Thermodynamic Terms Links

The derivation is a primary example of application of first principles in statistical thermodynamics, to link both microscopic and macroscopic domains for practical applications. For the reader s convenience, Table 5.1 gives the nomenclature used in Sections 5.1.1 and 5.1.2 as well as a listing (in parentheses) of the equations in which each term first appears. 

The sorption coefficient (K) in Equation (2.84) is the term linking the concentration of a component in the fluid phase with its concentration in the membrane polymer phase. Because sorption is an equilibrium term, conventional thermodynamics can be used to calculate solubilities of gases in polymers to within a factor of two or three. However, diffusion coefficients (D) are kinetic terms that reflect the effect of the surrounding environment on the molecular motion of permeating components. Calculation of diffusion coefficients in liquids and gases is possible, but calculation of diffusion coefficients in polymers is much more difficult. In the long term, the best hope for accurate predictions of diffusion in polymers is the molecular dynamics calculations described in an earlier section. However, this technique is still under development and is currently limited to calculations of the diffusion of small gas molecules in amorphous polymers the... 

The next question is a very important one, perhaps the one most central to the issue of the relation between the phase rule and the behavior of small systems. This is the question of how the equilibrium between solid and liquid forms of small systems can be described in a manner that links that behavior to the behavior of macroscopic systems. It is, in fact, straightforward to address in traditional, classical thermodynamic terms. 

This holds provided that no stable intermediate is involved in the reaction. If the rate at which ML dissociates into M and L (defined by the rate constant, kb) is slower than the rate at which M and L assemble into ML (defined by the rate constant kf), then there will always at equilibrium be a larger amount of ML present than of M and L, which translates into a large thermodynamic stability constant K, which equates with the ratio of these two kinetic terms. In effect, for such simple steps, there is a thermodynamic-kinetic link. At this level, however, we shall not dwell on this too deeply. 

The first and most important thermodynamic potential we need is entropy. One way to define entropy would be to simply say that the Z-term in Equation (3.14) does indeed exist, where entropy is called S, and Z = —S. This provides a useful analogy between pressure-volume and temperature-entropy, and we will see these terms linked together in many equations. They represent work and heat energy in many processes we will be considering. 

For reversible reactions one normally assumes that the observed rate can be expressed as a difference of two terms, one pertaining to the forward reaction and the other to the reverse reaction. Thermodynamics does not require that the rate expression be restricted to two terms or that one associate individual terms with intrinsic rates for forward and reverse reactions. This section is devoted to a discussion of the limitations that thermodynamics places on reaction rate expressions. The analysis is based on the idea that at equilibrium the net rate of reaction becomes zero, a concept that dates back to the historic studies of Guldberg and Waage (2) on the law of mass action. We will consider only cases where the net rate expression consists of two terms, one for the forward direction and one for the reverse direction. Cases where the net rate expression consists of a summation of several terms are usually viewed as corresponding to reactions with two or more parallel paths linking reactants and products. One may associate a pair of terms with each parallel path and use the technique outlined below to determine the thermodynamic restrictions on the form of the concentration dependence within each pair. This type of analysis is based on the principle of detailed balancing discussed in Section 4.1.5.4. 

It is clear that an appreciation of chemical stationary systems with respect to (1), (2) and (5) is virtually complete as explained above in terms of thermodynamic variables, although the analysis of cells is not, since the equations linking the functional variables are missing. Even if we are sure that the variables are known, we do not know their functional connections. 

The structure of hydrogels that do not contain ionic moieties can be analyzed by the Flory Rehner theory (Flory and Rehner 1943a). This combination of thermodynamic and elasticity theories states that a cross-linked polymer gel which is immersed in a fluid and allowed to reach equilibrium with its surroundings is subject only to two opposing forces, the thermodynamic force of mixing and the retractive force of the polymer chains. At equilibrium, these two forces are equal. Equation (1) describes the physical situation in terms of the Gibbs free energy. 

What we would like to do is use these thermodynamic properties to calculate an equilibrium elastic moduli. The bulk modulus is by definition the constant of proportionality that links the infinitesimal pressure change resulting from a fractional change in volume (Section 2.2.1). In colloidal terms this becomes... 

What happens upon equilibration with liquid water instead of water vapor According to Equation (6.13), the capillary radius would go to infinity for PVP —> 1. Thus, in terms of external conditions, swelling would be thermodynamically unlimited, corresponding to the formation of an infinitely dilute aqueous solution of ionomer. However, the self-organized polymer is an effectively cross-linked elastic medium. Under liquid-equilibrated conditions, swelling is not controlled by external vapor... 

Interestingly, the standard entropies (and in turn heat capacities) of both phases were found to be rather similar [69,70]. Considering the difference in standard entropy between F2(gas) and the mixture 02(gas) + H2(gas) taken in their standard states (which can be extracted from general thermodynamic tables), the difference between the entropy terms of the Gibbs function relative to HA and FA, around room temperature, is about 6.5 times lower than the difference between enthalpy terms (close to 125 kJ/mol as estimated from Tacker and Stormer [69]). This indicates that FA higher stability is mostly due to the lower enthalpy of formation of FA (more exothermic than for HA), and that it is not greatly affected by entropic factors. Jemal et al. [71] have studied some of the thermodynamic properties of FA and HA with varying cationic substitutions, and these authors linked the lower enthalpy of formation of FA compared to HA to the decrease in lattice volume in FA. 

There is a link between the two calculations in terms of residence time, r, which is influenced by thermodynamic issues. 

Specific properties of polysilanes have been linked to the method of synthesis.35 For example, in the case of anionic polymerization of poly[l-(6-methoxy-hexyl)-l,2,3-trimethyldisilanylene] a new type of chromism was induced in the polysilane film by the difference in the surface properties of substrates and was termed a surface-mediated chromism. The polysilane exhibited thermochromism with an absorption maximum at 306 nm at 23°C, but <15°C a band at 328 nm began to appear. A monolayer of the polysilane was transferred onto both a clean hydrophilic quartz plate and a hydrophobic one treated with hexamethyldisilazane by the vertical dipping method. With the hydrophobic plate, a broad UV absorption at 306 nm is obtained, whereas the absorption on a hydrophilic plate shifts to 322 nm. The conformation of the polysilane is preserved by hydrogen bonding between the silica surface and the ether section of the substituent on the hydrophilic plate. The polysilane is attached to the hydrophobic surface only by van der Waals forces, and this weaker interaction would not sustain the thermodynamically unstable conformational state that is attained on the water surface. 

The equations of constraint link the cost estimate through the system s thermodynamic performance to fuel costs. The thermodynamic analysis must relate the variables used to describe the system s performance to those used in the cost estimate. In this problem, costing equations are used which are generally in terms of stream and performance variables. Thus the thermodynamic analysis need only be in terms of these variables. Sixteen equations of constraint have been developed from a thermodynamic analysis of the cycle, and are given in Table III. 

Since thermodynamic efficiency is linked to the destruction of exergy and controllability is an issue of process dynamics, we start our investigation by writing a dynamic exergy balance. This can easily be done by differentiation of Eq. (A. 17) and adding exergy flow terms to allow for representations of open systems (Bejan, 1997). 

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