The measurement of heat capacities

An adiabatic calorimeter is designed to have negUgible heat flow to or from its surroundings. The calorimeter contains the phase of interest, kept at either constant volume or constant pressure, and also an electric heater and a temperature-measuring device such as a platinum resistance thermometer, thermistor, or quartz crystal oscillator. The contents may be stirred to ensure temperature uniformity. 

To make a heat capacity measurement, a constant electric current is passed through the heater cireuit for a known period of time. The system is the calorimeter and its contents. 

The electrical work Wei performed on the system by the heater circuit is calculated from the integrated form of Eq. 3.8.5 on page 88 Wei = At, where / is the electric current, / ei is the electric resistance, and At is the time interval. We assume the boundary is adiabatic and write the first law in the form 

Consider first an adiabatic calorimeter in which the heating process is carried out at constant volume. There is no expansion work, and Eq. 7.3.12 becomes 

An example of a measured heating curve (temperature T as a function of time f) is shown in Fig. 7.3. We select two points on the heating curve, indicated in the figure by open circles. Time t is at or shortly before the instant the heater circuit is closed and electrical heating begins, and time 2 is after the heater circuit has been opened and the slope of the curve has become essentially constant. 

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Spin-state transitions have been studied by the application of numerous physical techniques such as the measurement of magnetic susceptibility, optical and vibrational spectroscopy, the Fe-Mbssbauer effect, EPR, NMR, and EXAFS spectroscopy, the measurement of heat capacity, and others. Most of these studies have been adequately reviewed. The somewhat older surveys [3, 19] cover the complete field of spin-state transitions. Several more recent review articles [20, 21, 22, 23, 24, 25] have been devoted exclusively to spin-state transitions in compounds of iron(II). Two reviews [26, 27] have considered inter alia the available theoretical models of spin-state transitions. Of particular interest is the determination of the X-ray crystal structures of spin transition compounds at two or more temperatures thus approaching the structures of the pure HS and LS electronic isomers. A recent survey [6] concentrates particularly on these studies. 

A lock-in present in a bridge for cryogenics has usually a reference at frequencies between 10 and 1000 Hz. Electrical power delivered to the resistive sensors are between 10-16 -7-10-14 W. The maximum measuring rates are usually less than 10 readings for second. The latter fact poses severe limits in dynamic measurements as, e.g., the measurement of heat capacity (see Section 12.6.4). 

We saw in Section 3.2 that the knowledge of low-temperature specific heat is extremely important to understand the physical properties of a solid. The measurements of heat capacity are not, conceptually, more difficult than those of thermal conductivity. On the contrary, some problems such as the anisotropy of materials are not present, and the shape of the sample to be measured is usually unimportant. Nevertheless, from a technical... 

Figure 12.6 The measurement of heat capacity by DSC, under dynamic operating conditions.

It will be of interest to consider the thermal measurements since this area has been one of major activity in recent years. It is well known that the measurement of heat capacities at low temperatures provides a powerful tool for studying many solid-state phenomena, including the energy separation degeneracy of low lying energy levels in crystalline substances. The advantage of low temperature heat capacity measure-... 

A major application of DSC has been the measurement of heat capacity of a material, particularly at subambient temperatures. In simple terms, in the absence... 

Calorimeters with constant heat flow. Constant heat flow calorimeters are characterized by a constant temperature difference between the calorimetric vessel and the cover. To this group of calorimeters also belong the high-speed calorimeters for the measurement of heat capacities and the heats of modification transformation of substances, which are electrical conductors or semiconductors, where the heating is provided by their electrical resistance. 

Calibration of the calorimeter means the measurement of heat capacity, and depends on the type of calorimeter, the purpose of its use, as well as on the kind of the thermal effect. 

Reaction calorimeters are frequently calibrated using a known heat of a chemical reaction. No standard reaction is internationally accepted. For the measurement of heat capacities, drop calorimeters are frequently used and the calibration is made using a substance, the temperature dependence of which on heat capacity is known. As substances, metals like Cu, Ag, Au, and aluminum oxide in the form of sapphire are used. Calorimeters... 

Calorimetry of non-reacting systems involves the measurement of heat capacity dependencies on temperature, which enables us to calculate the enthalpies of phase transformations. Based on the prevailing mode of the heat exchange between their individual parts, calorimeters for this purpose can be classified as low-, medium-, and high-temperature calorimeters. In the measurement of thermodynamic parameters of molten electrolytes, mostly the last two types of calorimeters are used. 

The measurement of heat capacity and related quantities is known as calorimetry. Most often the constant-pressure heat capacity is measured some instruments measure the constant-volume heat capacity Cy. Often, what is actually measured is not the derivatives Cp and Cy but an energy change divided by a small but finite temperature change. In some cases, the original enthalpy increment data may be more useful than the approximate heat capacities derived from them. In addition to the lUPAC books referenced in Section 1.8.1, the monograph of Flemminger and Flohne... 

When a crucible is to be used more than once in a series of measurements it must be replaced each time in precisely the same position. One such example is in the measurement of heat capacity. Thermocouple assemblies are often designed to facilitate the repositioning of the crucible. 

Experimentally, the careful measurement of the heat capacity of low molar mass liquid crystals shows this "diverging" behavior at the nematic-isotropic transition, and also at the smectic-nematic transition For polymers, there is a limitation in the measurement of heat capacities, since the transition takes place in a relatively large temperature range, and thus, there is at each temperature a mixture of heat capacity and heat of transition. In order to solve this problem, we have built a tool for being able to make the difference between the first and the second order component of a DSC transition peak. The basic idea is that these two contributions of the total recorded heat power will behave in a different manner when the heating rate or the mass of sample are changed. 

From the measurement of heat capacity one can derive the free enthalpy as drawn schematically in Fig. 2.118. Since the heat capacity of the liquid is always larger than the heat capacity of the glass (see 2.117), the free-enthalpy curve of the liquid must have the larger curvature since (5 G/3T )p = -Cp/T (see Fig. 2.19) and when... 

This discussion of the measurement of heat capacity by anabatic calorimetry gives insight into the difficulties and the tedium involved. Today, computers handle the many control problems, as well as the data treatment. The rather involved experimentation is the main reason why adiabatic calorimetry is not used as often as is required by the importance of heat capacity for the thermodynamic description of matter. 

The main advantage of adiabatic calorimetry is the high precision. The cost for such precision is a high investment in time. For the measurement of heat capacities of linear macromolecules, care must be taken that the sample is reproducible enough to warrant such high precision. Both chemical purity and the metastable initial state must be defined so that useful data can be recorded. 

The next step is the analysis of a single, sinusoidal modulation in the DSC environment. In the top line of Fig. 4.91 Eq. (2) of Fig. 4.69 is repeated, the equation for the measurement of heat capacity in a standard DSC. The second line shows the needed insertions for T and AT for the case of modulation. When referred to T, the phase difference of AT is equal to 6 and its real part is the cosine, the derivative of the sine, as given by the top equation. Next, the insertion and simplification of the resulting equation are shown. By equating the real and imaginary parts of the equations separately, one finds the equations listed at the bottom of Fig. 4.91. These equations suggest immediately the boxed expression for C - C, in Fig. 4.92. 

A simple analysis of an irreversible first-order transition is the cold crystallization, defined in Sect 3.5.5. For polymers, crystallization on heating from the glassy state may be so far from equilibrium that the temperature modulation will have little effect on its rate, as seen in Fig. 4.122. The modeling of the measurement of heat capacity in the presence of large, irreversible heat flows in Fig. 4.102, and irreversible melting in Figs. 3.89 and 4.123, document this capability of TMDSC to separate irreversible and reversible effects. Little needs to be added to this important application. 

Heat Capacity Jump for Unfilled Polypropylene Resin The corresponding heat capacity jump ACp for unfilled polypropylene resin is shown in Fig. 12.11. The observed glass transition temperature range is about -15° to -5°C. Before and after onset of glass transition, heat capacity versus temperature behavior is linear but with different slopes. The measurement of heat capacity jump ACp is simply the difference of heat capacity values defined by end-points A and B for the two linear regions shown in Fig. 12.11. The resulting value of ACp is 0.105 J/g°C. 

The measurement of heat capacities as a function of temperature permits the calculation of all thermodynamic functions thus, the missing state functions can be calculated from Sv(T), Sp(T), Ev(1), and Hp T) if the change of pressure or volume dependent on temperature is also measured. The thermodynamic potential function corresponding to the selected free variables is thus obtained. The system is now completely described because the still missing dependent variables can be calculated as partial derivatives of the respective thermodynamic potential function. 

Mixing calorimetry with water as the calorimeter liquid was already introduced in the mid-eighteenth century for the measurement of heat capacities. Calibration was done likewise with water until the mid-nineteenth century. For this purpose, a given mass of water m (specific heat capacity =1 cal g at that time) of known temperature %, was poured into the calorimeter, and the subsequentiy established temperature of the mixture Tg was measured. Thus,... 

Topic 10 of volume 5 of the International Encyclopedia of Physical Chemistry and Chemical Physics gives a clear account of the perfect gas. Following an introductory chapter on thermodynamics, there is a chapter on the measurement of heat capacities which surveys the principles and limitations of various experimental methods. A large part of the work deals with the calculation of thermodynamic properties, including consideration of the ortho- and para-states, residual entropy, hindered internal... 

The present review describes recent developments in the experimental techniques for low-temperature (10 to 400 K) calorimetry and in the measurement of heat capacities of organic compounds during the past ten years. Data on heat capacities, enthalpies of phase changes, and entropies of organic compounds published before 1960 can be found in Landolt-Bornstein, Zahlenwerte und Funktionen, II. Band, 4. Teil (Berlin, Springer-Verlag, 1961). See also Chapter 2. 

Partington and Shilling s Specific Heat of Gases reviews the methods and results of heat capacity measurements made in the first quarter of this century. Masi, in an article published in 1954, covered the period from 1925 to 1952 and compared experimental and calculated values of heat capacities for many simple gases. He also briefly described ten possible methods of measuring heat capacities. In the last ten years Rowlinson devoted a chapter of his book, The Perfect Gas , to the measurement of heat capacities, and a detailed account of measurements and treatment of results in vapour-flow calorimetry was written by McCullough and Waddington in Experimental Thermodynamics , Volume I. ... 

Three main types of calorimeter have been described for the measurement of heat capacities by the vapour-flow method. In these methods the... 

Fig. 2.9 The absolute entropy (or Third-Law entropy) of a substance is calculated by extending the measurement of heat capacities down to T= 0 (or as close to that value as possible) and then determining the area of the graph of C/T against T up to the temperature of interest. The area is equal to the absolute entropy at the temperature T.

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