Excess thermodynamic functions enthalpy

From the experimental temperature dependence of A2 (and the corresponding inferred temperature dependence of juE the other basic excess thermodynamic functions can be determined using general thermodynamic relationships. This then provides a complete thermodynamic characterization of the system as a whole. Thus, for the determination of the excess molar enthalpy of the system at constant pressure, the following equation can be used (Prigogine and Defay, 1954) ... 

In this section it was shown that the excess entropy and excess enthalpy can be determined from various temperature derivatives of the excess Gibbs energy. These and other excess thermodynamic functions can also be computed directly from derivatives of the activity coefficients. Show that in a binary mixture the following equations can be used for such calculations ... 

Figure 4. Molecular-weight dependence of excess thermodynamic functions of dextran aqueous solutions at 37°C (9) excess virial coefficient, B (O) excess enthalpy coefficient, Bh (excess entropy coefficient, Bs (16).

As in the case of liquids, it is possible to define excess thermodynamic functions for gas mixtures. The enthalpy of mixing of two gases at constant temperature and pressure,... 

Vp is the specific pore volume of the material typical values are 200-400 cm /kg for zeolites and up to 1000 cm /kg for activated carbon, n is the actual number of molecules contained in the micropores the excess adsorption n subtracts fix>m n the number of molecules which would have been present in the micropores at the bulk density in the absence of adsorption. The (oversimplified) case when absolute adsorption is described by the Langmuir equation and the gets obeys the perfect gas law p = P/RT) has been worked out in detail for the isotherms and thermodynamic functions (enthalpy, entropy, etc.) [2]. 

The ideal solutions are characterized by zero excess thermodynamic functions. In thermodynamics, ideal solutions are often characterized by their excess entropy and enthalpy. Note that to obtain these, we need to assume differentiability of (4.91) with respect to temperature. This assumption is still quite weaker than the requirement that (4.91) be valid at all T and P. 

Now let us consider the non-electrolytes. Here we have two very distinct types of behaviour. There are the so-called hydro-phobic, and the hydrophilic effects. The hydrophobic effect can be shown schematically from a consideration of the thermodynamics of hydrocarbon solutions. Usually a non-ideal solution arises because the two components either strongly attract each other or strongly repel each other the effects are shown in the enthalpy. Figure 8 shows various types of behaviour, as reflected in the excess thermodynamic functions (Rowlinson, 1969). The drawn out lines are free energies, the broken lines are enthalpies and the dotted lines are the entropy curves. A positive free energy means a positive deviation from ideal behaviour. In normal systems AG follows the AH curve fairly benzene-MeOH. In... 

In this chapter, we shall consider the methods by which values of partial molar quantities and excess molar quantities can be obtained from experimental data. Most of the methods are applicable to any thermodynamic property J, but special emphasis will be placed on the partial molar volume and the partial molar enthalpy, which are needed to determine the pressure and temperature coefficients of the chemical potential, and on the excess molar volume and the excess molar enthalpy, which are needed to determine the pressure and temperature coefficients of the excess Gibbs function. Furthermore, the volume is tangible and easy to visualize hence, it serves well in an initial exposition of partial molar quantities and excess molar quantities. 

The difference in thermodynamic functions between a non-ideal solution and a comparative perfect solution is called in general the thermodynamic excess function. In addition to the excess free enthalpy gE, other excess functions may also be defined such as excess entropy sE, excess enthalpy hE, excess volume vE, and excess free energy fE per mole of a non-ideal binary solution. These excess functions can be derived as partial derivatives of the excess free enthalpy gE in the following. 

The heat capacity peak, characteristic of the transition, reflects the excess heat capacity arising from the enhanced enthalpy fluctuations that occur in the temperature range of the transition. In the case of a two-state transition, the thermodynamic functions are obtained in a straightforward way from the area QD under the peak (corrected for the baseline), which measures the overall enthalpy change resulting from the transition, and the overall heat capacity difference (ACp) ... 

Thermodynamic functions have been calculated for liquid binary Ga-Pb alloys in the composition range 10—90 atom % Pb. Enthalpies and excess entropies of mixing at 1000 K were reported. ... 

Flory and Krigbaum defined an enthalpy (Kj) parameter and an entropy of dilution ( /i) parameter such that the thermodynamic functions used to describe these long-range effects are given in terms of the excess partial molar quantities... 

This chapter deals with experimental methods for determining the thermodynamic excess functions of binary liquid mixtures of non-electrolytes. Most of it is concerned with techniques suitable for measurements in the temperature range 250 to 400 K and the pressure range 0 to 100 kPa. Techniques suitable for lower temperatures will be briefly reviewed. Techniques for measuring the molar excess Gibbs function G, the molar excess enthalpy and the molar excess volume will be discussed. The molar excess entropy can only be determined indirectly from either measurements of (7 and at a specific temperature = (If — C /T], or from the temperature dependence of G m [ S m = The molar excess functions have been defined by... 

Binary mixtures of non-aromatic fluorocarbons with hydrocarbons are characterized by large positive values of the major thermodynamic excess functions G , the excess Gibbs function, JT , the excess enthalpy, 5 , the excess entropy, and F , the excess volume. In many cases these large positive deviations from ideality result in the mixture forming two liquid phases at temperatures below rSpper. an upper critical solution temperature. Experimental values of the excess functions and of Tapper for a representative sample of such binary mixtures are given in Table 1. 

All the other thermodynamic excess functions are obtained by differentiation of with respect to T or P. The most important are the molar excess entropy 5 , enthalpy Pf, and volume ... 

During a solid-phase transformation, excess energetic terms are observed in addition to the transformation enthalpy. Even if there is no kinetic reason, AT is larger than that of the crystallization of the liquid. In order to obtain thermodynamic functions of the phase transfor-... 

The data for Hultgren, Orr, Anderson, and Kelley s compilation were prepared between 1955 and 1963. Information on 65 elements and 167 alloy systems is presented, and selected values for heat capacity, entropy, enthalpy, free energy functions , and vapour pressures of phases are given in tabular form. For alloys, the preferred values of integral, partial, and excess thermodynamic properties are listed or are presented as analytical functions. Phase diagrams and graphs are also included. 

As to the unreal limiting values of enthalpies [see Eq. (18)], in the past 10-15 years it has been proven both theoretically and experimentally that the enthalpies and entropies have finite values at total monolayer coverage. For example, in adsorption from solutions a total excess coverage is always formed, and, evidently, the changes in enthalpies (entropies) can be measured exactly. The experimental data prove [9] that the enthalpy of a total monolayer coverage can never become infinite. Since the infinite or finite character of thermodynamic functions is independent of the nature of the adsorptive system (gas/solid, vapor/solid, liquid/solid) the supposition of the classical isotherm equation concerning limiting values [Eqs (17) and (18)] should be rejected. 

The thermodynamic excess functions differ from the thermodynamic functions of mixing only for quantities which involve the entropy. For example, the excess enthalpy A is identical with the enthalpy of mixing given by (1.6.6). Furthermore the excess volume v is identical with the volume of mixing given by (1.6.7). The excess entropy (in terms of activity coefficients) is given by (cf. 1.6.5)... 

Lead, excess entropy of solution of noble metals in, 133 Lead-thalium, solid solution, 126 Lead-tin, system, energy of solution, 143 solution, enthalpy of formation, 143 Lead-zinc, alloy (Pb8Zn2), calculation of thermodynamic quantities, 136 Legendre expansion in total ground state wave function of helium, 294 Lennard-Jones 6-12 potential, in analy-... 

So far, we have seen that deviation from ideal behavior may affect one or more thermodynamic magnitudes (e.g., enthalpy, entropy, volume). In some cases, we are able to associate macroscopic interactions with real (microscopic) interactions of the various ions in the mixture (for instance, coulombic and repulsive interactions in the quasi-chemical approximation). In practice, it may happen that none of the models discussed above is able to explain, with reasonable approximation, the macroscopic behavior of mixtures, as experimentally observed. In such cases (or whenever the numeric value of the energy term for a given substance is more important than actual comprehension of the mixing process), we adopt general (and more flexible) equations for the excess functions. 

The (liquid 4- liquid) equilibria diagram for (cyclohexane + methanol) was taken from D. C. Jones and S. Amstell, The Critical Solution Temperature of the System Methyl Alcohol-Cyclohexane as a Means of Detecting and Estimating Water in Methyl Alcohol , J. Chem. Soc., 1930, 1316-1323 (1930). The G results were calculated from the (vapor 4- liquid) results of K. Strubl, V. Svoboda, R. Holub, and J. Pick, Liquid-Vapour Equilibrium. XIV. Isothermal Equilibrium and Calculation of Excess Functions in the Systems Methanol -Cyclohexane and Cyclohexane-Propanol , Collect. Czech. Chem. Commun., 35, 3004-3019 (1970). The results are from M. Dai and J.-P.Chao, Studies on Thermodynamic Properties of Binary Systems Containing Alcohols. II. Excess Enthalpies of C to C5 Normal Alcohols + 1,4-Dioxane , Fluid Phase Equilib., 23, 321-326 (1985). 

Once the species present in a solution have been chosen and the values of the various equilibrium constants have been determined to give the best fit to the experimental data, other thermodynamic quantities can be evaluated by use of the usual relations. Thus, the excess molar Gibbs energies can be calculated when the values of the excess chemical potentials have been determined. The molar change of enthalpy on mixing and excess molar entropy can be calculated by the appropriate differentiation of the excess Gibbs energy with respect to temperature. These functions depend upon the temperature dependence of the equilibrium constants. 

The thermodynamic characteristics of solutions are often expressed by means of excess functions. These are the amounts by which the free energy, entropy, enthalpy, etc. exceed those of a hypothetical ideal solution of the same composition (Denbigh, 1981). The excess free energy is closely related to the activity coefficients. The total free enthalpy (Gibbs free energy) of a system is ... 

We note with respect to this equation that all terms have the units of m moreover, in contrast to Eq. (10.2), the enthalpy rather than the entropy app on the right-hand side. Equation (13.12) is a general relation expressing as a function of all of its canonical variables, T, P, and the mole numb reduces to Eq. (6.29) for the special case of 1 mole of a constant-compo phase. Equations (6.30) and (6.31) follow from either equation, and equ for the other thermodynamic properties then come from appropriate def equations. Knowledge of G/RT as a function of its canonical variables evaluation of all other thermodynamic properties, and therefore implicitly tains complete property information. However, we cannot directly exploit characteristic, and in practice we deal with related properties, the residual excess Gibbs energies. 

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