Isobaric expansivity and isothermal compressibility

Knowledge of temperature and pressure dependence of physical-chemical properties is very useful to estimate the values of derived parameters, such as the thermal expansion coefficient, a, and the isothermal compressibility, Kj. 

The ILs do not expand appreciably at the commonly measured temperature range from 298.15 to 343.15 K. This small expansion with the temperature is best quantified by fhe volume expansivify, defined as 

Because fhe molar volumes and densities of ILs are usually linear functions of femperature, fhe isobaric expansivities are easy to obtain from linear fifs of fhe densify dafa. The values of ILs are in the range of 4 6 X 

10 K whereas fhe values of for most molecular organic liquids are 

Systematic density measurements at a wide range of femperafure and pressure [28,31,63,68,69] were helpful to obfain isofhermal compressibility, which is calculated using the isothermal pressure derivative of density according to Equation 1.2 

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Table 1.2 The isobaric expansivity and isothermal compressibility of selected compounds at 300 K.

Hofman, T. et al.. Densities, excess volumes, isobaric expansivity, and isothermal compressibility of the (l-ethyl-3-methylimidazolium ethylsulfate + methanol) system at temperatures (298.15 to 333.15) K and pressures from (0.1 to 35) MPa, /. Chem. Thermodyn., 40, 580, 2008. 

Standard molal volumes of minerals together with the coefficients of isobaric expansion and isothermal compressibility are available in several compilations (8,9). Standard partial molal volumes of uncomplexed ions are available at 25°C and 1 atm. (1) however, data are sparse for the coefficients of isobaric expansion and isothermal compressibility. 

The molar volume change in ionization reactions at higher temperatures and pressures cannot be calculated for most of the aqueous complexes because of a lack of data on isobaric expansion and isothermal compressibility coefficients. Entropy and heat capacity correlations have recently been used to generate equation of state parameters for estimating molal volumes of aqueous complexes at elevated temperatures and pressures (Sverjensky, 12). These coefficients are available for aqueous complexes only of univalent anions and, therefore, the pressure dependence of ionization constants at elevated temperatures cannot be estimated using Equation 4. 

Dielectric relaxation spectroscopy (DER), e.g. [103-105], DER monitors the mobility of dipolar groups in the polymer and also of small dipolar molecules (e.g. water) that may be dissolved in the polymer system. Corresponding to mechanical measurements, the maxima of dissipated energy indicate phase transition processes. Dilatometry, pVT measurements, e.g. [50,106]. These measurements unequivocally show a first order transition by a step in V T) and a bend if there is a glass transition. The important partial derivatives isobaric expansivity and isothermal compressibility can be derived from the corresponding measurements. The method is, however, quite time consuming and not widely used. 

Table 5.64 lists isobaric thermal expansion and isothermal compressibility coefficients for feldspars. Due to the clear discrepancies existing among the various sources, values have been arbitrarily rounded off to the first decimal place. 

If the heat capacity functions of the various terms in the reaction are known and their molar enthalpy, molar entropy, and molar volume at the 2) and i). of reference (and their isobaric thermal expansion and isothermal compressibility) are also all known, it is possible to calculate AG%x at the various T and P conditions of interest, applying to each term in the reaction the procedures outlined in section 2.10, and thus defining the equilibrium constant (and hence the activity product of terms in reactions cf eq. 5.272 and 5.273) or the locus of the P-T points of univariant equilibrium (eq. 5.274). If the thermodynamic data are fragmentary or incomplete—as, for instance, when thermal expansion and compressibility data are missing (which is often the case)—we may assume, as a first approximation, that the molar volume of the reaction is independent of the P and T intensive variables. Adopting as standard state for all terms the state of pure component at the P and T of interest and applying... 

The prediction of miscibility requires knowledge of the parameters T" (the characteristic temperature), p (the characteristic pressure) and V (the characteristic specific volume) of the corresponding equation of state which can be calculated from the density, thermal expansivity and isothermal compressibility. The isobaric thermal expansivity and the isothermal compressibility can be determined experimentally from p-V-Tmeasurements where these values can be calculated from V T) and V(p)j. The characteristic temperature T is a measure of the interaction energy per mer, V is the densely packed mer volume so that p is defined as the interaction energy per... 

Estimate the change of volume of water between two states by isobaric heating from T to T2 at Pi followed by isothermal expansion from Pi to P3 at T2 (a) by using van der Waals equation of state, (b) by using the thermal expansion and isothermal compression coefficients, (c) compare the results of part a and part b. 

TABLE 33 Volumetric Properties of Solvents at 25°C Densities,/ , Molar Volumes, V, Isobaric Expansibilities, a. Isothermal Compressibilities, and Intrinsic Volumes,... 

In an isothermal process, heat must be added during an expansion and removed during a compression to keep the temperature constant. We will describe this more fully as we now calculate the heat added or removed in isobaric, isochoric, and isothermal processes. 

Typical values of the isobaric expansivity and the isothermal compressibility are given in Table 1.2. The difference between the heat capacities at constant volume and constant pressure is generally negligible for solids at low temperatures where the thermal expansivity becomes very small, but the difference increases with temperature see for example the data for AI2O3 in Figure 1.2. 

Table 5.48 summarizes thermal expansion and compressibility data for amphibole end-members according to the databases of Holland and Powell (1990) and Saxena et al. (1993). Isobaric thermal expansion (a, K ) and isothermal compressibility (jS, bar ) may be retrieved from the listed coefhcients by applying the polynomial expansions... 

Table 5.64 Isobaric thermal expansion (K ) and isothermal compressibility (bar ) of feldspars...

As the enthalpy is a state function, the change in enthalpy A// is independent of the path selected for integration. The change between two states can be calculated by a chosen path, such as first an isothermal expansion from Pi to P2 at Pi followed by isobaric heating from Pi to P2 at P2 and isothermal compression from P2 to P3 at P2... 

Since LDL and HDL are two different liquids, the behavior of their thermodynamic response functions are quite different. The response functions of a system quantify how a given property, such as pressure, changes under the perturbation of a second property, such as T, under specific conditions, for example, constant volume and mole numbers. The basic response functions of a single component system are the isobaric specific heat, Cp T, P), isobaric thermal expansion coefficient, ap T, P), and isothermal compressibility, Kp T, P), all other response... 

Here (dPI8T)y is the isochoric thermal pressure coefficient that is seldom measured directly and is generally obtained by the last equality in (3.22). The magnitude of is >100 MPa, so that at ambient conditions and saturation vapor pressures, the last term, -P, in Equation 3.22 can be neglected. The isobaric expansibility, a, and isothermal compressibility, Kj, are available in Table 3.3. The differences U/Vat 25°C for the solvent listed here are shown in Table 3.8, with non-stiff solvents marked by italics font. The value of U/V-P for water is by far larger than for other structured solvents, but it diminishes with increasing temperatures [30] to become commensurate with P, of other solvents above 250°C. 

Two other important quantities are the isobaric expansivity ( coefficient of themial expansion ) and the isothermal compressibility k, defined as... 

We may first assume that isothermal compressibility fiy and isobaric thermal expansion coefficient a are independent, respectively, of T and P. Equations 1.91 and 1.99, integrated on T and P, respectively, give... 

Figure 8J (A) Isobaric thermal expansion, (B) its first r-derivative, (C) isothermal compressibility, and (D) isobaric heat capacity of H2O within the critical region, based on the equation of state of Levelt Sengers et al. (1983). From Johnson and Norton (1991), American Journal of Science, 291, 541-648. Reprinted with permission of American Journal of Science.

The schematic Ericsson cycle is shown in Fig. 4.27. The p-v and T-s diagrams of the cycle are shown in Fig. 4.28. The cycle consists of two isothermal processes and two isobaric processes. The four processes of the Ericsson cycle are isothermal compression process 1-2 (compressor), isobaric compression heating process 2-3 (heater), isothermal expansion process 3-4 (turbine), and isobaric expansion cooling process 4-1 (cooler). 

The density of the pseudoliquid is adjusted to reservoir pressure using the coefficient of isothermal compressibility and is adjusted to reservoir temperature using the coefficient of isobaric thermal expansion. 

The standard thermodynamic functions of solvation defined as above indicated by superscript differ from the generally tabulated standard thermodynamic functions indicated by superscript0 due to the constraints of fixed positions. They therefore lack the changes in the translational degrees of freedom, due to the compression from the volume of the gaseous state to that in the solution, not relevant to the solvation of the solute. Thus, AH = AH0 + RT (1 - ap7), where ap is the isobaric expansibility of the solvent, AS = AS0 + R( - ap7), AV = AV° - (RT/P°) -1 + ktP°), where kt is the isothermal compressibility of the solvent, and so on. 

The value of the molar volume of a solvent at other temperatures and pressures, not too far from the ambient, can be obtained by employing the isobaric thermal expansibility, ap, and the isothermal compressibility, kt. The former of these expresses the relative increase in volume on raising the temperature at a constant pressure and the latter expresses the relative decrease of the volume on raising the pressure at a constant temperature. These quantities are also temperature and pressure dependent, but over a limited range of these variables near ambient conditions they can be taken as being constant. 

Here, Cp is the heat capacity at constant pressure, aP is the isobaric thermal expansion, and kp is the isothermal compressibility Table 1.1 shows the molar heat capacities of some gas compounds. 

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