The Thermodynamic Criteria of Equilibrium

Use of the thermodynamic criterion of equilibrium in the derivation of the algebraic form of the equilibrium constant... 

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. 

A simple way (actually the only way) to determine this low temperature amorphous phase is to use a theory that correctly predicts the behavior of the liquid and extend it to low temperatures. One obviously should use the most realistic existing equilibrium theory to obtain the low temperature phase. The predictions are the two equations of state, S-V-T and P-V-T which are each derived from the Helmholtz free energy F which is in turn obtained from the partition function (F=-kTLnQ). In obtaining the S-V-T equation of state it is discovered that the configurational entropy Sc defined as the total entropy minus the vibrational entropy, approaches zero at a finite temperature (4), This vanishing of Sc is taken as the thermodynamic criterion of glass formation (5,6). 

The thermodynamic criterion of chemical equilibrium (f) for the general reaction ... 

The region of immunity [Fig. 1.15 (bottom)] illustrates how corrosion may be controlled by lowering the potential of the metal, and this zone provides the thermodynamic explanation of the important practical method of cathodic protection (Section 11.1). In the case of iron in near-neutral solutions the potential E = —0-62 V for immunity corresponds approximately with the practical criterion adopted for cathodically protecting the metal in most environments, i.e. —0-52 to —0-62V (vs. S.H.E.). It should be observed, however, that the diagram provides no information on the rate of charge transfer (the current) required to depress the potential into the region of immunity, which is the same (< —0-62 V) at all values of pH below 9-8. Consideration of curve//for the Hj/HjO equilibrium shows that as the pH... 

This criterion is good for establishish whether a process is under thermodynamic control. Care should be taken however to understand the term reversibility in this case. The folding of a protein is generally per se a chemically irreversible process, in the sense that the chemical equilibrium is overwhelmingly shifted towards the folded form - there is not a low activation energy barrier between the native folded and the unfolded form and a corresponding chemical equilibrium in the native state between the two forms. Thus, in the case of the thermodynamic hypothesis of... 

The following criterion of phase equilibrium can be developed from the first and second laws of thermodynamics the equilibrium state for a closed multiphase system of constant, uniform temperature and pressure is the state for which the total Gibbs energy is a minimum, whence... 

It is apparent that CMC values can be expressed in a variety of different concentration units. The measured value of cCMC and hence of AG c for a particular system depends on the units chosen, so some uniformity must be established. The issue is ultimately a question of defining the standard state to which the superscript on AG C refers. When mole fractions are used for concentrations, AG c directly measures the free energy difference per mole between surfactant molecules in micelles and in water. To see how this comes about, it is instructive to examine Reaction (A) —this focuses attention on the surfactant and ignores bound counterions — from the point of view of a phase equilibrium. The thermodynamic criterion for a phase equilibrium is that the chemical potential of the surfactant (subscript 5) be the same in the micelle (superscript mic) and in water (superscript W) n = n. In general, pt, = + RTIn ah in which... 

Thermodynamics is used to predict whether reactants have a spontaneous tendency to change into products. This tendency is associated with a decrease in the free energy or Gibbs energy of the system (G) to a minimum. As a consequence, the thermodynamic criterion for spontaneous change at constant temperature and pressure is AG < 0. Under standard conditions (concentrations = 1 M, and P = 1 atm), the standard Gibbs energy variation (AG°) is related with the equilibrium constant (A) by equation 11 ... 

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. 

Near thermodynamic equilibrium, similar linear relationships are also valid for elementary chemical processes, as well as for stepwise processes where the rates are proportional to the difference between the thermodynamic mshes of the initial and final reaction groups (see Section 1.4.2). Here, the criterion of proximity to thermodynamic equilibrium is relationship jA jl < RT, where Arij is the affinity for the transformation of reaction group i to reaction group j. In fact, while... 

The Onsager reciprocal relations are not satisfied in open strongly non equilibrium systems. As a result, the assumption on minimization of the entropy production rate is not substantiated. Therefore, the universal criterion of the system that is evolution far from equilibrium should be a generalization of the principle of the minimized entropy production rate in specific terms of nonlinear thermodynamics. 

This sensitivity of the IR and Raman spectra for both gases and liquids was one of the earliest observations of the spectral uniqueness of H bonding systems. It has been used so frequently to verify the presence of H bonded species that many workers consider it to be the primary criterion of H bonding. Of much greater importance is the spectral measurement of equilibrium constants at several temperatures. Such data permit calculation of the thermodynamic properties of H bond formation, AH, AS, and AF (see Section 7.3.1)". ... 

The specific examples in Section 14.5 demonstrate that when 1C > 1 the reaction has progressed far toward products, and when K 1 the reaction has remained near reactants. The empirical discussion in Section 14.6 shows how the reaction quotient Q and the principle of Te Chatelier can predict the direction of spontaneous reaction and the response of an equilibrium state to an external perturbation. Here, we use the thermodynamic description of K from Section 14.3 to provide the thermodynamic basis for these results obtained empirically in Sections 14.5 and 14.6. We identify those thermodynamic factors that determine the magnitude of K. We also provide a thermodynamic criterion for predicting the direction in which a reaction proceeds from a given initial condition. 

For the description of /-independent phases with k components, we therefore need f(k - 1) independent data on composition. To these we still have to add data on temperature and pressure, so we have altogether/( - l) + 2 intensive data, if the temperature and pressure are equal in the whole system. If the system considered is in equilibrium, the intensive criterion of the thermodynamic equilibrium must be fulfilled, fhus the chemical potentials of all the k components in all the/phases have to be equal. This criterion thus defines fhe number of binding conditions between the intensive variables. This number is kif- 1), because the number of binding conditions is one less than the number of phases. Then the difference between both the quantities defines the number of intensive variables, which are independent in a system with A components and/phases being in equilibrium -the variance, or the number of degrees of freedom v... 

There seems to be a law of nature that, in an equilibrium system, the chemical hardness and the physical hardness have maximum values, compared with nearby non-equilibrium states. However, it must not be inferred that these maximum principles are being proposed to take the place of estabished criteria for equilibrium. Instead, they are necessary consequences of these fundamental laws. It is very clear that the Principle of Maximum Hardness for electrons is a result of the quantum mechanical criterion of minimum energy. Similarly, Sanchez has recently derived the relationship (dB/dP) = 5 by a straightforward manipulation of the thermodynamic equation of state.The PMPH is a result of the laws of thermodynamics. 

Or, again, take the case of pure benzene on the one hand and a saturated solution of benzene in water on the other, both systems being at the same temperature A saturated solution of benzene is necessarily in equilibrium with pure liquid benzene itself because of the fact of saturation The conclusion to he drawn from the thermodynamic criterion considered is, that under these conditions, (8A)TV = o, and therefore, if we imagine one mole of benzene transferred from the pure benzene to the saturated solution, the work must be zero That is, there must be the same vapour pressure over the pure benzene as there is over its saturated solution in water, the vapour in each case being benzene vapour In the case of a hydrated salt on the one hand and the saturated solution of the salt on the other, the conditions are more complex We shall consider this point in Chap X in connection with the application of the Phase Rule to two component systems... 

In order to illustrate the restrictions on the thermodynamic properties of a system that the criterion for stable equilibrium imposes, we shall discuss a homogeneous system of two components. We consider the homogeneous phase to be divided into two portions, 1 and 2. At equilibrium, Eqs. (6-84) demand that... 

Remember the non-equilibrium thermodynamic with branching points after the instability threshold (Fig. 2), or the concepts of landscape metastability in the first case, the history becomes the leading criterion of transformation in the second, it is evident that, even when a succession to a climax may be considered valid at a single ecocoenotope scale, certainly it is not valid at a landscape scale. 

The requirements for a system to attain equilibrium depend on its interactions with the environment. As we have seen in Section 3.2, the second law of thermodynamics tells us that a process proceeds irreversibly if dS > SqlT and that an equilibrium is reached when dS = SqlT. The second law may be combined with expressions for U, H, F, or G to And the appropriate criterion for equilibrium. 

To explore whether/owrphases of a single substance could ever be in equilibrium (such as four of the many phases of ice), we think about the thermodynamic criterion for four phases to be in equilibrium. For equilibrium, the four molar Gibbs energies would aU have to be equal, and we could write... 

THE FUNDAMENTAL THERMODYNAMIC CRITERION OF PHASE AND CHEMICAL EQUILIBRIUM... 

Equations (7-8) and (7-9) are then used to calculate the compositions, which are normalized and used in the thermodynamic subroutines to find new equilibrium ratios,. These values are then used in the next Newton-Raphson iteration. The iterative process continues until the magnitude of the objective function 1g is less than a convergence criterion, e. If initial estimates of x, y, and a are not provided externally (for instance from previous calculations of the same separation under slightly different conditions), they are taken to be... 

Thermodynamics provide a straightforward method for quantifying this situation. The criterion for equilibrium is the equality of chemical potential in... 

---