State of function

In process simulation it is necessary to calculate enthalpy as a function of state variables. This is done using the following formulas, derived from the above relations by considering S and H as functions of T and p. 

The activation parameters from transition state theory are thermodynamic functions of state. To emphasize that, they are sometimes designated A H (or AH%) and A. 3 4 These values are the standard changes in enthalpy or entropy accompanying the transformation of one mole of the reactants, each at a concentration of 1 M, to one mole of the transition state, also at 1 M. A reference state of 1 mole per liter pertains because the rate constants are expressed with concentrations on the molar scale. Were some other unit of concentration used, say the millimolar scale, values of AS would be different for other than a first-order rate constant. 

Thermodynamics describes the behaviour of systems in terms of quantities and functions of state, but cannot express these quantities in terms of model concepts and assumptions on the structure of the system, inter-molecular forces, etc. This is also true of the activity coefficients thermodynamics defines these quantities and gives their dependence on the temperature, pressure and composition, but cannot interpret them from the point of view of intermolecular interactions. Every theoretical expression of the activity coefficients as a function of the composition of the solution is necessarily based on extrathermodynamic, mainly statistical concepts. This approach makes it possible to elaborate quantitatively the theory of individual activity coefficients. Their values are of paramount importance, for example, for operational definition of the pH and its potentiometric determination (Section 3.3.2), for potentiometric measurement with ion-selective electrodes (Section 6.3), in general for all the systems where liquid junctions appear (Section 2.5.3), etc. 

Electrode potentials are determined by the affinities of the electrode reactions. As the affinities are changes in thermodynamic functions of state, they are additive. The affinity of a given reaction can be obtained by linear combination of the affinities for a sequence of reactions proceeding from the same initial to the same final state as the direct reaction. Thus, the principle of linear combination must also be valid for electrode potentials. The electrode oxidation of metal Me to a higher oxidation state z+>2 can be separated into oxidation to a lower oxidation state z+>1 and subsequent oxidation to the oxidation state z+>2. The affinities of the particular oxidation processes are equivalent to the electrode potentials 2 0, i-o> and E2-. ... 

Remember that G, H and S are all thermodynamic functions of state, i.e. they depend only on the initial and final states of the system, not on the ways the last is reached. As we have seen, for AG = 0 the reaction has reached equilibrium (and in isolated systems AS has reached a maximum). If AG < 0 the reaction was spontaneous, but if AG > 0 the reaction could not have taken place unless energy was provided from other coupled source. If the source is external then the system is not isolated it is closed if there is no exchange of material or open if there is such exchange. In both cases the environmental changes must be taken into account. 

The various examples discussed have shown that electronics play their part in all basic functions of state-of-the-art washing machines and that it would be difficult to imagine a future without them. Above all, electronic control is crucial if washing machines are to be operated economically. Fig. 3.15 gives a final summary of the most important points. 

Because the terms on the right side are all constants or functions of state, so too is the term on the left, qlT. In Chapter 20, we learn that q T is equal to AS, the change in a state function called entropy. 

The Gibbs function is a function of state, so values of AG obtained with the van t Hoff isotherm (see p. 162) and routes such as Hess s law cycles are identical. 

The protons and oxide ions combine to form water. Again, the value of AGr for Equation (7.36) is negative, because the reaction is spontaneous. AG would be positive if we wrote Equation (7.36) in reverse. The change in sign follows because the Gibbs function is a function of state (see p. 83). 

It is appropriate at this point to define another function of state, the enthalpy H,... 

The second law of thermodynamics introduces a new function of state, the entropy, S, in order to quantify the spontaneity and direction of change for natural systems... 

Whether obtained from an actual experimentally feasible process or from a thought process, As i Gg, which is obtained from Eq. (2.9) by re-arrangement, pertains to the solvation of the solute and expresses the totality of the solute-solvent interactions. It is a thermodynamic function of state, and so are its derivatives with respect to the temperature (the standard molar entropy of solvation) or pressure. This means that it is immaterial how the process is carried out, and only the initial state (the ideal gaseous solute B and the pure liquid solvent) and the final state (the dilute solution of B in the liquid) must be specified. 

Because it is a function of state, As inGg may be considered to be made up additively of the contributions from the various stages in which the transfer of the solute particle from the gaseous state into the liquid solvent has been envisaged by the foregoing to take place. 

It has already been shown in Chapter 4 (section 4.2.1) that from the thermodynamic point of view the process described by Eq. (16.2) can be modeled by the sum of its partial processes (extraction steps), irrespective of whether they really proceed or not. That is because Gibbs free energy is the function of state and its total change does not depend on the reaction path. According to the complex formation-partition model [76], one can distinguish two main steps in extraction of metal ions ... 

The central quantities are the velocity correlation function of states of energy E at time t. C E,t), and the average square spreading of states of energy E at time t along the x direction AX E,t). The velocity correlation function is defined by ... 

Noncompetitive inhibition cannot be completely reversed by very high substrate concentrations. Monod et al. defined for an allosteric enzyme a function of state R (Eq. 9-71) which is the fraction of total enzyme in the R (B) conformation ... 

Figure 9-14 Fractional saturation Y and "function of state" R for hypothetical tetrameric enzymes following the MWC model. Curves are calculated for two different values of the apparent allosteric constant L (Eq. 9-70) and for c = 0.1 (Eq. 9-66). After Rubin and Changeux.86...

Fumarase. See Fumarate hydratase Fumarase-aspartase family 685 Fumarate 481s, 516s, 683s Fumarate hydratase (fumarase) 526, 683,688 acid-base catalysis 471 concerted reaction 685 Fumarase A 688 Fumarase B 688 Fumarase C 683 mechanism 683 - 685 pH dependence 684 rates of substrate exchange 684 turnover number of 683 Fumarate reductase 785 Fumarylpyruvate 690s Function of state R 476 Fungal infections 20 Fungi 20... 

After the formulation of defect thermodynamics, it is necessary to understand the nature of rate constants and transport coefficients in order to make practical use of irreversible thermodynamics in solid state kinetics. Even the individual jump of a vacancy is a complicated many-body problem involving, in principle, the lattice dynamics of the whole crystal and the coupling with the motion of all other atomic structure elements. Predictions can be made by simulations, but the relevant methods (e.g., molecular dynamics, MD, calculations) can still be applied only in very simple situations. What are the limits of linear transport theory and under what conditions do the (local) rate constants and transport coefficients cease to be functions of state When do they begin to depend not only on local thermodynamic parameters, but on driving forces (potential gradients) as well Various relaxation processes give the answer to these questions and are treated in depth later. 

The resulting equilibrium concentrations of these point defects (vacancies and interstitials) are the consequence of a compromise between the ordering interaction energy and the entropy contribution of disorder (point defects, in this case). To be sure, the importance of Frenkel s basic work for the further development of solid state kinetics can hardly be overstated. From here on one knew that, in a crystal, the concentration of irregular structure elements (in thermal equilibrium) is a function of state. Therefore the conductivity of an ionic crystal, for example, which is caused by mobile, point defects, is a well defined physical property. However, contributions to the conductivity due to dislocations, grain boundaries, and other non-equilibrium defects can sometimes be quite significant. 

We will introduce basic kinetic concepts that are frequently used and illustrate them with pertinent examples. One of those concepts is the idea of dynamic equilibrium, as opposed to static (mechanical) equilibrium. Dynamic equilibrium at a phase boundary, for example, means that equal fluxes of particles are continuously crossing the boundary in both directions so that the (macroscopic) net flux is always zero. This concept enables us to understand the non-equilibrium state of a system as a monotonic deviation from the equilibrium state. Driven by the deviations from equilibrium of certain functions of state, a change in time for such a system can then be understood as the return to equilibrium. We can select these functions of state according to the imposed constraints. If the deviations from equilibrium are sufficiently small, the result falls within a linear theory of process rates. As long as the kinetic coefficients can be explained in terms of the dynamic equilibrium properties, the reaction rates are directly proportional to the deviations. The thermodynamic equilibrium state is chosen as the reference state in which the driving forces X, vanish, but not the random thermal motions of structure elements i. Therefore, systems which we wish to study kinetically must first be understood at equilibrium, where the SE fluxes vanish individually both in the interior of all phases and across phase boundaries. This concept will be worked out in Section 4.2.1 after fluxes of matter, charge, etc. have been introduced through the formalism of irreversible thermodynamics. 

The fundamental question in transport theory is Can one describe processes in nonequilibrium systems with the help of (local) thermodynamic functions of state (thermodynamic variables) This question can only be checked experimentally. On an atomic level, statistical mechanics is the appropriate theory. Since the entropy, 5, is the characteristic function for the formulation of equilibria (in a closed system), the deviation, SS, from the equilibrium value, S0, is the function which we need to use for the description of non-equilibria. Since we are interested in processes (i.e., changes in a system over time), the entropy production rate a = SS is the relevant function in irreversible thermodynamics. Irreversible processes involve linear reactions (rates 55) as well as nonlinear ones. We will be mainly concerned with processes that occur near equilibrium and so we can linearize the kinetic equations. The early development of this theory was mainly due to the Norwegian Lars Onsager. Let us regard the entropy S(a,/3,. ..) as a function of the (extensive) state variables a,/ ,. .. .which are either constant (fi,.. .) or can be controlled and measured (a). In terms of the entropy production rate, we have (9a/0f=a)... 

Let us now consider the equalization of the component concentrations in an inhomogeneous multicomponent system. We may start with Eqn. (4.33) which relates the component fluxes, jk, to the (n-1) independent forces, Vyq, of the n-compo-nent solid solution. In local equilibrium, the chemical potentials are functions of state. Hence, at any given P and T... 

The time evolution of a system may also be characterized according to the degree of perturbation from its equilibrium state. Linear theories hold if local equilibrium prevails, that is, each volume element of the non-equilibrium system can still be unambiguously defined by the usual set of (local) thermodynamic state variables. Often, a crystal is in (partial) equilibrium with respect to externally predetermined P and 7j but not with external component chemical potentials pik. Although P, T, and nk are all intensive functions of state, AP relaxes with sound velocity, A7 by heat conduction, and A/ik by matter transport. In solids, matter transport is normally much slower than the other modes of relaxation. 

As an illustration, consider the isothermal, isobaric diffusional mixing of two elemental crystals, A and B, by a vacancy mechanism. Initially, A and B possess different vacancy concentrations Cy(A) and Cy(B). During interdiffusion, these concentrations have to change locally towards the new equilibrium values Cy(A,B), which depend on the local (A, B) composition. Vacancy relaxation will be slow if the external surfaces of the crystal, which act as the only sinks and sources, are far away. This is true for large samples. Although linear transport theory may apply for all structure elements, the (local) vacancy equilibrium is not fully established during the interdiffusion process. Consequently, the (local) transport coefficients (DA,DB), which are proportional to the vacancy concentration, are no longer functions of state (Le., dependent on composition only) but explicitly dependent on the diffusion time and the space coordinate. Non-linear transport equations are the result. 

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