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Stefan Mathias Sarge

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Herausgeber: Wiley-VCH
Kategorie: Wissenschaft und neue Technologien
Sprache: Englisch

Beschreibung

Clearly divided into three parts, this practical book begins by dealing with all fundamental aspects of calorimetry. The second part looks at the equipment used and new developments. The third and final section provides measurement guidelines in order to obtain the best results.
The result is optimized knowledge for users of this technique, supplemented with practical tips and tricks.

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Seitenzahl: 487

Veröffentlichungsjahr: 2014

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Leseprobe

Contents

Cover

Related Titles

Title Page

Preface

List of Quantities and Units

Introduction: Calorimetry: Definition, Application Fields and Units

Definition of Calorimetry

Application Fields for Calorimetry

First Example from Life Sciences

Second Example from Material Science

Third Example from Legal Metrology

Units

Greek Symbols

αCubic expansion coefficientK−1αDegree of reaction1βHeating rateK s−1 (or K min−1)ηdynDynamic viscosityPa sλThermal conductivityW m−1 K−1μChemical potentialJ mol−1φPhase shiftradϕElectric potentialVΦHeat flow rateWΠPeltier coefficientVρDensitykg m−3 (or g cm−3)σStandard deviationSame as corresponding quantityσSurface tensionJ m−2σBStefan–Boltzmann constant ((5.670373±0.000021) × 10−8 W m−2 K−4)W m−2 K−4ϑTemperature°CΘTemperature°CτTime constantsωAngular velocityrad s−1ωAngular frequencyrad s−1

Indices and Subscripts

AAmplitudeactActivationamAmorphappApparatusCContainercalCalorimeterclbCalibrationcombCombustioncondCondensatecrystCrystallizedeffEffectiveelElectricexpExperimentalFFurnace (surroundings)FFinal statefinFinal statefusFusiongGlass transitionIInflectionIInitial StateinInletiniInitial stateliqLiquidMMeasuring systemmMolarmaxMaximumnStandard conditionsoutOutputpAt constant pressurerReactionrefReference stateRReferenceRpResponsesSuperiorSSampleSSurfaceslnSolutionTAt constant temperaturethThermaltrsTransitionVAt constant volumevapVapor, vaporizationwWaterWWall

Introduction

Calorimetry: Definition, Application Fields and Units

Definition of Calorimetry

Calorimetry means the measurement of heat. In the past, the term heat was associated with various concepts. Nowadays one no longer speaks of different energies (e.g., heat energy, electrical energy, and kinetic energy) coexisting in a substance or system independent of one another. According to the modern view, there is only one single energy (the internal energy) stored in a body, which – only during an exchange – appears in a variety of energy forms such as heat energy, electrical energy, or kinetic energy. Accordingly, the form of energy known as heat can only be conceived as coupled with a change of energy. Heat is always associated with a heat flow. In other words, heat is the amount of energy exchanged within a given time interval in the form of a heat flow. Calorimeters are the instruments used for measuring this heat.

Application Fields for Calorimetry

Calorimetry has been well known for centuries as a very effective method in natural sciences. The precise measurement of heat capacity, heat of fusion, heat of reaction, heat of combustion, and other caloric quantities has built the basis for progress in thermodynamics and classical physical chemistry. The classical methods of calorimetry have not changed very much during the past century, and the scientific interest in and the knowledge about them have dropped accordingly. Only the progress in microelectronics and computer science during the past few decades has made it possible to develop new types of calorimeters and open new fields of application. As a result, there is now an increasing interest in calorimetry as a very easy and powerful method for different kinds of investigation.

Modern calorimetry is successfully used in many fields, such as material science, life sciences (biology, medicine, and biochemistry), pharmacy, and food science, for quality control, safety investigations, and the determination of the energy content of fuels. Some examples illustrating this are presented below.

First Example from Life Sciences

Disregarding, for example, frictional heat, every living organism produces heat because of the chemical reactions involved in the metabolism of the cells. Depending on the temperature and other parameters (atmosphere, nutrients, and respiration), the heat production differs. If the temperature and the surroundings are kept constant, the heat production remains constant, too. Different microbiological organisms such as microbes and bacteria may produce different amounts of heat, but a single cell produces – under the above conditions – a characteristic amount of heat. If this quantity is known for certain organisms – say, bacteria – it is possible with a microcalorimeter to determine the amount of bacteria in a sample from the heat flow rate they produce (approximately 1–3 pW per cell, James, 1987). Furthermore, it was shown that the heat flow–time profiles of bacteria in a suitable nutrient solution are very specific and can be used to identify a bacterial species with a microcalorimeter much faster (within 15–24 h) than with a traditional culture medium (Trampuz et al., 2007). Possible applications of this calorimetric method could be the quick testing of the possible contamination of donated blood products or the faster identification of the bacterium causing blood poisoning and the more successful treatment of very dangerous sepsis.

Second Example from Material Science

Polymer processing frequently implies molding of nonamorphous polymers that often crystallize partially on cooling. Unfortunately, crystallization changes the material properties of certain polymers dramatically for the worse and should therefore be avoided. This can be achieved by quick cooling into a supercooled state (below the crystallization range). To investigate the crystallization behavior of such polymers, quick heating and quick cooling are required. Unfortunately, the heating and cooling rates of differential scanning calorimeters (DSC s) are limited to less than 100 K min−1 because of the thermal inertia of the equipment. For many polymers, this is not fast enough to avoid crystallization and to come to a really amorphous state. To overcome this problem, so-called chip calorimeters have recently been developed with the help of modern chip processing technology. Such ultrasmall calorimeters have a very small mass and very good thermal conductivity and, therefore, nearly no thermal inertia. The very low time constants of such calorimeter chips together with very small (1 ng) and very thin (10 μm) samples allow heating and cooling rates up to 106 K s−1 (Minakov et al., 2007). At heating and cooling rates higher than 103 K s−1, recrystallization can be avoided for several polymers, and the melting kinetics along with the superheating behavior and their influence on material properties can be investigated with such a chip calorimeter.

Third Example from Legal Metrology

The commercial value of fuels in general and fuel gases in particular depends on the amount of energy contained in a given amount of fuel. Traditionally, this amount of energy per unit amount, the calorific value, has been determined by gas calorimetry (Hyde and Jones, 1960). In recent times, gas chromatography has been used to infer the calorific value of natural gas from its composition. But biogases and other nonconventional fuel gases such as landfill gas and shale gas contain components not included in natural gas. As a consequence, gas chromatography fails or becomes prohibitively expensive. As a simple, quick, and cheap method for the determination of the calorific value of such gases, gas calorimetry is again applied, this time computer controlled and fully automatically (Haug and Mrozek, 1998). In this particular calorimeter, fuel gas flow and combustion airflow are controlled by nozzles. The two gas streams are mixed and burned. The resulting heat is transferred to a constant flow of air, whose temperature is measured at the entrance and the exit of the heat exchanger. This temperature increase is a measure of the energy content of the gas. To account for the influence of different fuel gas properties on the flow through the nozzle, the density of the gas is measured and used for converting the primary output of the calorimeter, the Wobbe number, to the desired quantity, the inferior calorific value of the fuel gas.

Units

As illustrated by the earlier examples, there is no direct method for the measurement of heat. Consequently, heat has to be determined by means of its effects. The older unit quantity of heat – the calorie – was therefore defined in terms of a measurement instruction:

One 15 °C calorie () is the amount of heat required to raise the temperature of 1 g of water from 14.5 to 15.5 °C under standard atmospheric pressure.1

Because heat is merely one form of energy, as are the electrical and mechanical energies (Mayer, 1842; Joule, 1843; Colding, 1843, according to Dahl, 1963), a special unit for heat is unnecessary. Today, in the International System of Units (SI), heat is expressed in the unit of energy:

Conversion between the old unit (cal) and new SI unit (J) is made as follows:

The latter is the International Steam Table calorie (calIT), one of the two “calories” still in use of a number of older calories (e.g., the “15 °C water calorie” corresponding to 0.9996801 calIT) (Stille, 1961).

The second calorie still in use in some parts of the world is the US National Bureau of Standards calorie (calNBS or calthermochem):

Note

1. Temperature is treated here as a directly measurable quantity, which, strictly speaking, is not the case.

1.1 Compensation of the Thermal Effect

The heat released from a sample during a process flows into the calorimeter and would cause a temperature change of the latter as a measuring effect; this thermal effect is continuously suppressed by compensating the respective heat flow. The methods of compensation include the use of “latent heat” caused by a phase transition, thermoelectric effects, heats of chemical reactions, a change in the pressure of an ideal gas (Ter Minassian and Milliou, 1983), and heat exchange with a liquid1) (Regenass, 1977). Because the last three methods are confined to special cases, only the compensation by a physical heat of transition and by electric effects are briefly discussed here.

1.1.1 Compensation by a Phase Transition

Around 1760, Black2) (Robison, 1803) realized that the heat delivered to melting ice serves for a transition from the solid to the liquid state at a constant temperature. Indeed, although the melting of ice requires a steady supply of heat, the temperature of the ice–water mixture only begins to rise after all the ice has melted. Black is said to have been the first to have used this “latent heat of fusion” of ice for the measurement of heat. His “phase transition calorimeter” was very simple. He placed a warm sample in a cavity inside a block of ice and sealed the cavity with an ice sheet. After the sample had assumed the temperature of ice, he determined the mass of the melted ice by weighing.

The principle of this method is that the heat ΔQ exchanged with the calorimeter is not measured as a heat flow but causes a phase transition in a suitable substance (e.g., ice). If the specific heat of transition qtrs of the respective substance is known, the heat involved can be determined because it is proportional to the mass of the transformed substance Δm:

The mass of the transformed substance Δm is determined either directly by weighing or indirectly (e.g., by measuring the volume change due to the difference between the densities of the two phases).

The first usable calorimeter involving a phase transition – the “ice calorimeter”– was developed by Lavoisier and Laplace (1780). Figure 1.1 schematically shows the design of this device. The sample chamber is completely surrounded by a double-walled vessel containing pieces of ice. This inner ice jacket is surrounded by a second double-walled vessel filled with an ice–water mixture (outer ice jacket). The whole system is in thermal equilibrium at 0 °C. The basic idea in this calorimeter is that the measuring system proper (i.e., the inner ice jacket) is insulated by the outer jacket, in which any disturbing influence of heat from the environment on the inner ice jacket is compensated by an ice–water phase transition in the outer jacket. Only heat released inside the sample chamber serves for the melting of ice in the inner ice jacket. Because there is no temperature difference between the inner and outer jackets, no heat exchange between them takes place. Lavoisier and Laplace designated the measured heat as the “mass of melted ice.” The specific heat capacities of solids and liquids, as well as heats of combustion and the production of heat by animals, were measured this way. These measurements were carried out in winter when low environmental temperatures allowed experiments to be carried out over longer periods of time (up to 20 h) in order to measure relatively small heat from animals, for example. The ice calorimeter of Lavoisier and Laplace is very bulky; moreover, the inner ice jacket must be carefully prepared before each test. In addition, it suffers from a systematic error that stems from the influx of relatively warm air at the lid. This air cools down in the calorimeter, thus releasing heat, and escapes with the downward flow of ice water. Another systematic error that affects the accuracy results from the fact that the water layer located in the inner jacket between the pieces of ice may attain a local temperature of several degree C, depending on the magnitude of the heat and the rate of its release. These layers may rise because of density differences (Figure 1.2) and transfer part of their heat to the outer jacket (lid). This latter heat thus escapes being measured.

Figure 1.1 Calorimeter of Lavoisier and Laplace (according to Kleiber, 1975).

Figure 1.2 The density of water as a function of temperature.

Bunsen (1870) was the first to describe an ice calorimeter that was free from these errors and allowed precise and reliable measurements (see Section 7.8.1.1).

A disadvantage of all phase transition calorimeters stems from the fact that the experimental temperature is determined by the transition temperature. Consequently, a variety of experimental temperatures can only be obtained in such calorimeters by using substances other than water. Very high sensitivity can be attained by using the liquid–gas phase transition (e.g., liquid nitrogen–gaseous nitrogen). The advantages of phase transition calorimeters lie in their relatively simple construction, their great sensitivity, and the possibility of enclosing the calorimeter in a vessel in which a phase transition identical to that occurring in the calorimeter takes place. This approach compensates for disturbances from the surroundings and creates “adiabatic” conditions (see Section 5.3).

For quantitatively determining heats with phase transition calorimeters, the specific heat of transition of the phase changing substance must be known. To calibrate such a calorimeter, a known amount of (electrical) energy Eel is supplied to the inside of the calorimeter via a heating wire, and the mass of substance undergoing the transition is measured. The energy supplied divided by the mass of substance transformed gives the specific heat of transition:

1.1.2 Compensation by Electric Effects

This method was applied for the first time by von Steinwehr (1901) and Brönsted (1906). This measuring procedure is best illustrated by the experiment performed by Brönsted. The device shown schematically in Figure 1.3 served for the measurement of the endothermic heat of solution of a salt in water. An amount of salt is dissolved endothermically in a Dewar vessel containing water, and the contents are steadily mixed by means of a stirrer. An electric heater yields a heat output that is regulated so as to leave the solution temperature unchanged. If the voltage U (t) and the current I (t) are constantly recorded, then (the electrically generated compensatory heat) equals the heat of solution of the salt.

Figure 1.3 Brönsted's calorimeter (shown schematically).

A major advantage of this procedure is that the conditions of heat transfer to the surroundings (heat leakage; see Chapter 4) remain unchanged during the measurement. Consequently, a highly sensitive but not necessarily calibrated thermometer can be used as a zero change indicator. The only correction to be made is to the heat generated by the stirrer; this is determined separately in a blank run. Because a resistor can only produce heat, this method is restricted to the measurement of endothermic effects. The compensation of exothermic effects is possible, in principle, by the use of electric cooling using the Peltier effect. The applications of Peltier cooling are discussed in greater detail in Chapter 7. Calorimeters involving an electrical compensation of the thermal effect are advantageous because the calibration of the calorimeter is necessary only for the determination of the sensitivity of the apparatus, and the electrical quantities voltage U and current I can be measured with great accuracy.

1.2 Measurement of Temperature Differences

Every exchange of heat needs a temperature difference to enable a heat flow. Therefore, the measurement of heats and heat flow rates can be reduced to the measurement of temperature differences (i) as a function of time, that is, , inside a calorimeter, or (ii) as a function of position, , along a heat conducting path.

1.2.1 Measurement of Time-Dependent Temperature Differences

The oldest method for the indirect determination of heat consists of measuring the change of temperature of a given mass of water after the introduction of a hot sample. This approach to the measurement of heat emerged with the development of reproducible, graduated thermometers (approximately since 1700); it is based on the principle that a given heat, ΔQ, always changes the temperature of a given mass of water by the same amount, that is, ΔT = ΔQ /Cw. This is true only if the heat capacity of water Cw does not depend on temperature, which is only roughly correct. A way out of this difficulty would be to confine the use of the calorimeter to a narrow temperature range, say, between 14.5 and 15.5 °C, to give a historical example. By means of this “mixing calorimetry” method, Wilcke (1781) and Crawford (1788) determined the specific heat capacities of various substances.

Calorimeters with a liquid calorimeter substance can be made in a variety of designs (see Chapter 7). Using a version known as a combustion calorimeter, Crawford (1788) also measured the heats of combustion of various substances and compared these with the heat generated by a guinea pig.

The calorimeter substance used for this purpose does not have to be water. Other liquid and even solid substances are also suitable. If the calorimeter substance is a solid (usually a metal), the calorimeter is referred to as “aneroid” (nonliquid).

In all calorimeters based on this principle (Figure 1.4), the heat exchange between the sample and the calorimeter substance alters the temperature of the latter from T (tini) = Tini to T (tfin) = Tfin. The change of temperature ΔT = Tfin − Tini is measured. The quantity of heat to be determined is ΔQ = Ccal · ΔT. The “heat capacity”Ccal of the calorimeter, which represents the sum of the heat capacity of the calorimeter substance and the heat capacities of other instrument components (stirrer, thermometer, vessel) involved to a greater or lesser extent in the temperature change ΔT, must be known. This “heat capacity” is an instrument-associated factor (determined by a proper calibration procedure) that can assume a variety of values depending on the experimental conditions (e.g., the magnitude of ΔT, the duration of the investigation, etc.). Moreover, part of the entire heat ΔQ ′ escapes directly into the surroundings by heat transfer via “heat leaks” and does not contribute to the temperature change ΔT. This heat loss must be determined. Hence, there is the need for a calibration of the calorimeter. For this purpose, a known quantity of heat ΔQ is introduced, and the resulting temperature change ΔT is measured to get Ccal = ΔQ /ΔT.

Figure 1.4 Calorimeter for the measurement of a time-dependent temperature difference (classical calorimeter with liquid calorimeter substance).

The effective “heat capacity” of the calorimeter Ccal determined in this manner represents a calibration factor valid only for specific conditions of the respective experiment. The historic term for this quantity is “water value,” namely, the mass of water possessing the same heat capacity as that of the calorimeter components, partly or entirely, involved in the temperature change. Thus, the water value reflects the sensitivity of the calorimeter because it indicates the heat necessary to obtain a given temperature change. A large water value consequently means low sensitivity, and vice versa. The term “energy equivalent” is also used. The term “heat capacity” will be used here in the sense of an instrument-associated or calibration factor. The experimental determination of this factor is best performed using electric heating because electric energy can be conveniently released at the desired site and measured with great accuracy.

The advantage of these calorimeters is their simple construction. However, precise determinations require the use of special arrangements (see Chapter 7).

One of the major applications of calorimeters with measurement of temperature differences as a function of time is the determination of specific heat capacities by measuring the rise of the sample temperature following the supply of a known amount of electric energy. A number of versions of these devices are described in Chapter 7.

1.2.2 Measurement of Local Temperature Differences

This method consists of the simultaneous measurement of temperature at two positions, x1 and x2, usually as their difference. It is illustrated by the following examples.

1.2.2.1 First Example: Flow Calorimeter

Two liquids capable of reacting with one another and possessing the same known temperature T1 (at position x1) flow into a reaction tube (Figure 1.5). There they react. At the measuring position for T2 (i.e., at position x2), where the reaction is assumed to be already completed, the liquid flows out of the tube. The calorimeter operates continuously. With the establishment of a thermal steady state between the liquid-containing reaction tube and the surroundings, a constant temperature difference ΔT = T (x2) − T (x1) = ΔT (x1, x2) is established that is proportional to the heat of reaction. The proportionality factor has to be determined by proper calibration. This can be done in a subsequent experiment in which the collected reaction product flows with the same flow rate and temperature around an electric heater inside the reaction tube.

Figure 1.5 Calorimeter for the measurement of a local temperature difference (flow calorimeter).

1.2.2.2 Second Example: Heat Flow Rate Calorimeter

In this type of calorimeter, a sample container is connected to a thermostat via a certain heat conducting body (e.g., a bar) (Figure 1.6). Initially, the entire device has the temperature T0. However, the occurrence of a thermal process (reaction) in the sample alters its temperature. This generates an equalizing heat flow rate Φ =dQ /dt through the bar and under ideal circumstances only through the bar (no heat leaks). In the steady-state case, the heat flow rate between two adjacent cross sections of the bar is associated with a temperature difference ΔT = Φ/G (see Section 4.1), where G is the thermal conductance between the sites where the temperature is measured located at a distance Δx from one another. Thus, G =λ ·A /Δx (where λ is the thermal conductivity of the bar material and A the cross-sectional area of the bar). In the steady state, the temperature difference between two adjacent cross sections of the bar is thus proportional to the heat flow rate (for non-steady-state conditions, see Section 4.1). To determine an unknown heat flow rate, this temperature difference is measured as a function of time: ΔT (t) =T (x2, t) − T (x1, t). If the entire system reverts to the constant temperature T0, the entire heat has flowed through the bar. The total heat exchanged through the bar can be determined from the integral together with the thermal conductivity and the geometry of the bar. This, however, applies only to the ideal case in which the entire heat flows only through the bar without any losses caused by radiation, convection, or parasitic heat transfer (heat leakage). Under actual experimental conditions, there is always a certain leakage of heat, and the integral is proportional to the heat exchanged with an unknown proportionality factor K (T), which must be determined by proper calibration, whereupon .

Figure 1.6 Calorimeter for the measurement of a local temperature difference (heat flow calorimeter).

1.3 Summary of Measuring Principles

A brief overview of measuring principles is given below.

i. Measurement of the heat exchanged by compensation, that is, suppression of any temperature change of the calorimeter caused by the thermal effect of the sample.

1. Compensation by a phase transition and measurement of the mass of transformed substance.

2. Compensation by electric cooling (Peltier effect) or heating (Joule effect) and measurement of the respective electric energy. Compensation principleBy endothermic effectBy exothermic effectPhase transition (solid–liquid; liquid–gaseous)Phase transition (liquid–solid; gaseous–liquid)Electric cooling (Peltier effect)Electric heating (Joule effect)

ii. Measurement of the heat exchanged by measurement of a temperature difference.

1. Measurement of a time-dependent temperature difference and of the effective heat capacity of the calorimeter.

2. Measurement of a local temperature difference along a well-defined heat conducting path and of a calibration factor.

This classification covers all the types of calorimeters that are of relevance in practice. It provides the basis for Chapter 7, which describes instruments operating in accordance with these methods.

It is noteworthy that any exact measurement of heat consists essentially of the measurement of electric energy or is traceable to electric energy determinations because the latter form of energy is easy to release, can be measured with great accuracy, and is directly connected to the base unit of the SI (Système international d'unités) for the electric current, the ampere. Accordingly, all calorimeters are calibrated either directly by the use of electricity or by means of precisely known heats of reaction or transition, which in turn are measured in electrically calibrated or electrically compensated calorimeters.

Notes

1. For example, “Bench Scale Calorimeter” developed by Ciba-Geigy Ltd., Switzerland, and commercialized by Mettler-Toledo (Schweiz) GmbH, Switzerland, as Reaction Calorimeter RC1.

2. Black only reported his findings verbally; see Encyclopaedia Britannica (2003) or Ramsay (1918).

References

Brönsted, J.N. (1906) Studien zur chemischen Affinität. II. Z. Phys. Chem., 56, 645–685.

Bunsen, R. (1870) Calorimetrische Untersuchungen. Ann. Phys. Chem., 141, 1–31.

Crawford, A. (1788) Experiments and Observations on Animal Heat and the Inflammation of Combustible Bodies; Being an Attempt to Resolve These Phenomena into a General Law of Nature, 2nd edn, Johnson, London.

Encylopaedia Britannica (2003) The New Encyclopaedia Britannica, vol. 2, Encylopaedia Britannica, Chicago, pp. 251–252.

Kleiber, M. (1975) The Fire of Life. An Introduction to Animal Energetics, Robert E. Krieger Publishing Company, Huntington, NY.

Lavoisier, A. L. and De Laplace, P.-S. (1784) Mémoire sur la chaleur. Mem. Acad. Sci. Année 1780, 355–408.

Ramsay, W. (1918) The Life and Letters of Joseph Black, M.D., Constable, London.

Regenass, W. (1977) Thermoanalytische Methoden in der chemischen Verfahrensentwicklung. Thermochim. Acta, 20, 65–79.

Robison, J. (ed.) (1803) Lectures on the Elements of Chemistry Delivered in the University of Edinburgh, by the Late Joseph Black, M.D., William Creech, Edinburgh.

Ter Minassian, L. and Milliou, F. (1983) An isothermal calorimeter with pneumatic compensation – principles and application. J. Phys. E: Sci. Instrum., 16, 450–455.

von Steinwehr, H. (1901) Studien über die Thermochemie sehr verdünnter Lösungen. Z. Phys. Chem., 38, 185–199.

Wilcke, J.C. (1781) Om eldens specifica myckenhet uti fasta kroppar, och des afmätande. Kongl. Swenska Wetensk. Acad. Handl., II, 49–78.

Measuring Instruments

Various calorimeters are described in Chapter 1. As pointed out in that chapter, heat cannot be measured directly but only indirectly through its effects. Before a more detailed treatment of calorimeters in the second part of this book, the measuring instruments required in calorimetry will be discussed. Two categories of instruments are introduced here. The first category comprises instruments that provide a quantitative measure of the changes of quantities that are the consequences of exchanged heats, and this discussion is provided in a manner that is as simple and precise as possible. The second category describes instruments necessary for sample preparation, for the calibration of a calorimeter, and for performing a calorimetric experiment correctly.

Sufficient precision and calibration capability are essential requirements necessary for the traceability of the measurement results to the International System of Units (SI). Total measurement uncertainties for the final result of the calorimetric experiment in the order of 10−2 are common, 10−3 is state of the art, and lower uncertainties require high expenditure for the calorimeter and ancillary instruments and high effort in performing the measurements.

2.1 Measurement of Amount of Substance

First, the amount of substance investigated in any calorimeter must be precisely known to obtain quantitative caloric data. The sample mass is normally determined with a balance. Section 1.1.1 describes calorimeters in which the exchanged heat serves for the phase transition of a substance (e.g., ice into water). Thus, the measurement of heat is reduced to the measurement of the amount of a substance. This quantity can be measured either directly by weighing or indirectly by determining the changes of volume or pressure associated with the phase transition. For other calorimeters, the volume and/or the pressure of a substance must be known, too.

2.1.1 Weighing

Analytical balances can have a high degree of accuracy, a range of seven orders of magnitude, and a resolution down to 10−7 g, working in an electronic compensation manner. Mass comparators compensate for most of the weight by means of reference weights and thus allow weighing with very high resolution in a narrow range. Many models permit the continuous registration of weight changes in the course of measurement. The accuracy of these instruments is so high that nonsystematic weighing errors can be neglected in the analysis of errors involved in calorimetric measurements because other error sources, such as heat leaks or an incomplete separation of the transformed substance, are nearly always dominant.

Nevertheless, correct mass determinations require some effort. Electrostatic charges, magnetic forces, adsorption or desorption especially of water, other contaminations, air drafts caused by temperature differences between the sample and the balance, and so on must be excluded. In mass determinations, a buoyancy correction is necessary when the density of the sample differs from that of the calibration weights. For further information, see, for example, Debler (2000).

2.1.2 Volume Measurement

Volume measurement (i.e., measurements of volume changes) can be easily carried out by the displacement of an incompressible medium (e.g., mercury). Small volume changes result in a large displacement of the meniscus of a mercury column in a capillary tube. This approach provides high resolution, and by careful execution (constant cross section of the capillary, elimination of parallax, avoidance of any temperature differences), it also ensures accuracy. Because temperature changes affect the volume of all substances and the thermal expansion coefficient depends also on temperature, strictly isothermal conditions must be created to obtain good repeatability as well as satisfactory calibration. The main drawbacks and sources of error of such displacement instruments stem from the necessarily nonisothermal experimental conditions. The conventional liquid-in-glass thermometer is, in effect, an instrument for the measurement of volume changes. It will be discussed in greater detail later in the description of instruments for the measurement of temperature. If the volume change takes place at constant temperature, as is the case with phase transition calorimeters, the volume change can be measured with high accuracy. Volume changes of 10−6 cm3 can easily be read on a thin capillary. (A volume change of this magnitude, using 5 cm3 of mercury, reflects a temperature change of 0.001 K.) The creation of isothermal conditions is crucial for an accurate determination of volume changes.

For the most direct link to the SI, it is advantageous to weigh the amount of liquid or gas. Some of the most precise calorimetric measurements have been performed with weighing of the fluid; for example, the determination of the heat capacity of sapphire, a widely used calibration material, has been performed with a Bunsen ice calorimeter by weighing the amount of molten water (Ginnings, Douglas, and Ball, 1950). The amount of methane burned for the measurement of its heat of combustion has been determined by direct weighing of a gas cylinder (Dale et al., 2002).

2.1.3 Pressure Measurement

The considerations for nonisothermal conditions apply also to the measurement of pressure changes, which usually consist of measurement of volume changes. However, pressure changes are far more conveniently measured by means of pressure sensors. Here, too, the pressure measurement is based on a deformation, however small and elastic, of the sensor. The displacement is measured by resistive, capacitive, or inductive transducers whose output signals are received and displayed by means of carrier frequency measuring bridges. At an uncertainty level of 0.1–1%, these instruments provide a continuous recording of the pressure course at a resolution of less than 1 Pa in the atmospheric pressure range (105 Pa).

2.1.4 Flow Measurement

Some flow calorimeters (continuous calorimeters) make use of air as a heat transfer medium; in other cases, gases or liquids react with each other or are products of the reaction. In the latter case, a possible approach to the measurement of amounts of substances consists in allowing the newly formed phase (usually a gas) to leave the system via a flow meter. Here the flow rate provides a measure of the quantity of substance transformed per unit time. Usually a pressure difference is the measurand as in capillary flow meters or is caused by the back pressure of the measuring instrument; however, the possibility of pressure rises (caused by a “buildup”) in the vessel must be taken into account. Other techniques for measuring amounts of gas make use of displacement gas meters, turbine meters, or ultrasonic meters. In these cases, the volume flow is the measured quantity. For measuring the mass flow, Coriolis or thermal mass flow meters can be used. In any case, it is very difficult to reduce the uncertainty of flow measurements below approximately 1%. This can only be achieved in exceptional cases when great effort is made to calibrate the meter with fluids of similar and known thermophysical properties (e.g., heat capacity, thermal conductivity, viscosity, density, etc.).

2.2 Measurement of Electric Quantities

A precisely known heat can be released with relative ease by means of an electric current flowing through a resistance. For determining the heat released, it is sufficient to measure the potential difference and current at the resistance and the time the current flows.

It is possible to display voltages and currents with a resolution of 10−6 and more by means of modern digital voltmeters. A relative accuracy of 10−4 is easy to achieve. However, this applies first and foremost to the measuring instrument. For a highly accurate measurement of low voltages, it is necessary to eliminate the influence of contact resistances and thermal electromotive force s (EMFs) in the circuit. If only one measurement (the voltage) is to be made, it is advisable to use a constant current source where the current can be adjusted and stabilized electronically to 10−5. The heat released at the resistance equals the product of the current, the potential difference (voltage) at the resistance, and the time interval during which the current flows (a correction for the heat released in the current carrying wires must, however, be taken into consideration):

The time interval involved is usually measured by means of an electronic clock that is triggered by the current. The resulting uncertainty of the time measurement is usually not limited by the clock but by the jitter of the trigger signal and is rarely better than 10 ms.

Thus, electrically generated heat can be released and measured without much expense and with a relative uncertainty of 10−3 to 10−4. This illustrates the efficiency of calibration with electrically created heat and explains its extensive application in calorimetry.

2.3 Measurement of Temperatures

The measurement of a change of temperature in a body as a result of an exchange of heat has remained the fundamental approach of calorimetry from its earliest days to the present time. Similar to heat, temperature changes can only be measured indirectly, that is, by means of their effects. These may consist of a change of volume, resistance, the spectral distribution of emitted light, or the contact potential of metals. All of these measurements refer, in effect, to differences, which raises the question of the zero point of temperature and of the temperature scale. Thermodynamics has shown the existence of an absolute temperature and its independence from any thermometer. For the present purposes, it is sufficient to state that there is a thermodynamic temperature scale, but the experimenter has nothing to do with it. And the experimenter is not concerned with the primary standards derived from this scale. The commercial measuring instruments used in experimental work are based on or calibrated against such standards. Their accuracy with respect to the standards is an essential requirement, as are the possibility of calibration, the constancy of sensitivity, and so on. The relationships involved will not be discussed in greater detail here. Briefly, all temperature measurements are associated with a thermometer whose principles and operation must be understood before entering into a discussion of possible errors in the course of measurement (for further reading, see Eder, 1981; Quinn, 1990; Nicholas and White, 2001; Bernhard, 2004).

2.3.1 Thermometers

2.3.1.1 Liquid-in-Glass Thermometers

Liquid-in-glass thermometers were the earliest devices used for the measurement of temperature (Table 2.1). They operate on the basis of a change of the volume (length, thickness) or pressure of a body as a result of a change of temperature. Variously graduated liquid-in-glass thermometers were already in production as long ago as 1700 (Fahrenheit, 1709; Réaumur, 1730; Celsius, 1742). The classical thermometer displays the volume change of a liquid (usually alcohol or mercury) located in a capillary glass container as a function of temperature. Because thermal expansion coefficients are usually related to temperature in a nonlinear manner, such thermometers have to be calibrated point by point, taking the following factors into account.

Table 2.1 Methods of temperature measurement.

Before a reading is taken, some time must be allowed until the entire thermometer substance and the glass container have assumed the temperature to be measured and the corresponding mechanical equilibrium. Because glass also has a thermal expansion coefficient, its increase in volume leads to an increase in the volume available to the thermometer liquid. Consequently, various temperatures are read from the same thermometer according to the depth to which it is immersed in the bath whose temperature is to be measured. Most liquid thermometers are calibrated for the case of a fully immersed thermometer body. Moreover, the external pressure affects the internal volume of the thermometer and consequently its reading. The main cause of erroneous measurements is an inhomogeneous distribution of temperature. Homogeneity of temperature sets in rather slowly. The hysteresis of the thermal expansion of the glass also has to be taken into account. An accurate measurement therefore requires long measuring times; in other words, the liquid thermometer is a thermally sluggish measuring system, particularly on decreasing temperatures.

Because of these error sources, liquid thermometers are rarely used nowadays for precision measurements (Hall and Leaver, 1959).

2.3.1.2 Gas Thermometers

Gas thermometers, which likewise operate on the basis of a volume change, are still extensively used for accurate measurements, in particular for measurements on the thermodynamic temperature scale. Here, the thermometer substance is a gas that is almost ideal, whose expansion coefficient equals 1/273 K−1 and is independent of temperature.

From the ideal gas equation p ·V = n ·R ·T (where p is pressure, V is volume, n is amount of substance, R is universal gas constant, and T is temperature), temperature can be inferred, either in terms of a pressure measurement at constant volume or as a volume measurement at constant pressure. Both procedures are precise and reliable (see Section 2.1). Here, too, however, attention must be given to errors resulting from changes of the container volume (e.g., owing to changes of the ambient pressure).

2.3.1.3 Vapor Pressure Thermometers

Vapor pressure thermometers constitute a special measuring device used mostly in low-temperature calorimetry. These instruments measure the vapor pressure of a known liquid substance – usually a liquefied gas – which is always associated with a given temperature. Owing to the close and definite relationship between temperature and the vapor pressure of a given substance (Figure 2.1), this method permits accurate measurements of temperature.

Figure 2.1 Vapor pressures p of various gases as a function of temperature T.

All thermometers described above occupy a relatively large volume and require a long time interval to reach a steady-state distribution of temperature. In other words, they are hardly suitable for local or rapid measurements of temperature.

2.3.1.4 Resistance Thermometers

Temperature affects the electric conductivity of metals and semiconductors. Consequently, it can be determined by measuring the electric resistance and referring to a calibration table. Platinum is particularly suitable for resistance thermometers owing to its high melting point and remarkable chemical inertness, which results in highly reproducible measurements. The use of appropriate manufacturing techniques results in quite small platinum resistance thermometers (as small as 1 mm in diameter) that respond rapidly to temperature changes owing to their small heat capacity.

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