General Conditions for Equilibrium

The equilibrium state is reached when there is no change with time in any of the system s macroscopic properties. The phase rule by Gibbs gives the general conditions for equilibrium between phases in a system. It is assumed that the equUibrium is only influenced by temperature and pressure, that is, surface, magnetic, electrical, and magnetic... 

By combining the above relationships with the general condition for equilibrium, the following relationship is obtained ... 

General Conditions for Equilibrium Since the total differential of the enthalpy is given by... 

General Conditions for Equilibrium where can be written in the form of Eq. (6-93) as... 

General Conditions for Equilibrium implies, among other things, that the equations... 

As noted before, one of the general conditions for equilibrium and steady state is that the forward and backward fluxes of a process are equal (F+ = F ). The specific condition to fulfill a chemical equilibrium is fi+ = n- (Eq. 4.92). Although a chemical equilibrium occurs when two or more reversible processes occur at the same rate, and such a system can be said to be in steady state, a system that is in steady state might not necessarily be in a state of equilibrium, because some of the processes involved are not reversible. A system in a steady state has numerous properties that do not change over time. The concept of steady state has relevance in many fields, in particular thermodynamics. Hence, steady state is a more general situation than dynamic equilibrium. If a system is in steady state, then the recently observed behavior of the system will continue into the f uture. In stochastic systems, the probabilities that various different states will be repeated will remain constant. We will generalize Eq. (2.142) as follows (see Chapter 2.8.4.1 and (4.88)) ... 

If the averages defined by (36) are independent of the time, the ensemble is in statistical equilibrium and dg/dt = 0. As a sufficient condition for equilibrium it is, according to (37) therefore necessary that [g, H] = 0. In general therefore, an ensemble is in statistical equilibrium if the density operator commutes with the Hamiltonian. 

In the isotope edited/ filtered spectra of a protein-ligand complex, the species actually observed is generally the complex itself. This is an important difference from transferred NOE or saturation difference techniques, where the existence of an equilibrium between free and bound species - and a certain rate of exchange between them - is essential (Chapts. 13 and 16). The general conditions for isotope filtering/editing are therefore identical to those required for standard protein NMR sample concentrations are usually limited by availability and solubility of the components to the order of 1 mM. Considerably lower concentrations will reduce the sensitivity of the experiments to unacceptable levels,... 

Then the general thermodynamic condition for equilibrium in a reaction involving charged particles and electrons is... 

At constant T and P (easy to experimentally control), the condition for equilibrium is AG = 0. The Gibbs energy function (Eq. 2.11) is a generally more useful function than the internal energy function (Eq. 2.6) that requires constant S and V at equilibrium (AU = 0). 

Because dc may be brought outside the summation sign and is not zero, this gives as the general condition for chemical equilibrium ... 

We can summarize the main results from the previous investigations in the following points (i) soliton solutions have been found under general conditions for an electron-positron plasma and by assuming quasi-neutrality in an electron-ion plasma (ii) sub-cycle nondrifting solitary waves represent an equilibrium in a multicomponent warm plasma that is, half-wavelengths of the EM radiation can be trapped inside a plasma density well (iii) the... 

The general condition for the stability of equilibrium state with respect to thermal, volume, and fluctuations in the numbers of moles is obtained by combining Eqs. (12.7), (12.9), and (12.12)... 

When ions of some species / are in equilibrium across a membrane, its chemical potential outside (o) is the same as that inside (i), that is, ju,° equals fjij. Differences in the hydrostatic pressure term generally make a negligible contribution to the chemical potential differences of ions across membranes, so VjP can be omitted from jUy in the present case. With this approximation and the definition of chemical potential (Eq. 2.4 without the pressure and the gravitational terms), the condition for equilibrium of ionic species / across the membrane (jlJ = /jlj) is... 

The general condition for phase equilibrium is the equality of chemical potentials. For practical reasons this is replaced by the equality of the fugacity/of each of the... 

In this section, we discuss the thermodynamics of systems imdergoing chemical reactions. First we derive the conditions for equilibrium. Then, we generalize the Gibbs phase rule for reacting systems. Afterwards, we examine the calculation of the and introduce the equilibrium constant. Finally, we examine the influence of pressure and temperature on chemical reaction equilibria. 

For the present we restrict the discussion to systems that produce only expansion work so that = 0. With this restriction, the general condition of equilibrium is... 

The equations of state discussed so far, the ideal gas law, the van der Waals equation, and others, were relations between p, V, and T obtained from empirical data on the behavior of gases or from speculation about the effects of molecular size and attractive forces on the behavior of the gas. The equation of state for a liquid or solid was simply expressed in terms of the experimentally determined coefficients of thermal expansion and compressibility. These relations applied to systems at equilibrium, but there is a more general condition of equilibrium. The second law of thermodynamics requires the relation, Eq. (10.19),... 

In this chapter we apply the general criteria for equilibrium developed in Chap. 6 to systems in which chemical reactions may occur. In Sec. 8-1, we present a general discussion of chemical equilibrium in homogeneous and heterogeneous systems. The concept of a progress variable is introduced, and the conditions for chemical equilibrium are derived. The equilibrium constant is defined, and some of its properties are developed. A discussion of the Le Chatelier-Braun principle applied to chemical reactions is presented. In Sec. 8-2, the results of Sec. 8-1 are applied to chemical reactions in mixtures of real gases. 

Equations of a relation between the parameters of a system based on a phase Gibbs rule follows from the principle of detailed equilibrium in its different displays. Thus, for a multi-phase system the principle of a detailed equilibrium requires the equilibrium of any two phases with each other. This permits to separate them and to consider them separately from others. General conditions for an equilibrium in an isolated system are reduced to partial conditions of thermal (temperature of all phases is equal), mechanical (at plain... 

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