Isobaric coefficients

The partial differentials in the above relation may be recast as follows " as wiU be established in Section 1.12 by independent arguments, one may set dS/dT)y = Cy/T, and (dS/dV)j = dP/dT)y , where Cy is the heat capacity at constant volume and composition, and where the appropriate Maxwell relation has been introduced as the second relation. Next, set up the identity dP/dT)y = — dV/dT)p ,/ dV/dP)j. also, replace the numerator and denominator by the definitions —aV and — V, where a and are the isobaric coefficient of expansion and the isothermal compressibility, respectively. Last, set dS/dni)j y = 5, as defining the differential entropy at constant temperature, volume, and composition of species i. In this revised notation, Eq. (1.10.3c) assumes the form... 

We may replace dV/dT)p , with aV, where a is the isobaric coefficient of thermal expansion. So, we have found a second route for determining the enthalpy carry out the integrations called for in Eq. (1.12.11). 

Here the enthalpy of the substance is referred to that of a reference standard for that material chosen at any convenient reference temperature T and pressure Pi. This formulation is particularly useful for condensed phases. In the approximation of Dulong and Petit, the reference value for atomic solids at sufficiently high temperatures is approximated by 3nR. The partial derivative for volume in the integrand may be replaced by aV, where a is the isobaric coefficient of thermal expansion, which is fairly insensitive to pressure changes. For such materials, we cast the above in the form... 

CO is the equilibrium volume at the reference temperature, T, and a is the isobaric coefficient of thermal expansion. 

To illustrate calculations for a binary system containing a supercritical, condensable component. Figure 12 shows isobaric equilibria for ethane-n-heptane. Using the virial equation for vapor-phase fugacity coefficients, and the UNIQUAC equation for liquid-phase activity coefficients, calculated results give an excellent representation of the data of Kay (1938). In this case,the total pressure is not large and therefore, the mixture is at all times remote from critical conditions. For this binary system, the particular method of calculation used here would not be successful at appreciably higher pressures. 

Application of the algorithm for analysis of vapor-liquid equilibrium data can be illustrated with the isobaric data of 0th-mer (1928) for the system acetone(1)-methanol(2). For simplicity, the van Laar equations are used here to express the activity coefficients. 

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

Here Q(t) denotes the heat input per unit volume accumulated up to time t, Cp is the specific heat per unit mass at constant pressure, Cv the specific heat per unit mass at constant volume, c is the sound velocity, oCp the coefficient of isobaric thermal expansion, and pg the equilibrium density. (4) The heat input Q(t) is the laser energy released by the absorbing molecule per unit volume. If the excitation is in the visible spectral range, the evolution of Q(t) follows the rhythm of the different chemically driven relaxation processes through which energy is... 

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. 

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... 

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. 

Besides equilibriumconstants, additional thermodynamic data were included, if available, although little emphasis was put on their completeness. The data for primary master species comprise the standard molar thermodynamic properties of formation from the elements (AfG standard molar Gibbs energy of formation AfH°m standard molar enthalpy of formation ApSm- standard molar entropy of formation), the standard molar entropy (5m), the standard molar isobaric heat capacity (Cp.m), the coefficients Afa, Afb, and Afc for the temperature-dependent molar isobaric heat capacity equation... 

Similarity Relations for One-Dimensional, Constant-Area Channel Flow with Chemical Reactions. Similarity relations between stagnation temperature and mass fractions obtain during flow in a channel of constant cross section, provided a binary mixture approximation is used for the diffusion coefficient, the Lewis number is set equal to unity, the Prandtl number is set equal to 3/4, and a constant value is employed for the species and average isobaric specific heats. [The assumption that the species (cPii) and average (cp = 2YiCp,i) isobaric specific heats are... 

In the present study, systems composed of two solvents and a salt are treated as ternary systems. Data on the vapor pressure depression of the solvent by the salt for isothermal systems and on the boiling point elevation of the solvent in the presence of salt for isobaric systems are used to develop the parameters for the solvent-salt binaries. For such binaries only the activity coefficients for the solvent are considered. The parameters for all three binary sets are generated from the binary data by a regression subroutine. 

We therefore believe that the pic darret always arises from reactions of alkylperoxy radicals rather than of H02 radicals, and indeed that it can act as a peroxy radical tracer occurring when a critical peroxy radical concentration is achieved. Its disappearance at a certain temperature and reappearance at a higher temperature (see 250-mm. Hg isobar in Figure 4) after the region of negative temperature coefficient is in agreement with this view. 

Derive a general expression for the matrix coefficients ajk. (For simplicity assume isothermal, isobaric conditions.)... 

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