Criterion of equilibrium

When a system is left to itself, it would either 

If this is the case, the initial state is thtequilibrium state. 

The system will spontaneously move to the equilibrium state. 

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The criterion of equilibrium of a system maintained at constant temperature is therefore ... 

The necessary and sufficient criterion of equilibrium in a mechanically isolated system at a given temperature is ... 

The decrease in Gibbs free energy as a signpost of spontaneous change and AG = 0 as a criterion of equilibrium are applicable to any kind of process, provided that it is occurring at constant temperature and pressure. Because chemical reactions are our principal interest in chemistry, we now concentrate on them and look for a way to calculate AG for a reaction. 

An irreversible change is always spontaneous in an isolated system because no external force can interact with the system. Only at equilibrium can a change in an isolated system be conceived to occur reversibly. At equilibrium any infinitesimal flucmations away from equilibrium are opposed by the natural tendency to remrn to equilibrium. Therefore, the criterion of reversibility is a criterion of equilibrium, and the criterion of irreversibility is a criterion of spontaneity for an isolated system. 

We can apply the criterion of equilibrium expressed in Equation (9.17) to chemically reacting systems. Consider the reaction... 

Structural and molecular biologists often study the temperature dependence of the equilibrium position of a reaction or process. The Gibbs free energy undoubtedly provides the correct thermodynamic criterion of equilibrium. An understanding of this parameter can be achieved from either a macroscopic level (classical thermodynamics) or a molecular level (statistical thermodynamics). Ultimately, one seeks to understand the factors influencing AG° for a specific reaction. 

In summary, the first-derivative conditions (5.10) imply uniform values of the derivative (intensive) properties of S throughout the system. In this way, the system-wide uniformity of temperature, pressure, and other intensive properties is obtained from the Gibbs criterion of equilibrium as a deduction, not an assumption. 

Other stability conditions are obtained from the negativity of second derivatives with respect to V or N. (More generally, determinants of such second derivatives must also be negative in order to guarantee stability with respect to arbitrary combinations of energy, volume, and mass changes.) In summary, we can say that the Gibbs criterion of equilibrium for a closed system is equivalent to conditions of uniform intensive properties 7, P,... 

Let us now attempt to re-express the Gibbs criterion of equilibrium in alternative analytical and graphical forms that are more closely related to Clausius-like statements of the second law. For this purpose, we write the constrained entropy function S in terms of its... 

Constraint Conditions Thermodynamic Potential Criterion of Spontaneity Criterion of Equilibrium... 

We have seen that the criterion of equilibrium is AStot = 0. It follows from Eq. 17 that, for a process at constant temperature and pressure, the condition for equilibrium is... 

This fundamental equation for the entropy shows that S has the natural variables U, V, and n . The corresponding criterion of equilibrium is (dS) 0 at constant U, V, and n . Thus the entropy increases when a spontaneous change occurs at constant U, V, and ,. At equilibrium the entropy is at a maximum. When U, V, and , are constant, we can refer to the system as isolated. Equation 2.2-13 shows that partial derivatives of S yield 1/T, P/T, and pJT, which is the same information that is provided by partial derivatives of U, and so nothing is gained by using equation 2.2-13 rather than 2.2-8. Since equation 2.2-13 does not provide any new information, we will not discuss it further. 

This criterion of equilibrium provides a general method for determination of equilibrium states. One writes an expression for G as a function of the numbers of moles (mole numbers) of the species in the several phases, and then finds the set of values for the mole numbers that minimizes G subject to the constraints of mass conservation. This procedure can be applied to problems of phase, chemical-reaction, or combined phase and chemical-reaction equilibrium it is most useful for complex equilibrium problems, and is illustrated for chemical-reaction equilibrium in Sec. 15.9. 

At the equilibrium state differential variations can occur in the system at constant T and P without producing any change in G. This is the meaning of the equality in Eq. (13.52). Thus another general criterion of equilibrium is... 

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