Maxwell relations general

Equations (2.15) or (2.16) are the so-called Stefan-Maxwell relations for multicomponent diffusion, and we have seen that they are an almost obvious generalization of the corresponding result (2.13) for two components, once the right hand side of this has been identified physically as an inter-molecular momentum transfer rate. In the case of two components equation (2.16) degenerates to... 

Equation (A 1.25) is known as the Maxwell relation. If this relationship is found to hold for M and A in a differential expression of the form of equation (A 1.22), then 6Q — dQ is exact, and some state function exists for which dQ is the total differential. We will consider a more general form of the Maxwell relationship for differentials in three dimensions later. 

Solution As shown in Sidebar 5.5, the desired derivative can be generally expressed [with the help of the Maxwell relation (5.49c)] as... 

The dielectric constant of coal is strongly dependent on coal rank (van Krevelen, 1961 Speight, 1994, and reverences cited therein). For dry coals the minimum dielectric constant value is 3.5 and is observed at about 88% w/w carbon content in the bituminous coal range. The dielectric constant increases sharply and approaches 5.0 for both anthracite (92% carbon) and lignite (70% carbon). The Maxwell relation which equates the dielectric constant to the square of the refractive index for a polar insulators generally shows a large disparity even for strongly dried coal. 

The optimal Reynolds number defines the operating conditions at which the cylindrical system performs a required heat and mass transport, and generates the minimum entropy. These expressions offer a thermodynamically optimum design. Some expressions for the entropy production in a multicomponent fluid take into account the coupling effects between heat and mass transfers. The resulting diffusion fluxes obey generalized Stefan-Maxwell relations including the effects of ordinary, forced, pressure, and thermal diffusion. 

Applying Eq. (A.7) to thermodynamic state functions (instead of a general function /) gives rise to the celebrated Maxwell relations. They can be used to express certain quantities that are hard to measure or control in a laboratory experiment in terms of mechanical variables such as a set of stresses and strains and their temperature and density dependence. 

In general, for a function of state /, that is completely determined by variables X and y, df = A dx + B dy. Cross-differentiation in df gives (dA/dy = (dBtdx)y, known as a Maxwell relation. Similarly, cross-differentiation in dU, dH, dF, and dG yields a wide variety of Maxwell relations between differential quotients. For instance, by cross-differentiation in dG we find... 

Generally, the increased complexity of the model based on Stefan-Maxwell relations does not justily improvement in model accuracy, which in most cases is redundant. Complex equations can be tolerated in numerical modelling however, in anal3dical modelling, clarity and simplicity of the resulting expressions are of the highest priority. Below we will use a simple Pick s law of diffusion to describe species transport in GDLs and in catalyst layers. 

In Sect. 4 we present several adsorption isotherms which are solutions of the Maxwell relations of the Gibbs fundamental equation of the multicomponent adsorbate [7.15]. These isotherms are thermodynamically consistent generalizations of several of the empirical isotherms presented in Sect. 3 to (energetically) heterogenous sorbent materials with surfaces of fractal dimension. In Sect. 5 some general recommendations for use ofAIs in industrial adsorption processes are given. 

U = U S, V Nic) and S = S U, V,Nk) are functions of the indicated variables. James Clerk Maxwell (1831-1879) used the rich theory of functions of many variables to obtain a large number of relations between thermodynamic variables. The methods he employed are general and the relations he obtained are called the Maxwell relations. 

It is left as an exercise to construct the seven remaining generalized thermodynamic functions whose double differentiation leads to a collage of aU possible additional Maxwell relations. The construction of the hnear phenomenological equations based on the choice of the independent variables of Eq. (5.11.10) is also left as an exercise. 

In general, the dielectric constant associated with a varying field is a complex number, but the imaginary part vanishes in two limiting cases for zero frequency and infinite frequency. The high-frequency dielectric constant e o is to be associated only with the displacements of the electrons from their equilibrium positions, and should satisfy the Maxwell relation Sea = where n is the refractive index. In addition to the electronic displacements, the static dielectric constant sq contains contributions from the atomic polarization and, in case of polar media, the orientation of the molecules. 

For the case of nonequilibriiun systems, the generalized Maxwell relations assume the forms 9Afc(0 9A,(0... 

A. Nisbet, Physica 21, 799 (1955) (extends the Whittaker-Debye two-potential solutions of Maxwell s equations to points within the source distribution a full generalization of the vector superpotentials, for media of arbitrary properties and their relations to such scalar potentials as those of Debye). 

Another attempt to correlate transport and self-diffusivities has been based on a generalization of the Stefan-Maxwell formulation of irreversible thermodynamics [111-113]. By introducing various sets of parameters describing the facility of exchange between two molecules of the same and of different species, the resulting equations are more complex than eqs 27 and 28 They may be shown, however, to include these relations as special cases... 

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