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For larger temperature differences such calorimeters can also be used as heat-flux calorimeters, using only the last two terms in Eq. (4). Because of the small losses, Tian–Calvet calorimeters have found application for the measurement of slow, biological reactions.

4.2.6 Modern Calorimeters

With the exception of the combustion calorimeters, not very many calorimeters have in the past been developed commercially. Calorimeters were usually built one at a time. Several of these unique calorimeters are described in the references. The choice of commercial calorimeters made in this section serves to illustrate the variety of available calorimeters.

A schematic of the operation of the Sinku Riku ULVAC SH-3000 adiabatic, scanning calorimeter is shown in Fig. 4.38. The calorimeter is detailed on the right. It is a miniaturization of the classical calorimeter in Fig. 4.30. The sample is indicated by 1. It is heated by supplying constant power, outlined in the block diagram on the

Fig. 4.38

left by . Temperature is measured by the block marked by . The rise in temperature of the sample holder, 2, inside the calorimeter is sensed by the thermocouple, . The adiabatic deviation is detected by the multi-junction thermocouple, marked and controlled by . The adiabatic deviation is used to raise the temperature of the adiabatic enclosure 3, and heater 4 (right). The instrumentation to , provide minicomputer control. Cooling water is provided to the surface of the calorimeter. The temperature range of the calorimeter is claimed to be 100 to 800 K with two different calorimeter models. The sample mass can be several grams. Adiabatic control is good to ±0.002 K, and heat input is accurate to ±0.5%.

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Fig. 4.39

Figure 4.39 illustrates a Calvet calorimeter, built by Setaram. A cross section through the low-temperature version of the calorimeter is shown. This instrument can operate between 80 and 475 K. Other calorimeters based on the same principle are available for operation up to 2000 K. The key features are the heat flow detectors which carry practically all heat to and from the sample. Although the Peltier effect could be used to quantitatively compensate heats evolved or absorbed, it is usually more precise to detect only the heat flow. The heating and cooling feature through the Peltier effect is then used for introducing initial temperature differences or for quick equilibration of sample and thermostat. The major feature of these Calvet calorimeters is their extremely good insulation. The special feature of the specific model is its ability to be cooled with liquid nitrogen. The temperature of the thermostat can be kept isothermal, or it can be programmed at rates from 0.1 to 1 K min 1. As little as 0.5 W of heat flow is detectable. The calorimetric sensitivity is 50 J. The cells may contain as much as 100 cm3 of sample. These specifications make the calorimeter one of the most sensitive instruments, and make it suitable for the measurement of slow changes, as are found in biological reactions or dilute-solution effects.

In Fig. 4.40 a stirred liquid bench-scale calorimeter is displayed. It closely duplicates laboratory reaction setups. The information about heat evolved or absorbed is extracted from the temperature difference between the liquid return (TJ) and the reactor (TR). This difference is calibrated with electric heat pulses to match the observed effect at the end of a chemical reaction. In a typical example, 10 W heat input gives a 1.0 K temperature difference between TJ and TR. The sample sizes may vary from 0.3 to 2.5 liters. The overall sensitivity is about 0.5 W. The calorimeter can be operated between 250 and 475 K. Heat loss corrections must be made for the stirrer and the reflux unit. The block diagram in Fig. 4.40 gives an overview of the data handling.

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Fig. 4.40

4.2.7 Applications of Calorimetry

Many applications of the various types of calorimetry described in Sects. 4.2 4.4 are given throughout the book. In this section four examples of calorimetric results are given: a) purity determination, b) determination of thermochemical functions by measurement and computation, c) and d) measurement of thermodynamic functions of carbon allotropes and paraffins.

Purity Analysis [15–18]. Purity analysis is an example of quantitative handling of a eutectic phase diagram with two components A and B. The analysis is based on the ideal mixing equation applied to the phase equilibrium of low-molar-mass compounds, as derived in Sect. 2.2.6 and expanded to macromolecules in Chap. 7 and shown in Fig. 4.41. The unknown impurity (B, component 2) is assumed not to crystallize together with the major phase (A, component 1). Its concentration x2, the lowering of the melting point of the crystals A, Tm, and the fraction melted at any the given temperature, F, must be evaluated. Figure 4.41 shows in the top-left the equation needed to estimate the purity of the sample.

If half of A is melted (F = 0.5), the concentration of the impurity is double the overall concentration x2, and the freezing-temperature lowering, Tmo Tm, is double that at the liquidus line, the line in the phase diagram indicating the end of melting shown in Fig. 4.9. At the eutectic temperature all B and sufficient A melt, to reach the eutectic concentration with the corresponding increase in enthalpy as can be deduced from the upper diagram in Fig. 4.9. This is followed by a broad melting range of the remaining pure component A that ends at the liquidus line, as described by the equation and in Fig. 4.9 (F = 1). For small impurities the beginning of melting is hard to find because of lack of equilibrium on crystallization, and the end of melting is not

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Fig. 4.41

easily detected, because of calorimeter-lag, particularly when using scanning calorimetry (see Sect. 4.3). These difficulties are removed by step-wise increasing of the temperature. After each melting step of, let us say 0.1 K, thermal equilibrium is awaited before one continues with the next step. The successive partial heats of fusion of A are 1, 2, 3, etc. The value of F, after melting the nth fraction, n, is then approximated as given in Fig. 4.41. The term E represents the unmeasured heat of fusion before the first step 1 was made. The melting temperature after step n is given by the top right equation. Between any two successive steps, n 1 and n, the remaining unknown E can then be eliminated as indicated in the boxed equation for x2. To reach highest accuracy, the two steps should be close to, but not include, the final portion of the melting peak.

When using continuous DSC for purity determination, the data must be corrected for instrument lag and F must be corrected for the omitted portion E as shown in Fig. 4.42 for testosterone. Computer programs exist to optimize the fit to a linear curve. Over-correction would give a downward deviation instead of the upward deviation. This purity determination is only applicable if there is solubility of A and B in the melt, but no solubility of B in crystals of A (eutectic system).

Thermochemistry. Measurements of changes in enthalpy during chemical reactions are the basic tools of the branch of thermal analysis called thermochemistry. The thermodynamic description for a sample reaction of hydrogen with oxygen to water is given in Fig. 4.43. The top equation shows the differential change of the enthalpy with temperature, pressure, and composition, as discussed in Chap. 2. The integration of the enthalpy for the case of constant pressure and composition can be done with help of the experimental heat capacity Cp, obtainable by adiabatic calorimetry or DSC. The boxed chemical equation represents the reaction to be discussed. The next equation expresses the differential changes in H with composition

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Fig. 4.42

Fig. 4.43

at constant T and p for the indicated reaction, usually measured by bomb calorimetry (Fig. 4.30). The correlations between the differential changes of the concentrations during the chemical reaction and the integrated change for one mole of gaseous water are given next. It is easy to calculate U from H as long as all gases can be treated as ideal gases (i.e., pV = nRT, p V = nRT). The experimental data were taken at

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298.15 K, the standard thermochemical reference temperature. The value of U(298) (= 240.59 kJ mol 1) is only little different from H.

Absolute values for Ho, the partial molar enthalpies, are not known. For chemical changes of interest it is, however, not necessary to know the absolute value of Ho for the elements since the same value enters into the computations for all compounds containing the same element. One can thus fix Ho of the elements arbitrarily to any value. Simplest is to set Ho of the stable element at 298.15 K equal to zero. The H at 298 K and atmospheric pressure of a reaction leading from elements to a compound is then called the enthalpy of formation and written Hfo(298), as shown in Fig. 4.43. The enthalpies of formation are widely tabulated. Furthermore, since all enthalpies are functions of state, one can make a compound in any sequence of reactions and obtain the same heat of reaction, as long as the initial and final states are the same. By adding the appropriate enthalpies of reactions, one can also get the enthalpies of formation which are not directly measurable. Figure 4.44 illustrates the evaluation of the enthalpy of formation of ethane, C2H6, which cannot be made out of C and H2. Enthalpies of formation at other temperature can be calculated using the appropriate

Fig. 4.44

heat capacities for the compounds as well as for the elements. The enthalpies of formation of elements at temperatures higher or lower than 298 K are not zero.

Typical thermochemical data are listed in Table 4.3. Note that the table contains data on free enthalpy and entropy, but H is easily computed from: H = G T S. For wider ranges of temperature the heat capacity cannot be treated as a constant, as is discussed in Chap. 2. The data in Table 4.3 can also be used to compute equilibrium constants, as shown in any introductory text on physical chemistry. The combination of thermochemical measurements that produce heats of reaction at fixed temperatures and thermophysical measurements that yield heat capacities permits a

full description of the energetics of systems at all temperatures. The pressuredependence is usually of less importance, but can be treated in analogy to the

temperature dependence by evaluation of ( H/ p)T,n in addition to ( H/ T)p,n.

A simple method to obtain at least approximate heats of reaction for covalently bound molecules makes use of heats of atomization and bond energies, as outlined in Fig. 4.45. This method is particularly important for most small organic molecules and flexible, linear macromolecules, the main subjects of this course. To find the heat of reaction, the endothermic heats (+) of breaking all bonds of the reactants to make

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Fig. 4.45

atoms are added to the exothermic heats ( ) of reconnection of the atoms to form the products. This method is based on the assumption that covalent bonds have always the same bond energy, independent of the structure details, an assumption that is not always true. Table 4.4 contains a table of all common bonds and elements in covalent molecules, and Fig. 4.45 illustrates the calculation of the heat of reaction of propane. In case larger discrepancies occur, one can still use the table of Table 4.4 to discuss the deviation of the bonding from the norm. Poor fits are found for the following three examples: conjugated double bonds relative to isolated ones, differences between >C=O and O=C=O, and the presence of polar bonding.

Table 4.4. Heats of Atomization and Bond Energies

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Element Atomization H C C= C N N= N O O=

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a at atmospheric pressure, 101,2325 Pa, and 298 K, in kJ mol 1 of atom or bond indicated.

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Thermodynamic Functions of Three Carbon Allotropes [19]. Low temperature heat capacities of three allotropes of carbon are shown in Fig. 4.46. The data were derived from adiabatic calorimetry. Based on the discussion of heat capacities of Sect. 2.3 and the known chemical structure of the allotropes given in Fig. 2.109, the thermal behavior can be easily understood. Fullerene, C60, is a small molecule. All strong bonds lie in the surface of the almost spherical molecule. The low-temperature heat

Fig. 4.46

capacity involves the molecules as a whole. Because of the weak interaction and the large mass of the C60 molecules, there are six low-frequency intermolecular vibrations per C60 that account for the high low-temperature specific heat capacity. Diamond, in contrast, consists of a continuous, tightly-bonded, three-dimensional network and has, as a result only a negligible low-temperature specific heat capacity. Graphite with its two-dimensional strong network is intermediate in specific heat capacity since some of the trans-planar vibrations are of low frequency and contribute also at low temperature to the heat capacity.

Figure 4.47 shows that at somewhat higher temperature, the specific heat capacities of graphite and C60 become almost equal. The surface structure of the fullerene and the layer-structure of graphite are quite similar, they consist of rings of conjugated double bonds. Diamond lags behind in gaining heat capacity because it lacks the low-frequency, intermolecular C60 and inter-planar graphite vibrations. Since the sp3 bonds are somewhat weaker than the conjugated double bonds, diamond ultimately gains the higher heat capacity at about 1000 K. At sufficiently high temperature, perhaps about 2000 K, all three allotropes must have the same heat capacity since they all have the same number of vibrational modes of motion. Dulong Petit’s rule, CV = 3R = 24.9 J K 1 (mol of carbon atom) 1 suggests the same vibrational heat capacity for all three carbon allotropes.