Thermodynamic potentials entropy

Piezoelectric solids are characterized by constitutive relations among the stress t, strain rj, entropy s, electric field E, and electric displacement D. When uncoupled solutions are sought, it is convenient to express t and D as functions of t], E, and s. The formulation of nonlinear piezoelectric constitutive relations has been considered by numerous authors (see the list cited in [77G06]), but there is no generally accepted form or notation. With some modification in notation, we adopt the definitions of thermodynamic potentials developed by Thurston [74T01]. This leads to the following constitutive relations ... 

Since the entropy is the partial differential coefficient of with r constant, or with p constant, with respect to T, the magnitudes and thermodynamic potentials at constant volume and at constant pressure respectively. 

However, these potentials do not yet express the second law in the form most convenient for chemical applications. Open laboratory vessels exposed to the temperature and pressure of the surroundings are subject neither to constraints of isolation (as required for entropy maximization) nor to adiabatic constant-volume conditions (as required for energy minimization). Hence, we seek alternative thermodynamic potentials that express the criteria for equilibrium under more general conditions. 

Every one-, two- or three-dimensional crystal defect gives rise to a potential field in which the various lattice constituents (building elements) distribute themselves so that their thermodynamic potential is constant in space. From this equilibrium condition, it is possible to determine the concentration profiles, provided that the partial enthalpy and entropy quantities and jj(f) of the building units i are known. Let us consider a simple limiting case and assume that the potential field around an (planar) interface is symmetric as shown in Figure 10-15, and that the constituent i dissolves ideally in the adjacent lattices, that is, it obeys Boltzmann statistics. In this case we have... 

Equation 2.2-8 indicates that the internal energy U of the system can be taken to be a function of entropy S, volume V, and amounts nt because these independent properties appear as differentials in equation 2.2-8 note that these are all extensive variables. This is summarized by writing U(S, V, n ). The independent variables in parentheses are called the natural variables of U. Natural variables are very important because when a thermodynamic potential can be determined as a function of its natural variables, all of the other thermodynamic properties of the system can be calculated by taking partial derivatives. The natural variables are also used in expressing the criteria of spontaneous change and equilibrium For a one-phase system involving PV work, (df/) 0 at constant S, V, and ,. ... 

Fundamental equation 2.2-8 has been presented as the equation resulting from the first and second laws, but thermodynamic treatments can also be based on the entropy as a thermodynamic potential. Equation 2.2-8 can alternatively be written as... 

This indicates that any irreversible process, if occurring at constant entropy S and pressure p, is accompanied by a decrease in the enthalpy from the initial high level Hjnijal toward the final low level Hfiml of the system. From the foregoing we see that the internal energy and enthalpy may play the role of thermodynamic potentials for an irreversible process if occurring under the condition of constant entropy S. This condition of constant entropy, however, is unrealistic because entropy S contains both created entropy Sirr and transferred entropy Sm. 

This equation indicates that the affinity corresponds to the thermodynamic potentials of U and H under the conditions of constant entropy S and to the thermodynamic potentials of F and G under the conditions of constant temperature T. 

The next steps are the following Step 1 Passage to the entropy representation and specification of the dissipative thermodynamic forces and the dissipative potential E. Step 2 Specification of the thermodynamic potential o. Step 3 Recasting of the equation governing the time evolution of the np-particle distribution function/ p into a Liouville equation corresponding to the time evolution of np particles (or p quasi-particles, Up > iip —see the point 4 below) that then represent the governing equations of direct molecular simulations. 

At constant external pressure the enthalpy is the relevant thermodynamic potential for the Boltzmann distribution. A large area A implies that the distribution of the thicknesses is confined near the minimum of the enthalpy, while a small value of A corresponds to large thickness fluctuations. The first case corresponds to a low enthalpy, but a large entropy, whereas the second to a large enthalpy, but a low entropy. The real distribution is provided by the minimization of the Gibbs free energy with respect to A at constant external pressure II. 

The Gibbs free energy is a measure of the probability that a reaction occurs. It is composed of the enthalpy, H, and the entropy, S° (Eq. 5). The enthalpy can be described as the thermodynamic potential, which ensues H = U + p V. where U is the internal energy, p is the pressure, and V is the volume. The entropy, according to classical definitions, is a measure of molecular order of a thermodynamic system and the irreversibility of a process, respectively. 

Thermodynamics shows that it is not the change in the energy U but rather that in the free energy F = U — TS, or in the thermodynamic potential or free enthalpy G — U — TS + PV which determines the position of the equilibrium. In the temperature with which we are concerned, TS is, however, considerably smaller than U (this holds all the more for PV) and in addition the change of entropy S varies little for different compounds. 

Principles of thermodynamics find applications in all branches of engineering and the sciences. Besides that, thermodynamics may present methods and generalized correlations for the estimation of physical and chemical properties when there are no experimental data available. Such estimations are often necessary in the simulation and design of various processes. This chapter briefly covers some of the basic definitions, principles of thermodynamics, entropy production, the Gibbs equation, phase equilibria, equations of state, and thermodynamic potentials. 

The entropy balance equation is derived assuming that the gas is at local equilibrium. We also assume that the suspension of Brownian particles in the heat bath may be a multicomponent ideal solution, and the thermodynamic potential is expressed by... 

Entropy at constant energy and density of Brownian particles has a maximum, is uniform in velocity space, and equal to the thermodynamic potential of a Brownian ideal gas, so we have... 

A chemical reaction is an irreversible process that produces entropy. The general criterion of irreversibility is d S > 0. Criteria applicable under particular conditions are readily obtained from the Gibbs equation. The changes in thermodynamic potentials for chemical reactions yield the affinity A. All four potentials U, H, A, and G decrease as a chemical reaction proceeds. The rate of reaction, which is the change of the extent of the reaction with time, has the same sign as the affinity. The reaction system is in equilibrium state when the affinity is zero. 

The Gibbs stability theory condition may be restrictive for nonequilibrium systems. For example, the differential form of Fourier s law together with the boundary conditions describe the evolution of heat conduction, and the stability theory at equilibrium refers to the asymptotic state reached after a sufficiently long time however, there exists no thermodynamic potential with a minimum at steady state. Therefore, a stability theory based on the entropy production is more general. 

Thermodynamics is the basis of all chemical transformations [1], which include dissolution of chemical components in aqueous solutions, reactions between two dissolved species, and precipitation of new products formed by the reactions. The laws of thermodynamics provide conditions in which these reactions occur. One way of determining such conditions is to use thermodynamic potentials (i.e., enthalpy, entropy, and Gibbs free energy of individual components that participate in a chemical reaction) and then apply the laws of thermodynamics. In the case of CBPCs, this approach requires relating measurable parameters, such as solubility of individual components of the reaction, to the thermodynamic parameters. Thermodynamic models not only predict whether a particular reaction is likely to occur, but also provide conditions (measurable parameters such as temperature and pressure) in which ceramics are formed out of these reactions. The basic thermodynamic potentials of most constituents of the CBPC products have been measured at room temperature (and often at elevated temperatures) and recorded in standard data books. Thus, it is possible to compile these data on the starter components, relate them to their dissolution characteristics, and predict their dissolution behavior in an aqueous solution by using a thermodynamic model. The thermodynamic potentials themselves can be expressed in terms of the molecular behavior of individual components forming the ceramics, as determined by a statistical-mechanical approach. Such a detailed study is beyond the scope of this book. 

Since the energy, entropy and volume of M mols of a solution are each equal to M times the energy, entropy and volume of 1 mol of the solution, it follows from the defining equation of the thermodynamic potential ( = U-TS + pv) that the thermodynamic potential of M mols of the solution is equal to M times the thermod3mamic potential of 1 mol of the solution. We have therefore... 

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