect and the time constant. The apparatus function is obtained by generating a heat pulse in the sample and dividing the obtained heat flow rate function by the peak area (normalization) (see Section 6.3.4 and Figure 9.4). Such parameters as the sample mass, the magnitude of the heat pulse, and occasionally the heating rate are to be varied. If all the normalized curves obtained in this manner are identical, there is one single apparatus function, and the measured heat flow rate function of the calorimeter can be desmeared (see Section 6.3). The halfwidth of the apparatus function yields approximately the time constant of the calorimeter. The above calibration and test measurements should be repeated from time to time in order to detect any changes occurring in the calorimeter – as a result of soiling or electronic faults, for example. But for this purpose, one does not have to go through the entire range of parameter variation. Checking a few points of the calibration matrix is usually sufficient. 9.2.4 Performing the Experiment

Every scientific experiment needs some experience to be performed successfully. It is not possible to communicate the necessary knowledge within the scope of such a book, but some general remarks may be helpful to avoid the typical faults of beginners. The right way to attain experimental success is, however, learning by doing. 9.2.4.1 Preparation of the Sample The preparation of the sample to be investigated must, of course, meet the usual requirements of laboratory work: safety and security demands have to be followed; any risk must be avoided. This is especially important in case of materials with unknown properties. For calorimetric investigations, in particular for DSC measurements, there are some more issues to be raised:

The sample should be dry and free of the remains of solvents to avoid errors because of additional evaporation heat occurring during the experiment. The mass of the sample should be known with sufficient accuracy. To get an uncertainty of 1% for the heat exchanged needs a weighing uncertainty well below 1% only obtained with an accurately calibrated balance with proper resolution. It is a good idea to weigh the sample both before and after the experiment to detect possible weight losses. The sample container (crucible) should be compatible with the sample material and possible decomposition products; any reaction should be avoided. Aluminum crucibles are not suitable for aggressive chemicals; melting metals can form alloys with other metals and destroy the container and the calorimeter system. The crucibles must be thoroughly cleaned, also when disposable crucibles are used.

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Liquid samples must be measured in hermetically closed containers, preferably completely filled to avoid any evaporation of the liquid during the experiment. However, if the experiment requires heating of the sample, enough space must be provided to accommodate the thermally expanding sample. The size and form of solid samples should be optimized to get the best possible heat transfer conditions: a pressed tablet is better than a powder, and a flat foil is better for undisturbed heat transfer than an irregular grain with poor thermal contact to the crucible.

9.2.4.2 Calorimetric Measurement Before starting an experiment, the calorimeter must be carefully calibrated (see Section 9.2.3.1). The calibration should be verified from time to time (depending on the stability of the instrument). In case of higher accuracy demands, such verification is to be recommended before and after every experiment to be on the safe side regarding the reliability of the calorimetric results. Depending on the sensitivity of the calorimeter to temperature or voltage and other environment fluctuations, such influences should be eliminated. A temperature-controlled laboratory with a closed door and windows together with a mains voltage stabilizer or an uninterruptable power supply system is helpful in obtaining stable measuring conditions. After the insertion of the sample into the calorimeter, enough time must be given to the instrument to come to a stable state and thermal equilibrium before the measurement can be started. Proper measurement parameters must be chosen: in the case of scanning calorimetry, the initial temperature and the scanning rate must be chosen so as to give the calorimeter enough time to come to steady-state conditions before the event to be investigated starts. The main quantities of interest, namely, temperature, time, and heat flow rate, must be measured and stored for later evaluation. The data sampling rate must be chosen properly not to get too little or too much data for the event in question. Of course, the analog-to-digital converter must have the proper resolution and precision to fulfill the uncertainty demands of the measurement. It is always a good idea to repeat the measurement with the same sample under the same conditions (second run) to see possible differences and distinguish between reversible and irreversible reactions and events. Of course, every measurement cycle should be repeated several times to get a reliable basis for uncertainty estimation. For DSC investigations of time-dependent (kinetic) processes, the measurement should be repeated at different heating rates. This advice must be followed if the kinetics of a reaction have to be researched. If the heat flow rate function measured without any sample (empty crucible measurement) is nonzero for the calorimeter in question, the measurement cycle must be repeated with the same crucible but without the sample inside. If that is not possible, a new crucible of the same size and the same mass should be taken. This additional measurement is necessary in order to be able to correct for a


nonzero baseline in particular for evaluations when the exact heat flow rate of the sample itself is needed, for example, for the determination of heat capacity and thermodynamic state functions. 9.2.4.3 Evaluation of the Measurement The primary measurement result is usually a heat flow rate as a function of time. In the case of scanning calorimeters, this is usually displayed as a function of temperature. For higher accuracy demands, the empty crucible measurement should be subtracted, to eliminate apparatus influences from the raw measurement values before further evaluations are carried out. All of the commercially available calorimeters are connected to a computer that manages the measurement and provides software for standard and more advanced evaluations of the original measurements. Unfortunately, the respective evaluation method and the details often are not well documented by the manufacturer, and the customer has to trust in the results calculated by a “black box.” This is particularly true of special evaluations such as kinetic analysis, desmearing, and heat capacity determination. To avoid problems, the user is advised to understand the respective evaluation method (see Chapter 6) and the possible sources of error involved. It is always a good idea to test the evaluation software with a measurement on such a substance or reaction where the evaluation results are well known. Of course, the manufacturer should be asked about the details of the evaluation software if the manual does not contain enough information to understand what the computer is doing. If proper software is not on hand, the evaluation has to be done by the experimenter himself in the classical way with a pencil and paper. An alternative method is to use mathematics software for this purpose. Different evaluation methods exist for different purposes. Some are explained in Section 6.4, which may be helpful as a guideline for the problem in question. 9.2.5 Interpretation of the Results

The interpretation of measurements to solve a scientific problem is not an easy task and needs a lot of experience, which newcomers do not have. There is no other way than learning by doing. However, it is strongly recommended to study the literature to benefit from the experience of other scientists and to avoid at least one or the other mishap. Some general advice may, however, be helpful. A crucial point is to distinguish between real effects coming from the sample itself and artifacts produced by the apparatus or by the environment (temperature and line voltage fluctuations, electronic and computer problems). Real sample effects such as transitions and reactions are, as a rule, repeatable, whereas artifacts caused by environmental influences occur almost accidentally. If an event occurs again at the same temperature (or under the same measurement conditions), it is very likely that it is a real caloric event from the sample. If a very small caloric event has to be distinguished from arbitrary fluctuations (noise) of the calorimeter signal, it may

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be helpful to decrease the noise by averaging several measurements; this will improve the signal-to-noise ratio. It should be mentioned that changes of the heat transfer condition between the sample and the calorimeter (e.g., by vibrations or bumps) produce peaks in the output signal. The same is true if the sample moves inside the crucible. The latter happens often if a sample from a deformed material (containing internal strain) becomes soft on heating. Because this happens at a certain temperature, this movement artifact is not easy to distinguish from real events happening inside the sample. The right interpretation of the measured effect needs an additional measurement by which any movement of the sample can be excluded. 9.2.6 Uncertainty Estimation

Generally, there are two methods for the estimation of the measurement uncertainty: i) On the basis of a mathematical model of the measuring process, the propagation of the uncertainties of the input factors to the output quantity is calculated. This requires the identification and quantification of the main influencing factors and their incorporation into the mathematical model. ii) By use of reference materials with similar properties as the sample material, the influence of the uncertainty of the different unknown influencing factors on the uncertainty of the output quantity is implicitly and experimentally quantified. This requires repeated measurements under variation of the known influencing factors. In calorimetry, often a combination of the two approaches is preferable. The calorimeter is calibrated by means of reference materials with known physical properties. In the course of the calibration, the influencing factors are varied. As a result, the scatter of the calibration values reflects the uncertainty of the calibration, provided the uncertainty of the respective physical value is small (i.e., it is at least a factor of 3 smaller). This scatter is usually quantitatively expressed by the standard deviation of the mean value. Combining this information with the scatter of repeated measurements on the sample by use of Gauss’s law of error propagation renders the uncertainty of the particular experiment. If no repeated measurements can be made, the information about the probable scatter of the measurement can be inferred from historical data or the calibration measurements.

9.3 Calorimetric Applications

The above criteria for the handling of measurement problems in calorimetry will be illustrated in some detailed examples from different fields of science.


The selection has been made from a didactic point of view rather than from the scientific significance of the research in question. The aim is to show the possibilities and power of calorimetry in quite different fields of research. 9.3.1 Example from Material Science

Polymers are playing a more and more dominant part in material science. Polymeric materials have the advantage of being much lighter than metallic materials. On the other hand, the properties of polymer materials change heavily with changing temperature. For this reason, the temperature dependence of polymer properties is of great interest in material science; the following example may illustrate this. 9.3.1.1 Definition of the Problem to be Investigated Many polymers are amorphous; some are partially crystalline. For the latter, the degree of crystallinity and the size of the crystallites play an essential role with respect to the material properties. To determine the degree of crystallinity, one has to measure the specific heat of fusion of a sample and compare that value with the respective value of the 100% crystallized material, a value that for many polymers can be retrieved from literature. The degree of crystallinity attained with this method is, however, an average value valid for the temperature region around the melting point. To get the temperature dependence of the degree of crystallinity wc more accurately, one has to measure the enthalpy of the sample as a function of temperature and to compare that with the amorphous and crystalline enthalpy of that polymer in the same temperature region: w c ðTÞ ¼

ham ðTÞ hS ðTÞ ham ðTÞ hcryst ðTÞ

ð9:1Þ

where ham and hcryst are the specific enthalpy of the amorphous and crystalline phase, respectively. The specific enthalpy of the sample hS (relative to a reference state at Tini) can be calculated from the specific heat capacity cp(T) (see Section 3.1.6): ðT hS ðTÞ hS ðT ini Þ ¼

cp ðTÞdT

ð9:2Þ

T ini

To get the desired quantities, one has to measure the heat of fusion of the sample as well as the specific heat capacity of the sample in the respective temperature region. 9.3.1.2 Selection of the Calorimeter To measure the heat of fusion and the specific heat capacity of polymers as a function of temperature, the DSC is the appropriate choice because it enables the user to get reliable results in a rather short time. For the heat capacity

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measurements, there is, however, a need for a DSC that is able to measure this quantity with an uncertainty well below 5%. For the aforementioned problem, a power-compensated DSC from PerkinElmer was used. Because of the abovementioned demands on the precision of the measurements, the DSC was carefully calibrated in accordance with the well-known rules (see Section 9.2.3) (H€ ohne, Hemminger, and Flammersheim, 2003). 9.3.1.3 Calorimetric Experiments First, the heat (peak area) and the heat capacity (heat flow rate) calibration of the DSC were verified using the same heating rate as in the planned experiments. Indium was used as the reference substance for heat calibration and sapphire for heat capacity calibration. To determine the degree of crystallinity from the fusion peak area, a polyethylene (PE) sample (9 mg) was measured in scanning mode at a heating rate of 10 K min1. To determine the degree of crystallinity from the enthalpy function via the specific heat capacity, the following measurement cycle was programmed:

a) 2 min isothermal at 50 C b) Heating with 10 K min1 to 280 C c) 2 min isothermal at 280 C With this method, two measurements were performed, the first run with a polyethylene terephthalate (PET) sample of about 8 mg of mass and the second run with empty crucibles having the same mass as those used with the sample. 9.3.1.4 Evaluation of the Measurements From the result of the first method with the PE sample (Figure 9.5), the peak area was determined using a baseline linearly extrapolated from the melt. The resulting specific heat of fusion measured at constant pressure (167 J g1) was compared with the literature value of the fusion enthalpy of PE DfusH ¼ 4.11 kJ mol1 (¼ 293 J g1) from the ATHAS Databank (2011). From these values, the degree of crystallinity of that sample in the melting region can be calculated: wc(PE) ¼ 167 J g1/293 J g1 ¼ 0.57. Initially, this value describes the average degree of crystallinity in the melting region, but does not say anything about the crystallinity of the original material at room temperature, which may have changed during heating of the sample. To illustrate this, the second experiment was performed using a PET sample, which shows a larger effect than one would get with the PE sample. For this purpose, the heat capacity of the sample has to be determined precisely. The results of the two PET measurement cycles are plotted in Figure 9.6. Obviously, the sample run and the empty pan run do not coincide, and the baseline of the empty calorimeter is not a horizontal straight line. To correct this, the usual evaluation procedure was performed: vertical shifting of both curves until the ends of the initial isotherms are zero and linearly tilting


Figure 9.5 Measured heat flow rate function of a PE sample; the given heat is the melting peak area (power-compensated DSC; heating rate: 10 K min1).

(sloping) of both curves until the ends of the final isotherms are zero. The result is plotted in Figure 9.7. In the next step, the empty pan function is subtracted from the sample function. The result is the corrected heat flow rate into the sample. This function is divided by the sample mass and the heating rate (10 K min1) to calculate the (apparent) specific heat capacity function. The result is plotted in Figure 9.8 as a function of temperature.

Figure 9.6 Measured heat flow rate function for a PET sample (solid) and an empty crucible (dotted) with a power-compensated DSC (heating rate: 10 K min1).

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Figure 9.7 The measured functions from Figure 9.6 shifted and tilted for zero heat flow rate in isothermal equilibrium (power-compensated DSC; heating rate: 10 K min1).

According to Eq. (9.2), the specific enthalpy difference of the sample relative to the value at 50 C was calculated and plotted in Figure 9.9. The specific transition enthalpy of this sample is determined from the extrapolated enthalpy functions of the liquid and solid states. Because the enthalpy is a function of state (see Chapter 3), the enthalpy function of the measured PET sample must coincide in the liquid state (above 250 C) with the enthalpy of amorphous PET from literature (ATHAS Databank, 2011). In Figure 9.10, the specific enthalpy functions of the amorphous and the crystalline

Figure 9.8 Specific heat capacity of a PET sample as a function of temperature calculated from the data plotted in Figure 9.7.


Figure 9.9 Specific enthalpy difference of a PET sample calculated from the measured specific heat capacity in Figure 9.8. The heat of fusion is the difference between the extrapolated enthalpy functions of the liquid and solid state (dotted lines) at 250 C.

phase are given together with the measured enthalpy of the sample. The latter was shifted vertically to fit the amorphous values in the molten state. With Eq. (9.1), the degree of crystallinity as a function of temperature has been calculated; the result is plotted in Figure 9.11. 9.3.1.5 Interpretation of the Results Obviously, the degree of crystallinity of the PET sample changes between 100 and 160 C from about 18 to 35% on heating. Such behavior is well known for PET

Figure 9.10 Specific enthalpy functions of amorphous (dotted), crystalline (dashed), and measured (solid) PET. (The enthalpy of crystalline PET at 30 C was used as a reference state.)

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Figure 9.11 Temperature dependence of the degree of crystallinity of a PET sample calculated from the values in Figure 9.10. The error bars indicate the range of values in which the results lie with a probability of 95%.

and other polymers. The material crystallizes further on reheating after being stored below the glass transition temperature (about 75 C). Obviously, the crystallization is not completed on cooling from the melt; there was not enough time for the nucleation and crystallization process to complete. The metastable state is frozen at temperatures below the glass temperature, but some structures seem to be formed during vitrification that later serve as nuclei for further crystallization on reheating above the glass temperature. This is the reason why a simple determination of wc from the heat of fusion (see Figure 9.9) would result in a value that is too large, namely, 55.5 J g1/140 J g1 ¼ 0.40, whereas the initial degree of crystallinity of that sample is below 0.2 (see Figure 9.11). For polyethylene, the situation is different in such a way that the glass transition temperature is well below room temperature and the crystallization has enough time to complete on storing that material at ambient temperature. As a result, the degree of crystallinity does not change too much on heating, and the crystallinity value calculated from the heat of fusion (see Figure 9.5) can be used as a sufficient approximation for the value at room temperature. 9.3.1.6 Uncertainty Estimation The uncertainty of the crystallinity of polyethylene (see Figure 9.5) is calculated according to the Guide to the Expression of Uncertainty in Measurement (JCGM 100, 2008) (for details, see Section 6.5). The mathematical model of the measurement is wc ¼

K clb Q fus =m Dfus h

where Kclb is the calibration factor as determined with indium, Qfus is the measured heat of fusion (peak area) of the partly amorphous sample, m is the sample


mass, and Dfush is the specific enthalpy of fusion of the totally crystalline material. Because the indium measurement confirmed the existing calibration, the value of the calibration factor is unity, but its uncertainty is given by the repeatability of the instrument, which amounts to 1%. The uncertainty of the heat of fusion measurements is dominated by the respective baseline construction and estimated by repeated evaluation under variation of the peak start and peak end temperatures. It was found that the range of estimated values is (167 8) J g1 with a rectangular distribution (i.e., all values inside this interval have the same probability. To convert this rectangular distribution into a normal distribution, the interval is divided by the square root of 12). The uncertainty of the mass of the sample is 20 mg, and the uncertainty of the enthalpy of fusion of the totally crystalline material is estimated from the published literature to be 1%. The uncertainty of the crystallinity u(wc) is calculated by Gauss’ law of error propagation: uðw c Þ ¼ wc

ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 ðK clb Þ u2 Q fus u2 ðmÞ u2 ðDfus hÞ þ þ þ m2 K 2clb Q 2fus Dfus h2

and amounts to 0.035 or 6%. The result of the first experiment is, thus, given as “wc ¼ (0.57 0.04), where the value is located with approximately 95% probability (coverage factor k ¼ 2)”. The evaluation of the uncertainty of the crystallinity of PET (Figure 9.11) starts from the heat capacity measurements on the partially crystalline actual sample. The enthalpy of the sample is calculated by summation of the measured heat capacity over temperature: hS ðTÞ hð50 CÞ ¼

280 ð

cp ðTÞdT ffi 50

X

cp;n ðT n Þ DT

n

Thus, the uncertainty of the enthalpy is influenced by the uncertainty of the heat capacity measurement, which is estimated to be 5%, and the uncertainty of the temperature difference DT, which is assumed to be 1% (halfwidth with rectangular distribution). The given uncertainty for the measured heat capacity already includes any contribution from the mass determination. All uncertainties are expanded uncertainties with a coverage factor of 2. A correlation between the heat capacity at different temperatures is introduced to account for the continuous measurement process, which decreases depending on the magnitude of the temperature difference from 1 to 0 at a rate of 0.05 K1 and is zero for all temperature differences spanning the melting temperatures at 250 C. The reason is that the heat transfer conditions between the sample and the crucible change drastically when the sample melts; as a consequence, there is no correlation between the heat capacity value in the melt and in the liquid state. In the melt, the correlation between temperatures decreases again from 1 to 0 at a rate of 0.05 K1.

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The expanded uncertainty of the relative enthalpy is, therefore, lower than that of the heat capacity measurement and was calculated to be 5.1% at 50 C, 4.9% at 100 C, 4.7% at 150 C, 4.4% at 200 C, 3.9% at 250 C, and 3.6% at 280 C. The calculation was done numerically with the aid of a suitable computer program. As an example, the uncertainty of the enthalpy at 60 C, h(60), is calculated here analytically: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX n1 X m X u n @hð60Þ 2 @hð60Þ @hð60Þ uðx i ; xj Þ uðhð60ÞÞ ¼ t uðx i Þ2 þ 2 @x i @x i @x j i¼1 i¼1 j¼iþ1 with n ¼ 3, namely, cp(50) ¼ 1.00 J g1, cp(60) ¼ 1.05 J g1, and DT ¼ 10 K, u(cp) ¼ 2.5%, u(DT) ¼ 0.3% (rectangular distribution converted to normal distribution, xi and xj stand for the different heat capacity values). Here, the expanded uncertainties have been converted to standard uncertainties by division by 2. The common uncertainty contributions or covariances u(xi, xj) are expressed by their correlation coefficients r(xi, xj): uðxi ; x j Þ ¼ rðx i ; x j Þ uðx i Þ uðx j Þ ( 2 2 @hð60Þ 2 @hð60Þ 2 @hð60Þ 2 uðhð60ÞÞ ¼ u cp ð50Þ þ u cp ð60Þ þ uðDT Þ2 @cp ð50Þ @c p ð60Þ @DT þ2 þ þ

@hð60Þ @hð60Þ r cp ð50Þ; c p ð60Þ u cp ð50Þ u c p ð60Þ @c p ð50Þ @cp ð60Þ

@hð60Þ @hð60Þ r c p ð50Þ; DT u c p ð50Þ uðDT Þ @c p ð50Þ @DT

1=2 @hð60Þ @hð60Þ r c p ð60Þ; DT u c p ð60Þ uðDT Þ @c p ð60Þ @DT

ð9:3Þ The correlation coefficients between the heat capacities and the temperature differences are zero; between the adjacent heat capacity values, they are 0.95. It follows that 2 2 2 uðhð60ÞÞ ¼ fDT 2 u cp ð50Þ þ DT 2 u cp ð60Þ þ cp ð50Þ þ cp ð60Þ uðDT Þ2 þ 2½DT 2 r cp ð50Þ; cp ð60Þ u cp ð50Þ u cp ð60Þ g1=2 ¼ f102 0:0252 þ 102 0:026252 þ ð1:00 þ 1:05Þ2 0:052 þ 2½102 0:95 0:025 0:02625g1=2 ¼ 0:516 i 2:5%


The crystallinity of the sample is calculated according to Eq. (9.1): w c ðTÞ ¼

ham ðTÞ hS ðTÞ ham ðTÞ hcryst ðTÞ

The data for the enthalpy of the totally amorphous phase and the totally crystalline phase are taken from literature. Although the published data do not contain any information on uncertainty, from the underlying literature and knowledge of the competence of the institutes where the measurements were performed, an expanded uncertainty of 2% can be assumed. During the evaluation, the calculated enthalpy of the test sample was shifted to yield the same value as that of the amorphous phase at 280 C.

exp exp exp hS ðT Þ ¼ hS ðT Þ þ ham ð280Þ hS ð280Þ ¼ hS ðT Þ þ Dhshift exp

where hS ðT Þ is the experimentally determined enthalpy of the test sample and Dhshift is the enthalpy shift. For physical reasons, the enthalpy of the sample must equal the enthalpy of the amorphous phase in the molten state above 280 C. In terms of uncertainty, this means that the uncertainty of hS(280) equals that of ham

(280). Theyexp are 100% correlated. The uncertainty of the difference u ham ð280Þ hS ð280Þ is, thus, zero. But at other temperatures, the correlation of the two quantities decreases, that is, the uncertainty of the difference increases. Because of the lack of other information, it is assumed that at the lowest measurement temperature, at 50 C, the uncertainty of the difference is given by the uncertainties of the underlying enthalpy data: ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

exp u Dhshift ð50Þ ¼ u2 hS ð50Þ þ u2 ½ham ð50Þ ¼ 0:0252 þ 0:012 ¼ 0:027i 2:7%

This is expressed by a linearly decreasing correlation coefficient changing from 1 at 280 C to 0 at 50 C. An additional correlation between the enthalpy of the amorphous phase and the enthalpy of the crystalline phase of 50% was assumed because both data were derived from the same set of experiments and applying consistent thermodynamic calculus. In addition, the uncertainty in temperature measurement has to be taken into account. From experience, one can say that the calibration uncertainty of DSCs amounts to approximately 1 K. A further 0.5 K uncertainty contribution is added here to account for thermal lag of the polymer sample and possible poor thermal contact. This applies to both the measurement on the test sample and the literature values. This uncertainty in temperature is converted to an uncertainty in enthalpy by using an average enthalpy/temperature coefficient of 1.6 J g1 K1 for all three contributing phases. At 280 C, where the enthalpies of the two phases are equal, only the uncertainties of the literature data have to be taken into account; at lower

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temperatures, the uncertainty of the test measurement contributes with an increasing level. Again, this is expressed by a decreasing correlation coefficient between the temperature measurements of 1 at 280 C and 0 at 50 C. Any uncertainty contribution from the determination of the sample mass can be neglected because this is taken care of when shifting the measured enthalpy curve to the literature value at 280 C. The result of these calculations has been included in Figure 9.11 as the error bars. 9.3.2 Examples from Biology

Every living creature is subject to metabolism, a biochemical process that in every case is connected with a certain heat flow rate. Calorimetry is therefore a convenient method to investigate processes in animals as well as plants and is becoming more and more important in life sciences, which may be illustrated by the following example (Schmolz and Lamprecht, 2000).

9.3.2.1 Definition of the Problem to be Investigated Holometabolous insects (e.g., flies, bees, butterflies, and beetles) undergo two additional developmental stages between egg and adult, namely, larva and pupa. The latter is characterized by a complete metamorphosis from the larva to the adult insect without any uptake of energy in the form of food. During these developmental stages, the insect’s metabolism has to be very specific. Every type of metabolism is connected with (bio)chemical reactions and because of this with an uptake or release of heat that may be measured in a calorimeter. In this way, it is possible to follow the degree of metabolism in the different stages of development at least qualitatively. For quantitative investigations, the respective metabolism and its energy uptake or release must be known, which is seldom the case because of the very complex biochemical processes behind them. However, calorimetry is the simplest and most convenient method to investigate the metabolism of living creatures. 9.3.2.2 Selection of the Proper Calorimeter To measure the heat production of living creatures, the appropriate calorimeter must be chosen. The vessel must be large enough to include the respective creature and allow convenient surroundings to enable a stress-free development and life. It should be possible to ventilate the container with a simultaneous quantitative determination of the gas exchange. The calorimeter has to be sensitive enough to measure the heat production of, say, one single bee pupa in the temperature range between 10 and 35 C in an isoperibol mode of operation. The Calvet calorimeter (see Section 7.9.2) with 100 ml vessels from Setaram (France) proved to be suitable for this purpose. The vessels were modified properly to make a comfortable life of the respective animals possible. The calibration of the calorimeter was performed with an electrical heater inside the vessel.


Figure 9.12 Eight days’ heat production rate W of one honeybee worker pupa (according to Schmolz and Lamprecht, 2000).

9.3.2.3 Calorimetric Experiments To investigate the heat production during this metamorphosis, the pupa of a honeybee worker (Apis mellifera) – in a wax cell – was put into the calorimeter vessel. The raw data of the heat flow rate produced by the pupa during 8 days are plotted in Figure 9.12. Similar measurements were done starting with the young larva and proceeding until the adult bee hatched from the capped wax cell. The mean of several such measurements is shown in Figure 9.13. The given error bars characterize the heat production range from different individuals rather than the uncertainty of a single measurement, which is only a minor part of it.

Figure 9.13 Averaged specific heat production rates W during the metamorphosis of single honeybees in a wax cell from larva (L), prepupal stage (V), and molt (M1) (left part) via pupa P (middle part), a second molt

(M2), the resting adult (RA), and hatching (H) to an adult worker bee (A) (right part). The error bars indicate the heat production range of different individuals (according to Schmolz and Lamprecht, 2000).

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9.3.2.4 Evaluation of the Results Three very different life forms of holometabolous insects suggest similarly different metabolic rates during their development from egg to adult. The heat production rate changes accordingly (Figure 9.13): During the first days after hatching from the egg, the larva is fed with a nourishing fluid by nursing worker bees and grows rapidly, resulting in a large heat production rate coupled to the transformation of food energy into tissue. After days 4 and 5 of the larval stage, the growth rate surmounts the heat production. The weight gain is 540-fold from 0.32 to 173 mg, and part of the energy uptake is stored in the body of the larva. The specific heat production rate decreases and becomes minimum when the wax cell is closed and the feeding stops. During metamorphosis in the pupa stage, there is no energy uptake, and the metabolism is reduced when organs and tissues are deconstructed and partly digested. The metabolism and the heat production rate increase again when the imaginal cells start to divide and build the adult body (Schmolz and Lamprecht, 2000). 9.3.2.5 Calorimetry on Hornets In another experiment, a hibernating hornet queen was placed in a calorimeter vessel that allowed continuously controlled exchange of the atmosphere at a constant temperature of 15 C. The carbon dioxide content of the outflow was measured together with the heat flow rate. The result is given in Figure 9.14; the CO2 release of the hornet queen is obviously not continuous but clearly periodical. The same is true of the measured heat flow rate – the sum of metabolic heat gain and evaporative heat loss – that decreases significantly during the CO2 releasing phases. The discontinuous heat production rate of the hibernating hornet queen (Figure 9.14) is obviously connected with discontinuous ventilation. During the time the tracheae are closed (internal CO2 production), the heat production rate is maximum. The slowly increasing opening of the tracheae, visible as increasing CO2 release, is coupled with a decreasing heat production rate, which is caused by

Figure 9.14 Periodic heat production rate W (solid) and CO2 release rate V_ CO2 (dotted) of a hibernating hornet queen (according to Schmolz and Lamprecht, 2000).


some water evaporation (an endothermic process): an effect that is strongly increased when the tracheae finally open totally every 50 min at 15 C (Schmolz and Lamprecht, 2000). 9.3.2.6 Uncertainty Estimation The uncertainty of the given heat production rates is almost totally determined by the fluctuations of the heat production rate measured with different animals and may be characterized by the standard deviation of the mean. The uncertainty of the calorimeter itself is much lower and can, as a rule, be neglected.

To sum up, calorimetry proved to be a convenient and successful method to follow the metabolism of insects during their metamorphosis. 9.3.3 Example from Medicine

In medicine, calorimetry has not been used so far other than in some scientific research projects. Recent investigations have, however, shown that there is a chance to use sensitive calorimeters for diagnostic purposes in clinical practice. Sepsis, often caused by blood poisoning (septicemia), is a very dangerous medical condition that needs fast and purposeful treatment. One condition for that is gaining knowledge about the responsible bacterium as quickly as possible. The standard method to identify it is the bacterial culture, which has a high sensitivity and reliability but often needs too much time. Because of this, people are looking for a faster alternative to identify bacteria in blood. Recently, methods have been reported that allow identifying bacteria by following the growth of cultures in a sensitive calorimeter. As an example, Trampuz et al. (2007) reported about the identification of bacteria in infected platelets (PLTs), one of the transfusion products used in medicine. 9.3.3.1 Definition of the Problem to be Investigated Single microbes produce a very small metabolic heat of 1–3 pW per cell, which cannot be detected even with the most sensitive calorimeter. But the exponential replication of bacteria in culture allows their detection in sensitive so-called microcalorimeters. To decide whether a product is infected or not, it is sufficient to put it in a calorimeter vessel under ideal growth conditions at 37 C. If the material is aseptic, no signal will develop, but in the presence of germs, an exothermic heat flow rate will be measured. To identify the different relevant bacteria, it is necessary to follow the growth behavior of the culture and see whether they can be distinguished from one another. Furthermore, the detection limit (the minimum bacteria concentration) has to be determined, and the significance of the measurement for different contaminations has to be proved. 9.3.3.2 Selection of the Proper Calorimeter For the problem characterized above, an isoperibol calorimeter is needed. The sensitivity should be as large as possible to detect even low bacterial

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contaminations. In this study (Trampuz et al., 2007), the Thermal Activity Monitor TAM III (from TA Instruments) equipped with 48 channels (calorimetric systems) was chosen. This calorimeter has a detection limit of 0.225 mW, and the large number of calorimeter systems allows the measurement of many samples in parallel, a great advantage for the desired diagnostic purpose. 9.3.3.3 Calorimetric Experiment Outdated but otherwise clean and sterile platelets were used for the experiments. Suspensions of different bacteria were prepared in sterile saline (0.85% NaCl) and diluted to get different initial concentrations from 1 to 105 colony-forming units (CFU) per milliliter. Samples of 1 ml of PLTs with 0.1 ml inoculation culture at four bacterial concentrations were prepared. Several samples with different bacteriainfected PLTs were transferred into the calorimeter vessels. The measurements were performed at (37 0.02) C for several days. 9.3.3.4 Evaluation of the Measurements The TAM calorimeter supplies heat production rates as a function of time for every calorimetric system simultaneously. The measured heat flow rate versus time functions needs no further evaluation for the above-mentioned diagnostic purpose. As an example, Figures 9.15 and 9.16 combine the results for two of the six measured bacteria at four different bacterial contamination rates each. 9.3.3.5 Interpretation of the Results The growth rates of different bacteria obviously follow quite different time laws as far as both the shape and the timescale are concerned. The scheme of the growth development seems, however, to be the same for each bacterium independent of the degree of contamination. This allows identifying different bacteria from the shape of the heat production rate independent of the respective concentration of

Figure 9.15 Heat production rates W of platelets infected with Escherichia coli bacteria at concentrations of 105 (A), 103 (B), 10 (C), and 1 (D) CFU ml1 (according to Trampuz et al., 2007).


Figure 9.16 Heat production rates W of platelets infected with Candida albicans bacteria at concentrations of 105 (A), 103 (B), 10 (C), and 1 (D) CFU ml1 (according to Trampuz et al., 2007).

germs. The lower the contamination, the later the heat production develops, but already 10 CFUs ml1 can easily be detected, although the heat production rates and timescales differ very much. For Escherichia coli (Figure 9.15), contamination identification is possible after 12 h; Candida albicans (Figure 9.16) grow much slower. In any case, bacterial contamination of the platelets can be detected after 15 h at the latest, but a clear identification of the respective bacterium needs more time, depending on the degree of contamination and the type of bacteria, between 12 and 70 h. 9.3.3.6 Uncertainty Estimation In every case, the calorimeter sensitivity and precision are much higher than the measured heat flow rate function. This can be seen in the initial and final parts of the measurements, which coincide very well and may illustrate the uncertainty of a single measurement. The repeatability of the shape of the heat production function for different degrees of contamination is sufficient to make an identification of certain bacteria possible. The shape of the calorimetric curve may therefore be used as a fingerprint for the determination of bacteria in transfusion products. The measurements have shown that it is possible to identify certain bacteria in platelets. Calorimetry could in such cases be used as a diagnostic tool and is able to give faster results than the conventional bacteria culture method. 9.3.4 Example from Chemistry

Every chemical reaction is connected with an exothermic or endothermic heat exchange. It is therefore easy to follow chemical reactions in a calorimeter. Knowledge about the mechanism and kinetics is an essential point in chemical

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research, on the one hand, to optimize the reaction and, on the other hand, to know the thermal behavior to avoid hazardous runaway reactions. Numerous papers tackle the problem to determine the kinetics of a chemical reaction from calorimetric measurements. Many kinetic models exist for that purpose, but a meaningful application needs a great deal of experience to come to reliable results (see Section 6.4.2). Here we restrict ourselves to a not too complex example (Strey, H€ohne, and Anderson, 1998) to demonstrate the method. 9.3.4.1 Definition of the Problem to be Investigated Epoxies play an important part in engineering and technology as a composite material and as an adhesive. The prepolymer is a liquid that is mixed with a suitable cure agent whereupon the polymerization starts, for example:

Perfect curing of epoxy resins is essential for the mechanical properties of the polymer composite material. Curing is – as every chemical reaction – a temperature- and time-dependent process: the higher the temperature, the shorter the curing time. To ensure a certain degree of curing and the respective material properties, the kinetics of curing must be well known. There are two ways to determine the kinetics of a reaction in a calorimeter: either by measuring the heat flow rate at constant temperature or in scanning mode of operation. The former method is easier to evaluate because of simpler kinetics, but has the huge disadvantage that the reaction starts immediately after the transfer of the sample at ambient temperature into the calorimeter, which is at an elevated temperature, although the sample (and the calorimeter system) did not have time enough to equilibrate. With the scanning method, the sample is mounted at room temperature, and the system has enough time to equilibrate before the reaction starts on heating. The disadvantage comes from the much more complicated nonisothermal kinetics, which however can be handled, thanks to available professional kinetics software. 9.3.4.2 Selection of the Proper Calorimeter For the investigation of chemical reactions, different calorimeters exist. Reaction calorimeters, mostly built of glass, are suitable for liquid reactions only. For the aforementioned curing reaction, these and many other calorimeters cannot be used because the produced resin is solid, and cleaning of the calorimeter vessel is not possible. The differential scanning calorimeter is advisable for such curing


reactions because the crucibles need not be cleaned after each experiment. In this case, the authors used a power-compensated DSC (PerkinElmer) for the experiments. 9.3.4.3 Calorimetric Experiment Before every run, the epoxy was carefully mixed with the proper amount of the curing agent. A sample of 5–10 mg was immediately filled into a crucible, sealed (with a small hole in the lid to allow pressure compensation) and positioned in the DSC at ambient temperature. The DSC was programmed to heat the sample at a constant rate from ambient to 200 C followed by controlled cooling to room temperature and a second heating procedure at the same conditions. During the second run, no curing occurs, and the respective heat flow rate curve can be used as the baseline (see Section 6.4.4). By subtracting it from the first run, one immediately obtains the reaction heat flow rate function provided the heat capacity does not change too much during curing, which is usually the case. These experiments were performed at three different heating rates (5, 10, and 20 K min1). This is done to test the kinetic model; a suitable model must simulate the reaction and the measured function properly at every heating rate. The resulting functions (see, for example, circles in Figure 9.17) form the base for the kinetic analysis. 9.3.4.4 Evaluation of the Measurements To obtain kinetic parameters of more complex chemical reactions, nonlinear optimization methods have to be used. These methods need, however, some experience to yield reliable results. The procedure starts with the choice of a certain

Figure 9.17 Heat flow rate W (circles) measured at a heating rate of 20 K min1 and amount of substance n of the chemical species A and B together with the best fit from the overall evaluation (three heating

rates) of the epoxy curing reaction using a simple reaction of the nth-order model (according to Strey, H€ ohne, and Anderson, 1998).

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Figure 9.18 Heat flow rate W (circles) measured at a heating rate of 20 K min1 and amount of substance n of the chemical species A, B and C together with the best fit from the overall evaluation (three heating

rates) of the epoxy curing reaction using a model of two consecutive reactions (according to Strey, H€ ohne, and Anderson, 1998).

kinetic model (see Section 6.4.2). After choosing proper starting values of the respective model parameters, the measured curve is calculated, and the best fit is determined using the common methods of nonlinear parameter variation. The best results are achieved if the measured heat flow rate functions from three different heating rates are modeled simultaneously (overall evaluation). Strey et al. (1998) tried with the simple model of a reaction of the nth order first, but the result from the best fit was not satisfactory (Figure 9.17). Obviously, a more complex kinetic model must be chosen to describe the kinetics of this curing reaction. Actually, the model of a consecutive reaction corresponds better to the epoxy curing, namely, A ! B and B ! C with B an intermediate compound. With this model, the epoxy curing can be satisfactorily described. The fit is perfect for all three heating rates (Figure 9.18). The kinetic parameters determined this way read (private communication) as follows: For A ! B :

1st order;

E act ¼ 43 8 kJ mol1 ;

lnðk0 =s1 Þ ¼ 9:2 0:6;

Dr H ¼ 120 kJ mol1

For B ! C :

2nd order;

E act ¼ 62 5 kJ mol1 ;

lnðk0 =s1 Þ ¼ 16 2;

Dr H ¼ 188 kJ mol1

9.3.4.5 Interpretation of the Results The results can be interpreted as follows. The compound A means the monomer with hydroxyl groups. The hydroxyl group transforms with the help of the cure agent to carboxyl groups (B), and these monomers polymerize to the final resin (C). The first step is a rather fast transformation of one molecule, which is assumed to be of the first order. The second step is the polymerization where two molecules with respective active side groups react with one another; this reaction should be of the second order. The calculated amounts of the compounds during the reaction are also plotted in Figures 9.17 and 9.18.


9.3.4.6 Uncertainty Estimation The uncertainty of the kinetic analysis data is made up by different contributions. First, there is the uncertainty of the measurements. For the power-compensated DSC, the uncertainty of a single heat flow rate measurement of this type can be assumed to be lower than 2%. After subtraction of the second run, the uncertainty of the reaction heat flow rate function contains the uncertainty of the two measurements and the error caused by possible differences of the heat capacity of the reactants compared with the reaction product. The overall uncertainty amounts to 6% for the reaction heat flow rate functions of this curing reaction. Another source of error with DSC experiments is the smearing problem because of the thermal lag (see Section 6.3); the sample temperature is always behind the program temperature. As a result, the heat flow rate function is somewhat broadened and shifted to the right (higher temperature or larger time). To get the exact heat flow rate temperature–time function, the desmearing procedure should be performed. In the present case, the time constant of the heat exchange in the powercompensated DSC is lower than 10 s, a negligible value compared with the reaction times of about 500, 1000, and 2000s for the three heating rates. The respective thermal lag error does not increase the uncertainty of the measurement. Another contribution to the uncertainty of the kinetic data comes from the modeling. A model is assumed to describe a reaction sufficiently well, if the calculated heat flow rate functions – for at least three different heating rates – fit to the measured ones within the limits of the uncertainty estimation. This is the case for the two-stage model (Figure 9.18) but not for the simple nth-order model (Figure 9.17). Although the present model probably describes the curing reaction well, this does not tell us that it is the only true one. Other models may exist that describe the measurement just as well. Much experience is needed to decide what is correct, but the kinetic analysis of calorimetric measurements is, however, helpful. 9.3.5 Example from Combustion Calorimetry 9.3.5.1 Definition of the Problem to be Investigated One of the major sources of energy for the generation of heat, electricity, or power is natural gas. For any (large) consumer of natural gas, it is thus of commercial interest to determine the amount of energy contained in a certain quantity of gas with low uncertainty. This amount of energy is determined by multiplying the quantity of gas with its heat of combustion. The heat of combustion is usually measured by means of field calorimeters that rely on calibration gases for reliable and traceable measurements. The uncertainty of the heat of combustion realized by the calibration gas must thus be considerably smaller than the required measurement uncertainty, which is here approximately 1%. 9.3.5.2 Selection of the Proper Calorimeter Combustion calorimeters capable of achieving measurement uncertainty well below 1% are commercially not available. Other techniques for the determination

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of the heat of combustion of a gas, for example, gas chromatography, were excluded to keep the traceability chain to the base units as short as possible and to avoid possible errors caused by nondetected components in the gas. Thus, it was decided to build a suitable calorimeter from scratch. The calorimeter should be able to measure the superior heat of combustion close to its definition (ISO 6976, 1995), that is, at atmospheric pressure, at 25 C and with condensation of the water formed by the combustion. The design of the calorimeter is shown in Figure 9.19. The calorimeter consists of a burner and an electrical compensation heater in a double-walled heat pipe filled with Freon with a boiling point of 25 C. The Freon transports the heat, which is developed by the burner and the compensation heater, to a cooling unit that consists of water-cooled Peltier elements. A thermometer located in the vapor phase is used to control the power of the compensation heater so as to keep the temperature of the vapor constant independent of the heat of combustion of the gas. The flow of gas burned in the calorimeter is determined by means of mass

1 2 3 4 5 6 7 8

Thermostated H2O Peltier elements Control thermometer Freon gas Liquid Freon (heat pipe) Gas burner Compensation heater Isolation wall

Figure 9.19 Isothermal calorimeter for the measurement of the superior heat of combustion of gases (according to Alexandrov, 2002).


flow controllers. Before the fuel gas enters the calorimeter, it is mixed with argon and primary oxygen. The argon serves (i) to lift the flame from the burner nozzle to prevent cooling of the flame and soot formation and (ii) to control the extent of water saturation of the exhaust gases. Directly at the burner nozzle, the gas is mixed with secondary oxygen to ensure complete combustion. Primary and secondary oxygen are saturated with water. The gases entering the calorimeter are temperature controlled to 25 C, and the gases leaving the calorimeter are cooled down to 25 C before leaving the calorimeter. 9.3.5.3 Calorimetric Experiment A typical experiment consists of three phases. For 60 min the baseline of the calorimeter is recorded when only the compensation heater is active. Then the gas is burnt for 30 min, during which the power to the compensation heater is reduced to keep the system in equilibrium. After this period, a second baseline is recorded. The heat of combustion Qcomb is calculated according to the following equation: Q comb ¼

P 0 P comp V_ gas

where P0 is the baseline power, Pcomp is the compensation power, and V_ gas is the volume flow of the gas. In this particular case, pure methane was burned to determine exemplarily the measurement uncertainty of the calorimeter. Methane reacts with oxygen according to the following stoichiometry: CH4 ðgÞ þ 2O2 ðg; satÞ þ Ar ! CO2 ðg; satÞ þ 2H2 OðliqÞ þ ArðsatÞ

The flow of methane, argon, primary oxygen (saturated with water), and secondary oxygen (saturated with water) were 2.5, 2.5, 1.4, and 6.3 l h1, respectively. According to the stoichiometry of the particular reaction investigated here, this leads to complete condensation of the water formed by the combustion. Two series of experiments were performed. The measured data are 1st run :

_ CH4 ¼ 4:9844 104 g s1 m P 0 ¼ ð55:453 0:005ÞW P comp ¼ ð27:790 0:009ÞW

2nd run :

_ CH4 ¼ 4:9844 104 g s1 m P 0 ¼ ð56:096 0:006ÞW P comp ¼ ð28:428 0:013ÞW

The temperature of the Freon vapor was set and held at (24.800 0.001) C, the temperature of the fuel gas mixture was (24.960 0.002) C, and that of the exhaust gases was (24.972 0.002) C. 9.3.5.4 Evaluation of the Measurements The evaluation of the experiments is straightforward according to the above equation and gave for the first run a heat of combustion of Qcomb ¼ (55.498 0.016) MJ kg1 and for the second run Qcomb ¼ (55.508 0.010) MJ kg1. (The numbers in brackets indicate the standard deviations of the means.)

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9.3.5.5 Interpretation of the Results It was the purpose of these experiments to determine the measurement uncertainty of results obtained with this new calorimeter design. 9.3.5.6 Uncertainty Estimation First, a fundamental investigation was performed to prove that the new calorimeter design and its measuring principle were working correctly and according to the expectations. For this purpose, an additional electrical heater was put into the calorimeter instead of the burner. This allowed the investigation of the performance of the calorimeter without the additional complexity introduced by the chemical reaction and the influence of the mass flow controllers. It was found that the standard deviation of the mean power of the baseline power was 0.02% and during the compensation phase between 0.07 and 0.02%, depending on the magnitude of the compensation power. A small systematic deviation between the calorimetric measurement of the power and its electrical measurement between 0.04 and 0.08% was observed. It was assumed that this heat loss was caused by the electrical supply wires of the additional heater. From these experiments, it was concluded that the maximum error of the new calorimeter was less than 0.1%. For the determination of the measurement uncertainty, the main components of the calorimeter and their influence had to be identified and quantified. The main components are as follows:

i) The compensation heater with its power determination device (measuring voltage and current via the voltage drop across a calibrated resistor) with an uncertainty of 0.006%. ii) The mass flow controller for the fuel gas, which was gravimetrically calibrated for methane with an uncertainty of 0.035%. iii) The calorimeter with an intrinsic standard deviation of 0.036%. iv) The temperature difference between the fuel gas mixture and the exhaust gases was negligible. These data were combined according to the Guide to the Expression of Uncertainty in Measurement (GUM) approach (JCGM 100, 2008) and gave a combined relative standard uncertainty of 0.039% for the heat of combustion of methane. The result of this measurement could thus be expressed as Q comb ¼ ð55:503 0:043ÞMJ kg1

or

H s;V ¼ ð11:062 0:009ÞkWh m3

where Hs,V is the calorific value of methane under standard conditions for metering ( pn ¼ 1013.25 mbar, Tn ¼ 273.15 K). The uncertainty interval covers a probability of approximately 95% (coverage factor k ¼ 2). This value had to be compared with literature values. Just two series of precision measurement of the heat of combustion of methane have been performed, one by Rossini (1931) and the other by Pittam and Pilcher (1972). A combination of their results gives values for the heat of combustion of

References

55.515 MJ kg1 or 11.064 kWh m3 with a standard deviation of 0.013 kWh m3 i 0.12% (2s) (ISO 6976, 1995). The measurement uncertainties of these two experiments are not given in the respective publications and could not be inferred from the available information. Thus, the standard deviation is used here for comparison.

References Alexandrov, Y.I. (2002) Estimation of the uncertainty for an isothermal precision gas calorimeters. Thermochim. Acta, 382, 55–64. ATHAS Databank (2011) http://athas.prz. rzeszow.pl/ (February 14, 2013). Della Gatta, G., Richardson, M.J., Sarge, S.M., and Stølen, S. (2006) Standards, calibration, and guidelines in microcalorimetry. Part 2. Calibration standards for differential scanning calorimetry (IUPAC Technical Report). Pure Appl. Chem., 78, 1455–1476. H€ ohne, G.W.H., Hemminger, W.F., and Flammersheim, H.-J. (2003) Differential Scanning Calorimetry, 2nd edn, Springer, Berlin. ISO 6976 (1995) Natural Gas – Calculation of Calorific Values, Density, Relative Density and Wobbe Index from Composition, International Organization for Standardization, Geneva. JCGM 100 (2008) Evaluation of measurement data – Guide to the Expression of Uncertainty in Measurement. JCGM 100:2008 (GUM 1995 with minor corrections) 1st edn, 2008, corrected version 2010, Bureau International des Poids et Mesures, Sèvres. Pittam, D.A. and Pilcher, G. (1972) Measurements of heats of combustion by flame calorimetry. Part 8. Methane, ethane, propane, n-butane and 2-methylpropane. J. Chem. Soc., Faraday Trans. 1, 68, 2224–2229.

Rossini, F.D. (1931) The heats of combustion of methane and carbon monoxide. J. Res. Natl. Bur. Stand., 6, 37–49. The heat of formation of water and the heats of combustion of methane and carbon monoxide. A correction. J. Res. Natl. Bur. Stand., 7, 329–330. Sarge, S.M., Hemminger, W., Gmelin, E., H€ohne, G.W.H., Cammenga, H.K., and Eysel, W. (1997) Metrologically based procedures for the temperature, heat and heat flow rate calibration of DSC. J. Therm. Anal., 49, 1125–1134. Schmolz, E. and Lamprecht, I. (2000) Calorimetric investigations on activity states and development of holometabolous insects. Thermochim. Acta, 349, 61–68. Strey, R., H€ohne, G.W., and Anderson, H.L. (1998) Determination of kinetic parameters of polymerizations by differential scanning calorimetry. Thermochim. Acta, 310, 161–165. Trampuz, A., Salzmann, S., Antheaume, J., and Daniels, A.U. (2007) Microcalorimetry: a novel method for detection of microbial contamination in platelet products. Transfusion, 47, 1643–1650. Wads€o, I. and Goldberg, R.N. (2001) Standards in isothermal microcalorimetry (IUPAC Technical Report). Pure Appl. Chem., 73, 1625–1639.

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Keywords/Abstract Dear Author, Keywordsandabstracts will not beincludedin the printversion of yourchapter but only in the online version. Please check and/or supply keywords. If you supplied an abstract with the manuscript, please check the typeset version. If you did not provide an abstract, the section headings will be displayed instead of an abstract text in the online version. Abstract A guideline for performing calorimetric experiments describes the necessary steps for a successful outcome: (i) the problem to be solved must be defined, (ii) a suitable calorimeter must be chosen, tested and calibrated, and (iii) the experiment which comprises the preparation of the sample, the measurements, the evaluation, the interpretation of the results and the uncertainty estimation must be performed. Four examples, from material science the determination of the crystallinity of a polymer, from biology the identification of infected blood, from chemistry the determination of the kinetic parameters of a curing reaction, and from industry the performance evaluation of a gas calorimeter serve to illustrate the power and the pitfalls of modern calorimetry. Keywords: calibration; calorimeter; calorimeter selection; calorimeter testing; calorimetric practice; linearity; measurement; sensitivity; uncertainty estimation.