The effect of pressure on heat capacity

The data on heat capacities given in the handbooks, and in Appendix A, are, usually for the ideal gas state. Equation 3.13a should be written as  

The ideal gas values can be used for the real gases at low pressures. At high pressures the effect of pressure on the specific heat may be appreciable. 

Edmister (1948) published a generalised plot showing the isothermal pressure correction for real gases as a function of the reduced pressure and temperature. His chart, converted 

The ideal state heat capacity of ethylene is given by the equation  

The error in Cp if the ideal gas value were used uncorrected would be approximately 10 per cent. 

Edmister (1948) published a generalized plot showing the isothermal pressure correction for real gases as a function of the reduced pressure and temperature. His chart, converted to SI units, is shown as Eigure 3.2. Edmister s chart was based on hydrocarbons but can be used for other materials to give an indication of the likely error if the ideal gas specific heat values are used without corrections. 

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To generate isothermal compressibilities from sound speeds, it is necessary to have reliable expansibility and heat capacity data (equation 18). We have developed an iterative method to convert high pressure sound speed to isothermal compressibilities (84). The effect of pressure on the volume of a solution (3V/3P)T at a constant pressure is given by... 

The effect of pressure on the heat capacity of m-xylene and toluene has been tabulated, 5ft... 

Ehrenfest s concept of the discontinuities at the transition point was that the discontinuities were finite, similar to the discontinuities in the entropy and volume for first-order transitions. Only one second-order transition, that of superconductors in zero magnetic field, has been found which is of this type. The others, such as the transition between liquid helium-I and liquid helium-II, the Curie point, the order-disorder transition in some alloys, and transition in certain crystals due to rotational phenomena all have discontinuities that are large and may be infinite. Such discontinuities are particularly evident in the behavior of the heat capacity at constant pressure in the region of the transition temperature. The curve of the heat capacity as a function of the temperature has the general form of the Greek letter lambda and, hence, the points are called lambda points. Except for liquid helium, the effect of pressure on the transition temperature is very small. The behavior of systems at these second-order transitions is not completely known, and further thermodynamic treatment must be based on molecular and statistical concepts. These concepts are beyond the scope of this book, and no further discussion of second-order transitions is given. 

The methods discussed thus far neglect the effect of pressure on enthalpy, entropy, and heat capacity. Although efficiencies often are not known well enough to justify highly refined calculations,... 

Later, the pressure-scanning technique was used to investigate the thermophysical properties, isobaric molar heat capacity Cp (J K" mol" ), and Up, over extended T and p of several fluids or their mixtures, such as quinoline, n-hexane, 1-hexa-namine, and its binary mixtures with 1-hexanol, m-cresol, and its binary mixtures with quinoline, etc. As a rule, for simple liquids without strong intermolecular interactions, such as -hexane, for example, both the C -isotherms and the pressure effects (isotherms) on the isobaric heat capacity at pressures up to 700 MPa exhibit minima. It is worth recalling that the pressure effect on the Cp is related to the iso-baiic thermal expansibility ttp by the following equation (the effect of pressure on the Up is discussed in the next section) ... 

It is of interest to note that (d P/dT )v is zero for a van der Waals gas, as well as for an ideal gas hence, Cv should also be independent of the volume (or pressure) in the former case. In this event, the effect of pressure on Cp is equal to the variation of Cp — Cv with pressure. Comparison of equations (21.4) and (21.13), both of which are based on the van der Waals equation, shows this to be true. For a gas obeying the Berthelot equation or the Beattie-Bridgeman equation (d P/dT )v would not be zero, and hence some variation (f Cv with pressure is to be expected. It is probable, however, that this variation is small, and so for most purposes the heat capacity of any gas at constant volume may be regarded as being independent of the volume or pressure. The maximum in the ratio y of the heat capacities at constant pressure and volume, respectively, i.e., Cp/Cv, referred to earlier ( lOe), should thus occur at about the same pressure as that for Cp, at any temperature. 

If this parameter is assumed to be the same for all vibrations, one can obtain a bulk thermodynamic definition for y. The bulk Griineisen parameter is found to be about 4 for polymers from the effect of pressure on the velocity of sound. The data suggest that for the heat capacity only the interchain contribution should be taken into account. With this assumption, an order of magnitude calculation shows that the bulk Griineisen parameter for proteins is of the same order of magnitude as that of polymers. This suggests that the thermal expansion and the compressibility of proteins reflect primarily the movement between the secondary structures. These movements are reflected in the low frequency part of the... 

First some illustrations are given to support the concept of thin gas-filled layers between crystalline domains and then several examples are given to show that the gaseous fraction which Eyring assumed to comprise a liquid metal agrees well with certain physical measurements which others have found by experiment. These examples include heats of fusion, liquid coordination numbers, coefficients of thermal expansion of liquids, the effect of pressure on the melting point, and heat capacities of liquids. 

On the basis of some very crude approximations, this paper reports some computations for (1) the phenomena at the melting point, (2) the role of coordination number in liquids, (3) the coefficient of thermal expansion, (4) the effect of pressure on the melting point, and (5) the heat capacity of liquids to corroborate Eyring s theory that liquids can be represented as a mixture of a solid and a gaseous fraction. The results appear to be rather encouraging considering the lack of refinement in the calculations. 

Eq. (1.43) is the most general form of the solubility equation. In most situations (though not all) the effect of pressure on solubility is negligible so that the last term on the right-hand side of the equation can be dropped. In addition, the heat capacity term can also usually be dropped from the equation. This yields... 

When a liquid phase vaporizes to a vapor phase under its vapor pressure at constant temperature, an amount of heat called the latent heat of vaporization must be added. Tabulations of latent heats of vaporization are given in various handbooks. For water at 25°C and a pressure of 23.75 mm Hg, the latent heat is 44 020 kJ/kg mol, and at 25°C and 760 mm Hg, 44045 kJ/kg mol. Hence, the effect of pressure can be neglected in engineering calculations. However, there is a large effect of temperature on the latent heat of water. Also, the effect of pressure on the heat capacity of liquid water is small and can be neglected. 

The ideal solubility of a non-dissociating solute, assuming the effects of pressure and specific heat capacity change on melting are negligible is [7,8] ... 

In spite of the wealth of information available on the preparative and structural aspects of the lanthanide chlorides (1-3), experimental thermodynamic, and, in particular, high-temperature vaporization data are singularly lacking. The comprehensive estimates of the enthalpies of fusion, vaporization, heat capacities and other thermal functions for the lanthanide chlorides by Brewer et ah (4, 5) appear internally consistent, but the relatively few experimental measurements (6-/2) do not permit confirmation of the estimates due to the narrow temperature ranges of study. Additionally, the absence of accurate molecular data for the gaseous species has hampered third-law treatment of the limited experimental vapor pressure data available. The one reported study (12) of the vaporization of EuC12 effected by a boiling-point method lacks accuracy for these reasons. 

The effect of temperature on retention has been described experimentally,(4-8) but the functional dependence of k with temperature has only recently been described.W A thermodynamic model was outlined relating retention as a function of temperature at constant pressure to the volume expansivity of the fluid, the enthalpy of solute transfer between the mobile phase and the stationary phase and the change in the heat capacity of the fluid as a function of temperature.(9) The solubility of a solid solute in a supercritical fluid has been discussed by Gitterman and Procaccia (10) over a large range of pressures. The combination of solute solubility in a fluid with the equation for retention as a function of pressure derived by Van Wasen and Schneider allows one to examine the effect of solubility on solute retention. 

The various terms are interpreted as follows T(dSg/dT)p r represents the heat capacity, Cp r, of the adsorbate at constant pressure and surface occupancy r. The second term represents the mechanical work involved in the expansion of Vg on heating here the coefficient of expansion is relevant ap,r = V (dVg/dT)p r- In the third term we invoke the Maxwell relation that is specified in Eq. (5.2.8) of Table 5.2.1 T(dSg/dP)T,r = -T(dVg/dT)p r = —TVgap p, which again relates to mechanical work associated with the alteration of surface phase volume induced by pressure changes. The fourth term describes the contraction in volume of the surface phase due to the application of pressure. This effect is described by the isothermal compressibility fip.r = — V dVg/dP)T,r- The product —(pdAg obviously deals with the work of expanding the surface area. The sixth term is dealt with by use of the Maxwell relation from (5.2.8) from Table 5.2.1 T(dSg/dAg)T,p,ns = T d

temperature coefficient of the surface tension. We may therefore recast the above equation in the form... 

The magnitude of the variation of with pressure is thus determined by the variation of a with temperature. Since docjdT is always positive, the heat capacity at constant pressure decreases with increased pressure, but the effect becomes small at high temperatures since docjdT falls off more rapidly than T increases (c/. fig. 12.1). A similar formula is readily obtained for the effect of volume on c. ... 

Corrosion tests have inevitable limitations in their capacities to mimic actual service conditions of equipment. Standard, ambient pressure, immersion test procedures, with intermittent fluid refreshment, are available for both metallic and nonmetallic materials, but are limited to the ambient pressure boiling point of the fluid, and provide limited scope to simulate the effects of stress, geometry, heat transfer, and fluid flow. Such fesf procedures can be conducted at plant pressures and temperatures in autoclaves, and can be upgraded to focus on specific factors such as fluid flow and heat transfer. Even so a laboratory test, however elaborate, is a poor substitute for a test in the plant itself. 

Polymer chemists use DSC extensively to study percent crystallinity, crystallization rate, polymerization reaction kinetics, polymer degradation, and the effect of composition on the glass transition temperature, heat capacity determinations, and characterization of polymer blends. Materials scientists, physical chemists, and analytical chemists use DSC to study corrosion, oxidation, reduction, phase changes, catalysts, surface reactions, chemical adsorption and desorption (chemisorption), physical adsorption and desorption (physisorp-tion), fundamental physical properties such as enthalpy, boiling point, and equdibrium vapor pressure. DSC instruments permit the purge gas to be changed automatically, so sample interactions with reactive gas atmospheres can be studied. 

In writing this equation, it is very important that, unless otherwise stated, each reactant and product is understood to be the pure component in a separate and designated phase gas, liquid, or solid. A reaction is characterized by two important thermodynamic quantities, namely the heat of reaction and the Gibbs (free) energy of reaction. Furthermore, these two quantities are functions of temperature and pressure. Thermodynamic data are widely available in simulators and elsewhere, for more than a thousand components, for the calculation of these two quantities under standard state conditions, for example, at a reference temperature of 25°C and 1 bar with all components in a designated phase, usually as an ideal gas. The effect of temperature on the heat of reaction depends on the heat capacities of the reactants and products and the effect of temperature on those heat capacities. For... 

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