More on Heat Capacities

Recall that we defined two different heat capacities, one for a change in a system kept at constant volume, and one for a change in a system kept at constant pressure. We labeled them C and Cp. What is the relationship between the two  

We start with an equation that eventually yielded equation 2.22. The relevant equation is 

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Note that the derivative dV/dT has a p subscript, due to our specifying that this is for constant-pressure conditions. Also note that the expression is a partial derivative, because the quantities in the numerators depend on multiple variables. (Other derivatives have also been expressed as partial derivatives.) Because dH = dq, we can substitute on the left side of the equation to get 

The term dWdT)p has already been defined as the heat capacity at constant pressure, Cp. We now have a relationship between Cy and Cp  

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The heat capacities that have been discussed previously refer to closed, single-phase systems. In such cases the variables that define the state of the system are either the temperature and pressure or the temperature and volume, and we are concerned with the heat capacities at constant pressure or constant volume. In this section and Section 9.3 we are concerned with a more general concept of heat capacity, particularly the molar heat capacity of a phase that is in equilibrium with other phases and the heat capacity of a thermodynamic system as a whole. Equation (2.5), C = dQ/dT, is the basic equation for the definition of the heat capacity which, when combined with Equation (9.1) or (9.2), gives the relations by which the more general heat capacities can be calculated. Actually dQ/dT is a ratio of differentials and has no value until a path is defined. The general problem becomes the determination of the variables to be used in each case and of the restrictions that must be placed on these variables so that only the temperature is independent. 

For more detailed calculations (e.g. if the presence of free carbon must be assumed), the reader is referred to M. A. Cook The Science of High Explosives, Chapman Hall, London 1958 and, by the same author The Science of Industrial Explosives, copyright 1974 by IR-ECO CHEMICALS, Salt Lake City, USA. They contain basic data on heat capacities and equilibria constants concerned, as well as computing programs for hand and machine calculations. 

To address our efforts towards correlating the macroscopic-ally measured heat capacity data to their microscopic origin, the data base started in the 1960 s was updated and computerized about five years ago (.8). The data bank is now incorporated within ATHAS, Advanced Thermal Analysis, a laboratory for research and instruction. The data bank maintains a collection of more than 500 publications on heat capacities of polymers which includes all measurements ever reported. The publication list is updated every six months. From each publication the following information is retrieved ... 

For the same qualitative reasons k, like rj, is density independent. The thermal conductivity is more temperature sensitive than the viscosity because of the dependence on heat capacity. Its behavior is also quite different from that observed in liquids. In gases k increases with T, while in liquids the opposite behavior is normal. 

With the fascinating features of ionanofluids such as high thermal conductivity, high heat storage capacity and non-volatile nature, they can potentially be used as novel heat transfer fluids. Nonetheless, besides studying thermal conductivity measurement, more experimental and theoretical investigations on heat capacity and viscosity of ionanofluids are imperative in order to exploit their potential applications in numerous important fields. 

Many substances exist in two or more solid allotropic fomis. At 0 K, the themiodynamically stable fomi is of course the one of lowest energy, but in many cases it is possible to make themiodynamic measurements on another (metastable) fomi down to very low temperatures. Using the measured entropy of transition at equilibrium, the measured heat capacities of both fomis and equation (A2.1.73) to extrapolate to 0 K, one can obtain the entropy of transition at 0 K. Within experimental... 

Another way to improve the error in a simulation, at least for properties such as the energy and the heat capacity that depend on the size of the system (the extensive properties), is to increase the number of atoms or molecules in the calculation. The standard deviation of the average of such a property is proportional to l/ /N. Thus, more accurate values can be obtained by running longer simulations on larger systems. In computer simulation it is unfortunately the case that the more effort that is expended the better the results that are obtained. Such is life ... 

The most direct effect of defects on tire properties of a material usually derive from altered ionic conductivity and diffusion properties. So-called superionic conductors materials which have an ionic conductivity comparable to that of molten salts. This h conductivity is due to the presence of defects, which can be introduced thermally or the presence of impurities. Diffusion affects important processes such as corrosion z catalysis. The specific heat capacity is also affected near the melting temperature the h capacity of a defective material is higher than for the equivalent ideal crystal. This refle the fact that the creation of defects is enthalpically unfavourable but is more than comp sated for by the increase in entropy, so leading to an overall decrease in the free energy... 

Because of its small size and portabiHty, the hot-wire anemometer is ideally suited to measure gas velocities either continuously or on a troubleshooting basis in systems where excess pressure drop cannot be tolerated. Furnaces, smokestacks, electrostatic precipitators, and air ducts are typical areas of appHcation. Its fast response to velocity or temperature fluctuations in the surrounding gas makes it particularly useful in studying the turbulence characteristics and rapidity of mixing in gas streams. The constant current mode of operation has a wide frequency response and relatively lower noise level, provided a sufficiently small wire can be used. Where a more mgged wire is required, the constant temperature mode is employed because of its insensitivity to sensor heat capacity. In Hquids, hot-film sensors are employed instead of wires. The sensor consists of a thin metallic film mounted on the surface of a thermally and electrically insulated probe. 

Ultrasonic Spectroscopy. Information on size distribution maybe obtained from the attenuation of sound waves traveling through a particle dispersion. Two distinct approaches are being used to extract particle size data from the attenuation spectmm an empirical approach based on the Bouguer-Lambert-Beerlaw (63) and a more fundamental or first-principle approach (64—66). The first-principle approach implies that no caHbration is required, but certain physical constants of both phases, ie, speed of sound, density, thermal coefficient of expansion, heat capacity, thermal conductivity. 

Thermal Conductivity and Heat Capacity. Most fibers have similar thermal conductivities and heat capacities. The insulating characteristics of textiles are more related to fabric geometry than they are dependent on fiber thermal characteristics. 

A more recent compilation includes tables giving temperature and PV as a function of entropies from 0.573 to 0.973 (2ero entropy at 0°C, 101 kPa (1 atm) and pressures from 5 to 140 MPa (50—1400 atm) (15). Joule-Thorns on coefficients, heat capacity differences (C —C ), and isochoric heat capacities (C) are given for temperatures from 373 to 1273 K at pressures from 5 to 140 MPa. 

The ocean is an integral part of the climate system. It contains almost 96% of the water in the Earth s biosphere and is the dominant source of water vapour for the atmosphere. It covers 71% of the planet s surface and has a heat capacity more than four times that of the atmosphere. With more than 97% of solar radiation being absorbed that falls on the surface from incident angles less than 50" from the vertical, it is the main store of energy within the climate system. 

Different areas of the earth s surface react quite differently to heating by the sun. For example, although a sandy surface reaches fairly high temperatures on a sunny day, the heat capacity and conductivity of Scmd are relatively low the heat does not penetrate more than about 0.2-0.3 m and little heat is stored. In contrast, in a body of water, the sun s rays penetrate several meters and slowly heat a fairly deep layer. In addition, the water can move readily and convection can spread the heat through a deeper layer. The heat capacity of water is considerably greater than that of sand. All these factors combine to allow considerable storage of heat in water bodies. 

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