Isobaric expansibility

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

Mf = flame Mach number g = isobaric expansion ratio... 

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

The effect of temperature on the equation of state is introduced through the iso-baric thermal expansivity. It is generally assumed that isobaric expansivity and iso-baric compressibility work independently of each order and the volume as a function of T and p is then expressed as... 

Given A, V and T (or G, p and T), all other thermodynamic properties can be obtained by appropriate thermodynamic manipulations. For example, given the variation of volume with temperature (at given pressure) it is straightforward to calculate the isobaric expansivity a given by... 

It is useful to relate the expansivity to the volume dependence of the entropy. This is readily seen by manipulating the expression for the isobaric expansivity, a,... 

Gibbs energy minimization has also predicted negative isobaric expansion coefficients for certain crystalline zeolite framework structures, which subsequently were confirmed experimentally [6], Many solids show negative thermal expansion at very low temperatures, including even some alkali halides (Barron and White (Further reading)). Many other solids on heating expand in some directions and contract in others. 

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). 

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... 

Table 1.4 The Isobaric Expansivities, a, at 298.2 K of Several Ionic Liquids...

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. 

Kamlet-Taft) hydrogen bond donation ability ultrasound absorption coefficient isobaric expansibility... 

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. 

To provide an explicit example we introduce Curie s Law through the temperature dependence (condition H) ao = Ag/ T, xo = Ah/T where necessary we also set (dV/dT) i V/ T)p = aV. Here, a (with dimensions of reciprocal temperature, set in Roman lettering) represents the isobaric expansion coefficient, not to be confused with ao Ag as well as A/, are parameters. We ignore any possible temperature variation of a. Where these conditions apply Eqs. (5.8.11) simplify to... 

Nowadays, for a thermodynainicist, /pVT-calorimetry (further referred to as scanning transitiometry, its patented and commercial name ) is the most accomplished experimental concept. It allows direct determinations of the most important thermodynamic derivatives it shows how, in practice, the Maxwell relations can be used to fully satisfy the thermodynamic consistency of those derivatives. Of particular interest is the use of pressure as an independent variable this is typically illustrated by the relatively newly established pressure-controlled scanning calorimeters (PCSC). - Basically, the isobaric expansibility Op(p,T) =il/v)(dv/dT)p can be considered as the key quantity from which the molar volume, v, can be obtained and therefore all subsequent molar thermodynamic derivatives with respect to pressure. Knowing the molar volume as a function of p at the reference temperature, Tg, the determination of the foregoing pressure derivatives only requires the measurement of the isobaric expansibilities as... 

The combinatorial factor Q is related to the fraction of the three-dimensional space taken by polymeric chains and the other part taken by the solvent and is independent of the compactness (or otherwise) of the chain. The free volume factor Qfv depends on the experimental values of molar volume V (or segmental volume v), of isobaric expansivity a and of isothermal compressibility top. The configurational (or interactional, or potential... 

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