Thermodynamic equilibrium variable

The N equations represented by Eq. (4-282) in conjunction with Eq. (4-284) may be used to solve for N unspecified phase-equilibrium variables. For a multicomponent system the calculation is formidable, but well suited to computer solution. The types of problems encountered for nonelectrolyte systems at low to moderate pressures (well below the critical pressure) are discussed by Smith, Van Ness, and Abbott (Introduction to Chemical Engineering Thermodynamics, 5th ed., McGraw-Hill, New York, 1996). 

Constitutive relation An equation that relates the initial state to the final state of a material undergoing shock compression. This equation is a property of the material and distinguishes one material from another. In general it can be rate-dependent. It is combined with the jump conditions to yield the Hugoniot curve which is also material-dependent. The equation of state of a material is a constitutive equation for which the initial and final states are in thermodynamic equilibrium, and there are no rate-dependent variables. 

The variable / depends on the particular species chosen as a reference substance. In general, the initial mole numbers of the reactants do not constitute simple stoichiometric ratios, and the number of moles of product that may be formed is limited by the amount of one of the reactants present in the system. If the extent of reaction is not limited by thermodynamic equilibrium constraints, this limiting reagent is the one that determines the maximum possible value of the extent of reaction ( max). We should refer our fractional conversions to this stoichiometrically limiting reactant if / is to lie between zero and unity. Consequently, the treatment used in subsequent chapters will define fractional conversions in terms of the limiting reactant. 

Under these conditions, the distribution of the action variables (e.g., the momenta) [the vacuum, pg] tends irreversibly toward the thermodynamic equilibrium after a sufficiently long time. Under the same conditions, the correlations are determined by the vacuum (technically, they become functionals of the vacuum distribution pg) (see Appendix). 

Though the thermodynamic equilibrium constant is unaffected by pressure or inerts, the equilibrium concentration of materials and equilibrium conversion of reactants can be influenced by these variables. 

For a PVT system of uniform T and P containing N species and 71 phases at thermodynamic equilibrium, the intensive state of the system is hilly determined by the values of T, P, and the (N — 1) independent mole fractions for each of the equilibrium phases. The total number of these variables is then 2 + tt(N — 1). The independent equations defining or constraining the equilibrium state are of three types equations 218 or 219 of phase-equilibrium, N(77 — 1) in number equation 245 of chemical reaction equilibrium, r in number and equations of special constraint, s in number. The total number of these equations is Ar(7r — 1) + r + s. The number of equations of reaction equilibrium r is the number of independent chemical reactions, and may be determined by a systematic procedure (6). Special constraints arise when conditions are imposed, such as forming the system from particular species, which allow one or more additional equations to be written connecting the phase-rule variables (6). 

The production of species i (number of moles per unit volume and time) is the velocity of reaction,. In the same sense, one understands the molar flux, jh of particles / per unit cross section and unit time. In a linear theory, the rate and the deviation from equilibrium are proportional to each other. The factors of proportionality are called reaction rate constants and transport coefficients respectively. They are state properties and thus depend only on the (local) thermodynamic state variables and not on their derivatives. They can be rationalized by crystal dynamics and atomic kinetics with the help of statistical theories. Irreversible thermodynamics is the theory of the rates of chemical processes in both spatially homogeneous systems (homogeneous reactions) and inhomogeneous systems (transport processes). If transport processes occur in multiphase systems, one is dealing with heterogeneous reactions. Heterogeneous systems stop reacting once one or more of the reactants are consumed and the systems became nonvariant. 

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. 

In case of a genuine transformation, the phase considered must be in its internal thermodynamic equilibrium above and below the transformation temperature Ttr. This means that the physical properties of a one-component system depend only on two variables and must be independent of the path. 

As can be seen in Figure 5.6, thermodynamic equilibrium is achieved when the Gibbs energy is at a minimum ( = 0) and the difference in the updated mole fractions and previous mole fractions (Ax ) for each phase is less than IE-6. One of the crucial steps in obtaining a solution is creating a good initial estimate for the unknown variables. 

For systems that have not reached their stationary state (steady state or thermodynamic equilibrium), the behavior with regards to time cannot be determined without knowing the initial conditions, or the values of the state variables at the start, i.e., at time = 0. When the initial conditions are known, the behavior of the system is uniquely defined. Note that for chaotic systems, the system behavior has infinite sensitivity to the initial conditions however, it is still uniquely defined. Moreover, the feed conditions of a distributed system can act as initial conditions for the variations along the length. 

The phase diagrams of aqueous surfactant systems provide information on the physical science of these systems which is both useful industrially and interesting academically (1). Phase information is thermodynamic in nature. It describes the range of system variables (composition, temperature, and pressure) wherein smooth variations occur in the thermodynamic density variables (enthalpy, free energy, etc.), for macroscopic systems at equilibrium. The boundaries in phase diagrams signify the loci of system variables where discontinuities in these thermodynamic variables exist (2). 

Supersaturation is the thermodynamic driving force for both crystal nucleation and growth and therefore, it is the key variable in setting the mechanisms and rates by which these processes occur. It is defined rigorously as the deviation of the system from thermodynamic equilibrium and is quantified in terms of chemical potential,... 

Vectors, such as x, are denoted by bold lower case font. Matrices, such as N, are denoted by bold upper case fonts. The vector x contains the concentration of all the variable species it represents the state vector of the network. Time is denoted by t. All the parameters are compounded in vector p it consists of kinetic parameters and the concentrations of constant molecular species which are considered buffered by processes in the environment. The matrix N is the stoichiometric matrix, which contains the stoichiometric coefficients of all the molecular species for the reactions that are produced and consumed. The rate vector v contains all the rate equations of the processes in the network. The kinetic model is considered to be in steady state if all mass balances equal zero. A process is in thermodynamic equilibrium if its rate equals zero. Therefore if all rates in the network equal zero then the entire network is in thermodynamic equilibrium. Then the state is no longer dependent on kinetic parameters but solely on equilibrium constants. Equilibrium constants are thermodynamic quantities determined by the standard Gibbs free energies of the reactants in the network and do not depend on the kinetic parameters of the catalysts, enzymes, in the network [49]. 

Necessary Conditions for Stability. In a system with a fixed number of layers, such as the phospholipid bilayers, the equilibrium position (corresponding to the minimum of the free energy, F, of the whole system) is obtained when the free energy per unit area for the pair water/bilayer, f, is a minimum. This is no longer true when the number of pairs of layers is variable. In this case, at thermodynamic equilibrium one should use eq 3 c. From this equation, if the interactions between lamellae are known, one can calculate the surface tension y as a function... 

The zeroth law of thermodynamics involves some simple definition of thermodynamic equilibrium. Thermodynamic equilibrium leads to the large-scale definition of temperature, as opposed to the small-scale definition related to the kinetic energy of the molecules. The first law of thermodynamics relates the various forms of kinetic and potential energy in a system to the work which a system can perform and to the transfer of heat. This law is sometimes taken as the definition of internal energy, and introduces an additional state variable, enthalpy. 

---