Gibbs equation validity

Equation (46) is the Gibbs equation valid for S/G interfaees. As it ean be seen, the thiekness [i.e., the molar volume of the Gibbs phase (t )] is an important parameter function here. 

Assuming the validity of Adam-Gibbs equation for relaxation dynamics and the hyperbolic temperature dependence of heat capacity, the strength parameter is found to be inversely proportional to the change in heat capacity [see Eq. (2.10)] at the glass transition temperature [48,105]. 

For a small step in temperature, the fictive temperature Tf is never far from the actual temperature T hence r, as given by the Narayanaswamy or the Adam-Gibbs equations, doesn t vary much with time. Equation (4-27) then simplifies to the ordinary linear KWW equation, Eq. (4-1). For large AT, varies during the relaxation, and the asymmetry discussed earlier is predicted. Note, however, that in Eq. (4-27) is assumed to be a constant this is not strictly valid for large changes in temperature, but is usually acceptable even when AT is a few tens of degrees. 

Reversibility of Adsorption. Apparently, the data in Figure 10.13 imply that the Gibbs equation (10.2) does not hold for the protein. As we have seen, it is valid for the amphiphile. However, the slopes dll/d In c given in the figure differ only by a factor 2 between the two surfactants, whereas the values of Fm differ by two orders of magnitude. The explanation is not fully clear. Application of the Gibbs equation to polymers is anyway questionable, because it is generally not known what the relation is between concentration (c) and activity (a) of the surfactant. Moreover, proteins and other polymers are virtually always mixtures. 

It is known that for solutions containing molecular species the condition of a 1 is valid for concentrations up to 0.1 mol dm"3, and thus the use of the simplified Gibbs equation (II.5) is justified for sufficiently dilute solutions only. On the contrary, the magnitude of substance concentration in the interfacial layer, c = c(s), does not impose any restrictions on the use of eq. (II.5). 

Equation (II.8) is also valid for soluble surfactants as well. However, in the latter case the value of n can not be directly measured using Langmuir s method, but can be established from surface tension measurements as the drop in the surface tension n - -Aa = o0 - a(c) (refer to Chapter II, 2 regarding the identity between n and -Aa). If the T(p) or T(c) dependence is known, the two-dimensional pressure can be obtained by integrating the Gibbs equation, i.e. ... 

Another difficulty is caused by the application of Gibbs thermodynamic concept to nonequilibrium conditions of an adsorption layer, as it is done in many theoretical models (cf. Chapter 4). From non-equilibrium thermodynamics we learn (cf Eq. (2C.8) derived by Defay et al. (1966), Appendix 2C) that the diffusional transport causes an additional term to the Gibbs equation. However, this term seems to be negligible in many experiments. The experimental data discussed in Chapter 5, obtained from measurements in very different time windows, support the validity of Gibbs fundamental equation (2.33) also under non-equilibrium conditions. 

In the transition from thermodynamics to dynamics of adsorption transport and energetic aspects have to be distinguished. The main question on energetic aspects belongs to the validity of Gibbs equation under non-equilibrium conditions. Defay et al. (1966) demonstrated the outcome of non-equilibrium thermodynamics for this topic. This direction deserves further attention. Comparison of experimental data with theory enables us to determine the range of applicability of Gibbs theory to non-equilibrium systems. 

Though this interpretation of adsorption kinetics and pressure isotherms is very helpful in deriving structure-functionality relationships, the results are limited to the validity of the theoretical models or approaches. This concerns both the three-step concept of adsorption kinetics and the commonly used simplification of the original Gibbs equation [95], on which the determination of the surface area... 

In more general constitutive models the Gibbs equations (local equilibrium) are not valid and therefore explicit calculations of entropy are impossible. This seems to correspond to the nonuniqueness of entropy or to irreversibility of processes between nonequilibrium states [see below (1.37) and Rem. 20]. Such are some constitutive models in Sects. 2.1-2.3, but in particular models with long range memory [17, 23, 48]. Even the usefulness of entropy in situations far from equilibrium [11, 101, 114-120] seems questionable, the entropy inequality deduced and used in... 

Therefore, the free energy is a potential for entropy and pressure, i.e., Gibbs equations are valid... 

Equations (2.38) (which are in fact the Gibbs equation with equilibrium pressure P°, cf. (2.18), (2.19) of model A) are valid also in nonequilibrium processes and prove (in this uniform system B) an analog of the local equilibrium hypothesis of irreversible thermodynamics [12, 16]. 

This result gives the classical Gibbs equation only in equilibrium process (2.45), i.e., analog of local equilibrium hypothesis is not valid in this material model C. Two last members in (2.55) contribute to the entropy production. 

In equilibrium thermodynamics model A and in model B not far from equilibrium (and with no memory to temperature) the entropy may be calculated up to a constant. Namely, in both cases S = S(V, T) (2.6)2, (2.25) and we can use the equilibrium processes (2.28) in B or arbitrary processes in A for classical calculation of entropy change by integration of dS/dT or dS/dV expressible by Gibbs equations (2.18), (2.19), (2.38) through measurable heat capacity dU/dT or state Eqs.(2.6>, (2.33) (with equilibrium pressure P° in model B). This seems to accord with such a property as in (1.11), (1.40) in Sects. 1.3, 1.4. As we noted above, here the Gibbs equations used were proved to be valid not only in classical equilibrium thermodynamics (2.18), (2.19) but also in the nonequilibrium model B (2.38) and this expresses the local equilibrium hypothesis in model B (it will be proved also in nonuniform models in Chaps.3 (Sect. 3.6), 4, while in classical theories of irreversible processes [12, 16] it must be taken as a postulate). 

On the other hand, in material models C, D with longer range memory or also with memory in temperature, the entropy cannot be calculated because of (2.43)2, (2.9)2, we have no possibility to express, e.g., dS/dV, through measurable quantities because of more complicated (2.55) (as distinct from the finding of dS/dV from Gibbs equation (2.38) above). In model D, the Gibbs equation in the form (2.68) is valid, but it permits to calculate only the equilibrium (part of) entropy. This seems to correspond with property (1.37) in these models C, D. 

As follows then from (2.81) this fluid mixture without memory has zero entropy production E = 0 in every process and also Gibbs equations (2.18), (2.19) are valid here. In fact, this model of fluid mixture without memory is the same as model A of Sects. 2.1, 2.2 (the form of functions (2.85) depends on the whole mass m chosen). 

At the end we stress several characteristic features of the linear fluid. Though the thermodynamic quantities /, u, s, P as well as the transport coefficients k, C, rj are functions of p and T only, no relationships exist between these two groups. Therefore it is impossible to obtain transport coefficients from equilibrium measurements. But such measurements suffice to obtain the thermodynamic functions, because the Gibbs equation is valid (by (3.193), (3.194))... 

If we assume that the ideal gas studied Mfils the local equihbrium (and this is the usual case ideal gas may be from the linear fluid models discussed here, but it may be also from some nonlinear models fulfilhng this principle, e.g. those in [78]), then property (3.213) follows from state equation (3.212). Indeed, the local equilibrium means the validity of Gibbs equations (3.200)i, (3.198)i, from... 

We now deduce basic thermodynamic properties of the mixture of fluids with linear transport properties discussed in Sect. 4.5. Among others, we show that Gibbs equations and (equilibrium) thermodynamic relationships in such mixtures are valid also in any non-equilibrium process including chemical reactions (i.e. local equilibrium is proved in this model) [56, 59, 64, 65, 79, 138]. 

Therefore, the classical relations of thermochemistry were obtained. Especially, the Gibbs equations (4.201)-(4.206) are valid in arbitrary process in this chemically reacting mixture of fluids with linear transport properties, i.e. the principle of local equilibrium is valid in this mixture. But we show in the following relations that this accord with classical thermochemistry (e.g. [138]) is not quite identical indeed, if we differentiate (4.211) and use (4.22), (4.23) we obtain... 

Thus Gibbs equation is assumed to be valid for small elements with... 

Gibbs equation is used for calculating the rate of internal production of entropy. It may be noted that Gibbs equation provides a simple route for identifying fluxes and forces. Gibbs equation strictly holds for equilibrium. However, it is found to be valid even in non-equilibrium close to equilibrium. 

Although non-equilibrium thermodynamics has serious limitations in view of limited domain of validity of Gibbs equation, but as a first approximation, it leads to interesting results for systems close to equilibrium. It provides a base for further extension of studies for systems far from equilibrium and even beyond. 

In continuous systems, the situation is not so clear. On account of experimental difficulties, exhaustive tests as in the above case have not been performed. Nevertheless, experimental results on thermal diffusion and Dofour effect/Soret do support the theoretical predictions although in a much limited range, again on account of limited range of validity of Gibbs equation. 

The validity of Gibbs equation is extended by assuming its validity for small volume element by replacing extensive variables as follows ... 

VVc will consider fluctuations in a system 2 s such its states in which the rundamental Gibbs equation 1.1.1-3 remains valid, but the equilibrium with the environment (tlier-inuslal) is disturbed. According to the Boltzman equation, the total entropy both of the system S and of the thermostat S is... 

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