Gibbs relation

The simple approach described before involves approximations, particularly to obtain the stagnation pressure loss. The full determination of pf) m (7 o)5m from the various equations given above can lead to an approximation for the downstream entropy (.vsni). using the Gibbs relation applied between stagnation states. 

B, 109,15068 (2005). Spatially Heterogeneous Dynamics and the Adam-Gibbs Relation in the Dzugutov Liquid. 

C. Adam-Gibbs Relation Between the Configurational Entropy Sc and Relaxation in Glass-Forming Liquids... 

The surrounding fluid (Fig. 8-7) serves two purposes 1) it transmits the pressure to stress-load the surface and 2) it allows the surface to equilibrate chemically and thus provides juL in Eqn. (8.61) with physical meaning. Ideally, the Gibbs fluid has a vanishing buffer capacity for the solid so that after a change in an, the fluid becomes resaturated with respect to the solid before a noticeable amount of the solid or its surface dissolves. The key to verify Gibbs relation for solids under non-hydrostatic stress is therefore the existence of such an idealized fluid. 

Using all these variables the relations, which form the starting point for the further calculations, can be constructed. These relations are the energy density , the dissipation function R, the Gibbs-relation and the Gibbs-Duhem relation. To illustrate the idea of our model we split up e and R into several parts according to the different origin of the variables ... 

The set of basic equations is completed by the Gibbs-Duhem (the local formulation of the second law of thermodynamics) and the Gibbs relation (which connects the pressure P with the other thermodynamic quantities), which we will use in the following form ... 

Substituting Eq. (2.9) in the Adam-Gibbs relation for relaxation time Xad T) yields an expression for the strength parameter [18,41]... 

The Adam-Gibbs relation, namely, log [D(T, p)] versus (75c)-1, was found to be valid at each density [53]. 

Equation (1.90) is the total differential of the entropy as a function of the variables U and V only. To generalize this relation, we also consider the changes in the amounts of species. Using the mole amounts for the species, we have a general expression for the change of entropy from the Gibbs relation... 

Engineering systems mainly involve a single-phase fluid mixture with n components, subject to fluid friction, heat transfer, mass transfer, and a number of / chemical reactions. A local thermodynamic state of the fluid is specified by two intensive parameters, for example, velocity of the fluid and the chemical composition in terms of component mass fractions wr For a unique description of the system, balance equations must be derived for the mass, momentum, energy, and entropy. The balance equations, considered on a per unit volume basis, can be written in terms of the partial time derivative with an observer at rest, and in terms of the substantial derivative with an observer moving along with the fluid. Later, the balance equations are used in the Gibbs relation to determine the rate of entropy production. The balance equations allow us to clearly identify the importance of the local thermodynamic equilibrium postulate in deriving the relationships for entropy production. 

Assuming that the local thermodynamic equilibrium holds, we can write the Gibbs relation in terms of specific properties... 

Nonequilibrium thermodynamics estimates the rate of entropy production for a process. This estimation is based on the positive and definite entropy due to irreversible processes and of Gibbs relation... 

Entropy depends explicitly only on energy, volume, and concentrations because the Gibbs relation is a fundamental relation and is valid even outside thermostatic equilibrium. 

For a simple derivation of the dissipation function, consider an isothermal composite system with three compartments consisting of two external chambers (I and II) and a membrane compartment (m) in between. The volumes of the compartments are constant (dl = dVu = dVm = 0). The Gibbs relations for the compartments are... 

The phase rule of J. Willard Gibbs relates the variance (degrees of freedom) S for a nonchemically reactive system at heterogeneous equilibrium to the number of coexisting phases and the number of components (chemical species), C present. 

There is a thermodynamic relation between the adsorption and the surface-tension changes. Creation of concentration differences by accumulation of molecules in a boundary layer would, in itself, represent an increase in free energy, but it is compensated by the fact that the surface free energy is correspondingly lowered, since a is decreased. An equilibrium between the two effects is maintained. The simplest derivation of the well-known Gibbs relation which expresses the balance will now be given. 

Correlation between diffiision and entropy Adam-Gibbs relation 157... 

In liquids, one of the most celebrated equations that connects the microscopic dynamics of molecules with thermodynamics is the Adam-Gibbs relation. This relates the translational diffiisivity of the system to configurational entropy as follows ... 

Figure 10.3. Correlations between diffusion coefficient, configurational entropy, and tetrahedral order parameter (/h). Note that the left side of the y-axis represents the logarithm of diffusivity and the right side of ffie y-axis represents (th). The straight line fitting of the data validates the Adam-Gibbs relation between entropy and the diffusion coefficient, as discussed in the text The dashed line shows the correlation between (th) and configurational entropy. Adapted wifli permission from J. Phys. Chem. B, 114 (2010), 3633. Copyright (2010) American Chemical Society.

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